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THE JOY OFMA THEM TI CS

Discovering MathematicsAll Around You

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Wide World Publishing/Tetra

nature * science * architecture * arthistory * literature * philosophy * music

Copyright i 1986, 1987, 1989 by Theoni Pappas.All rights reserved. No part of this work may bereproduced or copied in any form or by any meanswithout written permission from Wide World Publishing/Tetra.

Revised Edition 1989

1st Printing 19862nd Printing 19873rd Printing May 19894th Printing September 19895th Printing March 19906th Printing September 19907th Printing May 19918th Printing September 19919th Printing December 199110th Printing July 199211th Printing April 199312th Printing December 199313th Printing October 1994

Wide World Publishing/TetraP.O. Box 476San Carlos, CA 94070

Printed in The United States of America

ISBN: 0-933174-65-9

Library of Congress Cataloging-in-Publication Data

Pappas, Theoni.The Joy of Mathematics / by Theoni Pappas. --Rev. ed.

p. cm.Includes index.ISBN: 0-933174-65-9: $10.951. Mathematics -- Popular works. 1. Title.

QA93. P365 1989510 -- dc2O 89-16503

CIP

I hope this book will serve as anintroduction to some of the many diversified

facets of mathematics, and that the reader

will be encouraged to explore in greater

depth the ideas presented.

- Theoni Pappas

". .the universe stands continually open to our gaze, but it

cannot be understood unless one first learns to comprehendthe language and interpret the characters in which it is

written. It is written in the language of mathematics, and

its characters are triangles, circles, and other geometricfigures, without which it is humanly impossible to

understand a single word of it; without these, one iswandering about in a dark labyrinth. "

-Galileo

INTRODUCTION

The Joy Of Mathematics and More Joy of Mathematics unveilconcepts, ideas, questions, history, problems, and pastimeswhich reveal the influence and nature of mathematics.

To experience the joy of mathematics is to realize mathematics isnot some isolated subject that has little relationship to the thingsaround us other than to frustrate us with unbalanced checkbooks and complicated computations. Few grasp the true natureof mathematics - so entwined in our environment and in ourlives. So many things around us can be described bymathematics. Mathematical concepts are even inherent in thestructure of living cells.

These books seek to help you become aware of the inseparablerelationship of mathematics and the world by presentingglimpses and images of mathematics in the many facets of ourlives.

The joy of mathematics is similar to the experience ofdiscovering something for the first time. It is an almost child-likefeeling of wonder. Once you have experienced it you will notforget that feeling - it can be as exciting as looking into amicroscope for the first time and seeing things that have alwaysbeen around you that you have been unable to see before.

When deciding how to organize The Joy Of Mathematics, at firstcertain divisions immediately came to mind-for example,mathematics and nature, mathematics and science, mathematicsand art, and so forth. But mathematics and its relationship to oursurroundings does not come already packaged into categories.Rather, mathematics and its occurrences are spontaneous withelements of surprise. Thus, the topics are randomly arranged toretain the true essence of discovery. The Joy Of Mathematics andMore Joy of Mathematics are designed to be opened at any point.Each section, regardless of how large or small, is essentiallyself-contained.

After one experiences the sheer joy of mathematics, theappreciation of mathematics follows and then the desire to learnmore.

"There is no branch of mathematics, however

abstract, which may not someday be appliedto the phenomena of the real world."

-Lobachevsky

TABLE OF CONTENTS

2 The evolution of base ten4 The Pythagorean Theorem5 Optical illusions and computer graphics6 The cycloid9 A triangle to a square problem

10 Halley's comet13 The impossible tribar14 The quipu16 Calligraphy, typography and mathematics17 The wheat and the chessboard problem18 Probability and ic20 Earthquakes and logarithms22 Parabolic ceiling and the Capitol24 Computers, counting and electricity26 Topo - a mathematical game28 Fibonacci sequence30 A twist to the Pythagorean Theorem31 Trinity of rings-a topological problem32 Anatomy and the golden section34 Catenary and parabolic curves35 The T problem36 Thales and the Great Pyramid37 Hotel Infinity38 Crystals - nature's polyhedra40 Pascal's triangle42 Mathematics of the billiard table43 The electron's path and geometry44 The Moebius strip and the Klein bottle47 A Sam Loyd puzzle48 Mathematics and paperfolding51 The Fibonacci trick52 The evolution of mathematical symbols55 Some geometric designs of Leonardo da Vinci56 Ten historic dates57 Napoleon's theorem58 Lewis Carroll-the mathematician60 Counting on fingers61 A twist to the Moebius strip62 Heron's theorem63 A look at Gothic architecture64 Napier's bones66 Art and projective geometry68 Infinity and the circle69 The amazing track70 Persian horses and Sam Loyd's puzzle72 The lunes74 Hexagons in nature76 The googol and the googolplex77 A magic cube78 Fractals-real or imaginary80 Nanoseconds-measuring time on computers

TABLE OF CONTENTS81 Geodesic dome of Leonardo da Vind82 Magic squares87 A special "magic" square88 The Chinese triangle89 The death of Archimedes90 A non-Eudlidean world93 Cannon balls and pyramids94 Conchoid of Nicomedes96 The trefoil knot97 The magic square of Benjamin Franklin98 Irrational numbers and the Pythagorean Theorem

100 Prime numbers102 The golden rectangle107 Making a tri-tetra flexagon108 Finding infinity in small places110 The five Platonic solids112 The pyramid method-making magic squares113 The Kepler-Poinsot solids114 The false spiral optical illusion115 The icosahedron and the golden rectangle116 Zeno's paradox-Achilles and the tortoise118 The mystic hexagram119 The penny puzzle120 Tessellations123 Diophantus' riddle124 The Konigsberg Bridge problem126 Networks128 Aztec calendar130 The impossible trio-three ancient construction problems133 Ancient Tibetan magic square134 Perimeter, area, and infinite series136 The checkerboard problem137 Pascal's calculator138 Isaac Newton and calculus139 Japanese abacus140 The proof of 1=2141 The symmetry of crystals142 The mathematics of music146 Numerical palindromes147 The unexpected exam paradox148 Babylonian cuneiform text149 The spiral of Archimedes150 The evolution of mathematical ideas152 The four color map problem takes a turn154 Art and dynamic symmetry156 Transfinite numbers159 Logic problem160 The snowflake curve162 Zero-when and where163 Pappus' theorem and the nine coin puzzle164 The Japanese magic circle & Gauss' problem

TABLE OF CONTENTS165 Spherical dome and water distillation166 The helix169 Magic "line"170 Mathematics and architecture172 History of optical illusions174 Trisecting the equilateral triangle175 The wood, water, and grain problem176 Charles Babbage, the Leonardo da Vinci of computers178 Mathematics and Moslem art179 A Chinese magic square180 Infinity and limits181 Counterfeit coin puzzle182 The Parthenon-an optical and mathematical design183 Probability and Pascal's triangle187 The involute curve188 The pentagon, the pentagram, and the golden triangle190 Three men facing a wall problem191 Geometric fallacy and the Fibonacci sequence192 Mazes195 Chinese "checkerboards"196 Conic sections198 The screw of Archimedes199 Irradiation optical illusion200 The Pythagorean Theorem and President Garfield202 The wheel paradox of Aristotole & Galileo's solution203 Stonehenge204 How many dimensions are there?206 Computers and dimensions207 The "double" Moebius strip208 Paradoxical curve-the space-filling curve209 The abacus210 Mathematics and weaving211 Mersenne's number212 The tangram puzzle213 Infinite vs finite214 Triangular, square, and pentagonal numbers215 Eratosthenes measures the Earth216 Projective geometry and linear programming218 The spider and the fly219 Mathematics and soap bubbles220 The coin paradox221 Hexaminoes222 The Fibonacci sequence and nature226 The monkey and the coconuts228 Spiders and spirals229 solutions235 bibliography239 index241 About the author

2 THE JOY OF MATHEMATICS

The Early forms of counting hadlu t. no positional base value

Evolutio n system. But around 1700 B.C.the positional base 60of Base Ten evolved. It was very helpfulto the Mesopotamians who

developed it to use in conjunction with their 360 day calendar.The oldest known true place value system is that devised bythe Babylonians, and was derived from the Sumeriansexigesimal system. Instead of needing sixty symbols to writethe numerals from 0 to 59, two symbols - Y for 1 and < for 10sufficed. Sophisticated mathematical computations could beperformed with it, but no symbol for zero had been devised. To

- = r 7 ?HINDU(Brahmi)-c. 300 B.C

HINDU(Gwalior)- 876 A.D.

HINDU(Devanagar)- 11th century

WESTARABIC(Ghobar)- 11th century

X r rVA IVA 9 -EASTARABIC- 1575

T* 3 '% 8 san 0EUROPEAN- 15th century

13 +5 6 It 9 *EUROPEAN- 16th century

L234 152 6?87 0COMPUTER NUMERALS- 20th century

THE JOY OF MATHEMATICS 3

indicate zero an empty position was left in the number. About300 B.C. a symbol for zero appeared, it or 4 , and the base 60system developed extensively. In the early A.D. years, theGreeks and Hindus began to use base 10 systems, but they did nothave positional notation. They used the first ten letters oftheir alphabet for counting. Then around 500 A.D. a Hinduinvented a positional notation for the base 10 system. Heabandoned the letters which had been used for numerals past 9,and standardized the first nine symbols. About 825 A.D., theArab mathematician Al-Khowavizmi wrote an enthusiasticbook about the Hindu numerals. The base 10 system reachedSpain around the 11th century when the Ghobar numerals wereformed. Europe was skeptical and slow to change. Scholarsand scientists were reticent to employ the base 10 systembecause it had no simple way to denote fractions. But it becamepopular when merchants adopted it, since it proved soinvaluable in their work and record keeping. Later, decimalfractions made their appearance in the 16th century, and thedecimal point was introduced in 1617 by John Napier.

Someday, as our needs and ways of computing change, will anew system evolve and replace base ten?

1 A positional base value system is a number system in which the location ofeach digit influences the value of that digit. For example, in base ten for thenumber 375, the 3 digit is not merely worth 3 but because it is in the hundredsplace it is worth 300.


Th e Anyone who has studiedLi algebra or geometry hasPythagorean heard of the Pythagorean

Theorem. This famousT neorem theorem is used in many

branches of mathematicsand in construction, architecture and measurement. In ancienttimes, the Egyptians used their knowledge of this theorem toconstruct right angles. They knotted ropes with units of 3, 4 and5 knot spaces. Then, using the three ropes, they stretched themand formed a triangle. They knew the triangle would alwaysend up having a right angle opposite the longest side(32+42=52).

i-ymnusuTuun mrwurem ;Given a right triangle, 4then the square of thehypotenuse of a righttriangle equals the sum ofthe squares of the two legsof the right triangle.

Its converse is also true.If the sum of the squares oftwo sides of a triangleequals the square of thethird side, then thetriangle is a right triangle.

Although this theorem is named after the Greekmathematician, Pythagoras (circa 540 B.C.), evidence of thetheorem goes back to the Babylonians of Hammurabi's time,over 1000 years before Pythagoras. Perhaps the name isattributed to Pythagoras because the first record of writtenproofs come from his school. The existence of the Pythagoreantheorem and its proofs appear throughout continents, culturesand centuries. In fact more proofs of this theorem haveprobably been devised than any other!


G raphics is another area inwhich people are exploring theuse of computer. The optical il-lusion below is a computer ren-dition of Schroder's staircase. Itfalls in the category of anoscillating illusion. Our mindsare influenced by past experienc-es and by suggestions. The mindat first will see an object oneway, and when a certain lengthof time has passed it will change its point of view. The timefactor is influenced by our attention, or how quickly we getbored with what we initially focus on. In Schroder's illusionthe staircase will appear to flip upside down.

OpticalIllusions

andComputer

Graphics


The CycloiThe HelenGeometry

the path of a fixed pointstraight line.

d- The cycloid is oneof the many fasci-

Of nating curves ofmathematics. It isdefined as -

the curve traced byon a circle which rolls smoothly on a

One of the first references to the cycloid appears ina book by Charles Bouvelles published in 1501. But it was inthe 17th century that a large number of prominent mathema-ticians (Galileo, Pascal, Torricelli, Descartes, Fermat,Wren, Wallis, Huygens, Johann Bernoulli, Leibniz, Newton)were intent on discovering its properties. The 17th century wasa time of interest in the mathematics of mechanics and motion,which may explain the keen interest in the cycloid. Alongwith the many discoveries at this period of time, there weremany argumentsasto who discovered what first, accusa-


tions of plagiarism, and minimization of one another's work.As a result the cycloid has been labeled the apple of discordand the Helen of geometry. Some of the properties of thecycloid discovered during the 17th century are:

1) Its length is four times the diameter of therotating circle. It is especially interesting to find thatits length is a rational number independent of it.

2) The area under the arch is three times the area ofthe rotating circle.

3) The point on the circle that is tracing the cycloidtakes on different speeds -in fact at one place, PS, it iseven at rest.

4) When marbles are released from different points ofa cycloid shaped container, they arrive at the bottomsimultaneously.

P1 P5

Each circle represents a quarter turn of the rotating circle. Note the length ofthe quarter turn from Pl to P2 is much shorter than from P2 to P3 .Consequently, the point must speed up from P? to P3 since it must travelfurther in the same length of time. The point is at rest at where it mustchange directions.


There are many fascinating paradoxes tied to the cycloid. Thetrain paradox is especially intriguing.

At any instant a moving train never moves entirely in thedirection the engine is pulling it. There is always part of thetrain moving in the opposite direction than that in which thetrain is moving.

This paradox can be explained by using the cycloid. Here thecurve formed is called a curtate cycloid - the curve traced by afixed point outside a revolving wheel. The diagram shows thatpart of the train wheel moves backwards as the train movesforward.


Ger m anrmat hemat ician, A triangle toDavid Hilbert (1862-1943),first proved that any polygon a squarecan be transformed into any

other polygon of equal areaby cutting it into a finite number of pieces.

This theorem is illustrated by one of the puzzles of therenowned English puzzlist, Henry Ernest Dudeney (1847-1930).Dudeney transforms an equilateral triangle into a square bycutting it into four pieces.

Here are the four pieces. Fit them back together. First make

them into an equluateral triar

See appendix for solution of a triangle to a square.


H alley s 0 rbits and paths are conceptsthat can be easily described mathe-

Com et matically by equations and theirgraphs. Studying graphs can some-

times reveal cycles and periods of paths. And so it must havebeen in the case of Halley's Comet.

Halley's comet as depicted in the Bayeaux tapestry

Until the 16th century, comets were an unexplainedastronomical phenomena which did not seem to obey the solarsystem laws of Copernicus and Kepler. But in 1704 EdmundHalley worked on the orbits of various comets for which datawas available. Some of the most comprehensive records wereon the comet of 1682. He noticed that its orbit passed throughthe same regions of the sky as the comets of 1607, 1531, 1456,and concluded they were a single comet orbiting the sun ellipti-cally about every 75 to 76 years. He successfully predicted itsreturn in 1758, and it became known as Halley's Comet. Recentresearch suggests that Halley's Comet may have been recordedby the Chinese as early as 240 B.C.


With each appearance, the spectacle of Halley's tail fades asevidenced by its last appearance in 1985-1986.

Comets are believed to originate from ice planetoids which circlethe sun in a spherical shell about 1 to 2 light years from the sun.These planetoids are composed of ice and particles of metal andsilicates. Here at the outskirts of our solar system, at freezingtemperatures, these planetoids circle the sun at speeds of 3 milesper minute, thus taking 30,000,000 years to orbit the sun.Occasionally gravitational interference from nearby stars slows aplanetoid causing it to fall toward the sun, thereby changing itscircular orbit to an elliptical one. Once it begins its ellipticalorbit about the sun, part of its ice becomes gaseous. This formsthe comet's tail which always points away from the sun becausethe tail is blown by the sun's solar winds. The comet's tail iscomposed of gases and small particles that are illuminated bythe sun. The comet would continually pass in an unchangingelliptical orbit about the sun if it were not for the gravitationalinfluence of Jupiter and Saturn. Each orbit brings the cometcloser to the sun, which melts more ice and extends the tail. Thetail makes the size of the comet appear much larger (a typicalcomet is about 10 km in diameter). In the tail travel meteors,originally embedded in icy layers of the comet. The meteors areremnants of the comet which remain in orbit after the comet hasdisintegrated, creating a meteor shower when its orbit coincideswith the Earth's orbit.


M any designs andillustrations have becomefamiliar and often taken forgranted. In the British Jour-nal of Psychology, February,1958, Roger Penrose pub-

lish;~"-CO tf I L]-i

He called it a three-dimensional rectangular structure. Thethree right angles appear normal, but are spatially impossible.

These three right an-gles seem to form a tri-angle, but a triangle isa plane object (notthree-dimensional)and its angles total 180degrees, not 270 de-grees.

More recently, Penrosehas been responsiblefor the theory of

twistors. Although twistors are notvisible, Penrose believes that spaceand time are interwoven by theinteraction of twistors.

twister

Can you determine why Hyzer's optical illusion is alsomathematically impossible?

Hyzer's optical illusion


The The Inca empire encom-passed the area around Cuzco,

Quipu most of the rest of Peru andparts of Ecuador and Chile.Although the Inca did not

have a written mathematical notation system or a writtenlanguage, they managed their empire (more than two-thousandmiles long) by use of quipus. Quipus were knotted ropes using apositional decimal system. A knot in a row farthest from themain strand represented one, next farthest ten, etc. The absenceof knots on a cord implied zero. The size, color and configurationof knots recorded information about crop yields, taxes,

This illustration of a Peruvian quipu was drawn by a Peruvian Indian, D. FelipePoma de Ayala, between 1583 and 1613. In the lower left hand corner, there is anabacus counting device used with maize kernels on which computations wereperformed and later transferred to the quipu.


population, and other data. For example, a yellow strandmight represent gold or maize; or on a population quipu the firstset of strands represented men, the second set women, and thethird set children. Weapons such as spears, arrows, or bowswere similarly designated.

Accounting for the entire Inca empire was done by a class ofquipu scribes who passed on the techniques to their sons. Therewere scribes at each administrative level who specialized inan individual category.

In the absence of written records the quipus served as a means ofrecording history - these historical quipus were recorded byarmantus (wise men) and passed on to the next generation,which used them as reminders of stories they had been told.

And thus these primitive computers - quipus - had knotted intheir memory banks the information which tied together theInca empire.

The Inca Royal Road extended 3,500 miles from Ecuador toChile. All aspects of goings on in the vast Inca empire werecommunicated along the Royal Road via chasquis (professionalrunners), who were assigned a two miles stretch. They were sofamiliar with their particular increment of road that theycould run it at top speed day or night. They would relayinformation until it reached the desired location. Their service,coupled with the use of quipus, kept the Inca emperor up to dateabout population changes, equipment, crops, possessions,possible revolts, and any other pertinent data. The informationwas relayed on a twenty-four hour basis and was very accurateand current.


Calligraphy, Architecture, engineer& ing, decoration and ty-Typography & pography are a few fields

.1 in which geometric prin-

M aith em a tics ciples are applied. Al-brecht Direr lived from

1471 to 1528. During his lifetime, he combined his knowledgeof geometry and his artistic ability to create many art formsand art methods. He systematized the construction of Romanletters, which was essential for accuracy and consistency oflarge letters on buildings or tombstones. These illustrations byDfirer show the use of geometric construction for writingRoman letters.

Today, computer scientists have used mathematics to designsoftware to produce quality typography and graphics. Anoutstanding example is POSTSCRIPT programming languagedeveloped by Adobe Systems (of Palo Alto, CA) to work withthe laser printer.

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How many grains of wheat

are needed if they are placedon a chessboard in the follow-ing manner?

.- -:.- :.- . I '.. -I .One grain in the first square,two in the second, four in the third, eight in the fourth square,and so on doubling the amount with each new square.

F-

See appendix for solution to the wheat & the chessboard problem.

The Wheat&the

Chessboard

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Mathematicians and scien-tists have always been in-trigued by 7t, but it acquired awhole new following when it

foiled a diabolic computer in aStar Trek episode. IT wears different hats - it is the ratio of acircle's circumference to the diameter, it is a transcendentalnumber (a number that cannot be the solution of an algebraicequation with integral coefficients).

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For thousands of years people have been trying to compute Itto more and more decimal places. For example, Archimedesapproximated TI to between 3 1/7 and 3 10/71 by increasing thenumber of sides of an inscribed polygon. In the Bible, in theBook of Kings and in Chronicles, It is given as 3. Egyptianmathematicians' approximation was 3.16. And Ptolemy, in 150A.D., estimated it as 3.1416.

Theoretically Archimedes' method of approximation could beextended indefinitely, but with the invention of calculus, theGreek method was abandoned and the use of convergent

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infinite series, products, and continued fractions were used toapproximate Xt. For example,

I = 4/(1+12/(2+32/(2+52/(2+72/( ... )))))

One of the most curious methods for computing t is attributedto the 18th century French naturalist, Count Buffon and hisNeedle Problem. A plane surface is ruled by parallel lines, all dunits apart. A needle of length less than d is dropped on theruled surface. If the needle lands on a line, the tossis considered favorable. Buffon'samazing discovery was that the ra-

no or favorable tosses to unravora-ble was an expression involving It.If the needle's length is equal to dunits, the probability of afavorable toss is 2/IT. The moretosses, the more closely did theresult approximate It. In 1901, the Italian mathematician M.Lazzerini made 3408 tosses, giving It's value as 3.1415929 -correct to 6 decimal places. But whether Lazzerini actuallycarried out his experiment has been questioned by Lee Badger1

of Weber State University at Ogden, Utah. In yet anotherprobability method to compute It, R. Chartres, in 1904 found theprobability of 2 numbers (written at random) being relativelyprime to be 6/It2 .

It's startling to discover the versatility of It, crossing as it doesthe wide spectrum of geometry, calculus and probability.

'See False calculation of 7X by experiment, by John Maddox. NATURE magazine,August 1, 1994, vol. 370, page 323.


Earthquakes T here seems to be ahuman need to describe

and natural phenomena inmathematical terms. Per-

Logarithms haps this is because wewant to discover methodsby which we can have

some control - possibly through prediction - over nature.And so it is with earthquakes. It may at first seem unusual to

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a seismograph of an earthquake

link earthquakes with logarithms, but the method used tomeasure earthquakes' magnitudes is responsible for theconnection. The Richter scale was devised in 1935 byAmerican seismologist Charles F. Richter. The scale measures

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the earthquake's magnitude by describing the amount of energyreleased at the focus of the earthquake. The Richter scale islogarithmic so every time the magnitude of the Richter scalegoes up 1 unit the amplitude of the earthquake's seismographiccurve increases 10 times, while the energy the earthquakereleases increases about 30 times. For example, an earthquake ofmagnitude 5 releases 30 times the energy than one of magnitude4. Thus, an earthquake of magnitude 8 releases approximately303 or 27,000 times the energy than an earthquake of magnitude5.

The Richter scale numbers range from 0 to 9, but theoreticallythere is no upper limit. An earthquake of magnitude greater than4.5 can cause damage; severe earthquakes have magnitudesgreater than 7. For example, the Alaskan earthquake of 1964 was8.4 on the Richter scale and the San Francisco earthquake of 1906-- ,< 7 Q a

Wval 1.0.

Today, scientistswho specialize inthe study of earth-quakes go into thefield of seismolo-gy, a geophysicalscience. Delicateand refined instru-ments and meth-ods are beingsought and de- This representation of the earliest known seismograph

was made in China in the 2nd century A.D. from avised in order to be bronze wine jar six feet in diameter. Eight dragons with

able to quantify bronze balls in their mouths surround the jar. In theevent of an earthquake a dragon would drop its ball

and predict earth- which would fall in the mouth of a toad below. The

quakes. One of the instrument would then lock, and thus, indicate the

first instruments direction of the earthquake.

and continually used instruments is the seismograph, which au-tomatically detects, measures and graphically picturesearthquakes and other ground vibrations.


I X . It seems rather amusinghe parabolic in our present high tech

i in of t e world to find that in thece ing of the 19th century, the Capitolwas coincidently de-

Cap i<tol signed with its own non-electronic eavesdropping

devices. The United States Capitol was designed in 1792 by Dr.William Thornton, and the structure was reconstructed in1819 after having been burned in 1814 by invading Britishtroops.

Ihe ceiling in Statuary M-all as it appears today In the Umited States %apUo.

To the south of the Rotunda (the huge domed hall), isStatuary Hall, so named because in 1864 each state was asked


to contribute statues of two of its famous citizens. TheHouse of Representatives met in Statuary Hall until 1857. Itwas in this room that John Quincy Adams, while a memberof the House of Representatives, discovered its acoustical phe-nomenon. He found that at certain spots one could clearlyhear conversations taking place on the other side of the room,while people standing between could hear nothing and theirnoise did not obscure the sounds coming from across theroom. Adam's desk was situated at a focal point of oneof the parabolic reflecting ceilings. Thus, he could easily eaves-drop on the private conversations of other House memberslocated near another focal point.

Parabolic reflectors function in the following manner:

sound bounces off the parabolic reflector (or, inthis case a ceiling's dome) and travels parallel to theopposite parabolic reflector, where it bounces to itsfocal point. Thus all sounds originating at a focalpoint pass through the opposite focal point.

A ----------- A

( o focus focusQ

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The Exploratorium in San Francisco, California has parabolicsound reflectors set up for public use. They are situated onopposite sides of a large room and their focal points areidentified. Two people can carry on a normal conversationspeaking at the focal points. Neither the number of people northe noise level in the room will hinder their ability to heareach other.


Computers, One communicateswith an electronic

Counting, & computer by using acomputer language.

Electricity The computer lan-guage in turn is trans-

lated into some base system in order to direct the electricalimpulses which power the computer. Base ten works verywell with our pen and pencil calculations, but another basesystem is needed for electronic computers. If a memory devicewere to operate in base ten it would have to assume tendifferent states for the ten numerals comprising base ten (0,1,2,3,4,5,6,7,8,9). Although this is possible with a mechanicalsystem, it is not feasible with electricity. On the other hand,the binary base system is a perfect candidate for electroniccomputers. Only two numerals are used in base two (thebinary base ). These are 0 and 1. These numerals can be easilyrepresented by electricity in one of three ways:

(1) by having the current either on or off,(2) by magnetizing a coil in one direction or the other,(3) by energizing or not energizing a relay

In any of the three cases, one state is taken by the numeral 0and the other state by the numeral 1.

Computers do not count the way people do - one, two, three,four, five, six , seven, eight, nine, ten, eleven, twelve,...Instead, they count one, ten, eleven, one hundred, onehundred and one, one hundred and ten, one hundred andeleven,...

Since, computers operate with electricity. Their mechanismsuse electricity to translate to symbols we can understand ontheir monitors. As electricity passes through the intricate partsof a computer, it can either turn a part on or off. On and off


are the only two possibilities for electricity which is why onlytwo digits, 0 and 1, and base two are used by computers.

one, two,1, 10,

three, four, five, six,11, 100, 101, 110,

13 -

seven, eight,...111, 1000,...

BASE TEN vs BASE TWO

When we write our numbers we use the digits 0, 1,2, 3, 4, 5, 6, 7,8, 9. This is called base ten because we use ten digits to formany number. The placement of the digit in the number stands forthat digit times a power of ten. When we write our numbers,each digit's value depends on its place in the number, forexample,

5374 does not mean 5+ 3+ 7 + 4, but it means5 thousands +3 hundreds + 7 tens +4 ones.

Each place in the number is a power of ten:thousand =1000=lOxlOxlO=103

hundred=100=1 OxI 02ten=10=101

one=1=100

Computers write their numbers using only the digits 0 and 1.Their system is called base two because only the first two digitsare used to form numbers and each place in the number is apower of two. The first place is the I's place, then comes the 2'splace, then the 2x2=4's place, then the 2x2x2=8's place, and soon.

2x2x2=8's 2x2=4's 2's l's23 22 21 20

so the number 1101 would mean1x8 +1x4 +0x2 +lxl = totals 13 in base ten system.


Topo - Topo is a gameof many changing

A Mathematical strategies. Anynumber of people

Game can play. Whenone is beginning tolearn, start with

only two players. There are three parts to the game.

I. building the regions of playII. assigning values on some or all regionsIII. capturing regions

I. Taking turns, each player draws a region adjacent in someway to a previously drawn region. Ten such regions aredrawn by each player, as illustrated in A.

II. Each player chooses a colored pen. Then, taking turns, eachplayer assigns a number value to a region until each playercolored regions total number value is 100. If a player wantsto just assign a region the number 100, then that playerwould have only one region.

III. Object of the game: The player ending with the mostregions is the winner. Note the number value of a region isirrelevant.

Moves: A region is captured when one or more adjacent regionsbelonging to another player total more than that region.

Once a region is captured, it is out of play; and should bemarked for the capturing player.

Continue capturing until no more regions can be captured.


Topo has some intriguing variations. The more one plays themore one discovers various strategies in building regions,assigning values, and capturing regions.


Fibonacci Fibonaccil, one of the leadingmathematicians of the Middle

Sequence Ages, made contributions toI arithmetic, algebra and geome-

try. He was born Leonardo daPisa (1175-1250), son of an Italian customs official stationed atBurgia in southern Africa. His father's work involved travel tovarious Eastern and Arabic cities, and it was in these regionsthat Fibonacci became familiarized with the Hindu-Arabicdecimal system, which had place value and used the zerosymbol. At this time Roman numerals were still being used forcalculating in Italy. Fibonacci saw the value and beauty of theHindu-Arabic numerals, and was a strong advocate of their use.In 1202 he wrote Liber Abaci, a comprehensive handbookexplaining how to use the Hindu-Arabic numerals; how addi-tion, subtraction, multiplication and division were performedwith these numerals; how to solve problems; and furtherdiscussion of algebra and geometry. Italian merchants werereluctant to change their old ways; but through their continualcontact with Arabs and the works of Fibonacci and othermathematicians, the Hindu-Arabic system was introduced andslowly accepted in Europe.

It seems ironic that Fibonacci is famous today because of asequence of numbers that resulted from one obscure problem inhis book, Liber Abaci. At the time he wrote the problem it wasconsidered merely a mental exercise. Then, in the 19th century,when the French mathematician Edouard Lucas was editing afour volume work on recreational mathematics, he attachedFibonacci's name to the sequence that was the solution to theproblem from Liber Abaci. The problem from Liber Abaci thatgenerated the Fibonacci sequence is:

1Fibonacci literally means son of Bonacci.


1) Suppose a one month old pair of rabbits (male andfemale) are too young to reproduce, but are matureenough to reproduce when they are two months old.Also assume that every month, starting from the secondmonth, they produce a new pair of rabbits (male &female).

2) If each pair of rabbits reproduces in the same way asthe above, how many pairs of rabbits will there be atthe beginning of each month?

cEzpair, mature enough to reproduceC=pair, too young to reproduce

no. of pairs1=Fl=lst Fib. no.1=F2=2nd Fib. no.2=F3=3rd Fib. no3=F4=4th Fib. no.5= F5=5th Fib. no.

Each term of the Fibonacci sequence is the sum of the twopreceeding terms and is represented by the formula:

Fn=Fn l+Fn_2

Fibonacci did not study this resulting sequence at the time, andit was not given any real significance until the 19th centurywhen mathematicians became intrigued with the sequence, itsproperties, and the areas in which it appears.

Fibonacci sequence appears in:I. The Pascal triangle, the binomial formula & probability

II. the golden ratio and the golden rectangleIII. nature and plantsIV. intriguing mathematical tricksV. mathematical identities

I


dealing with the squarestheorem dealt with anylegs and the hypotenuse.

A twist toPythagore,Theorem

al li. .tX . lI.- -- Us

on the legs and the hypotenuse, hisshaped parallelograms built on the

Q

Use any right triangle and the follow procedure:1) construct any size parallelograms on the two legs of the

triangle;2) extend the parallelograms' sides until they intersect in

point P;3) draw ray PA so that ray PA intersects segment BC at R and

IRQI=IPAI4) draw a parallelogram on the hypotenuse BC so that its two

sides are the same length and parallel to RQ.

Pappus concluded: The area of the parallelogram on thehypotenuse equals the sum of the areas of the other twoparallelograms.

the P appus of Alexandriawas a Greek mathemati-an cian of about 300 B.C. whoproved an interesting va-riation of the Pythagore-

-- - - -_rl - - - - - -- _-- -3 _ C


W hat happensif one ring isremoved?

Are any two rings linked?

Are all three rings linked?

Trinity of Rings- a topological

model


Anatomy L eonardo da Vinciextensively studied the

& the Golden proportions of the hu-man body. His drawing

Sectonow has been studiedin detail, and shown to

illustrate the use of the golden sectional This is one of his draw-ings in the book he illustrated for mathematician Luca Paciolicalled De Divina Proportione published in 1509.

1 The term golden section is also referred to as the golden mean, thegolden ratio, the golden proportion . It is the geometric mean when it islocated on a given segment as follows. Point B sections off segment ACso that ( I AC I / I AB I )=( I AB I / I BC I ). The value of the golden meanmay be determined as, 1+15 1.6

2 A B C

"S.t

/

- , 4� - �I 1� I


The golden section is also present in the unfinished work, St.Jerome, by Leonardo da Vinci, painted around 1483. The figureof St. Jerome fits perfectly into a golden rectangle, assuperimposed on this drawing. It is believed that this was notan accident, but that Leonardo purposely made the figureconform to the golden section because of his keen interest and useof mathematics in many of his works and ideas. In the words ofLeonardo -"...no human inquiry can be called science unless itpursues its path through mathematical exposition anddemonstration. "

St. Jerome. Leonardo da Vinci. Circa 1483

The catenary& the paraboliccurves

When weights are tied to the catenary curve at equal intervals,the chain becomes a parabolic curve. This is similar to a cablesuspension bridge, such as the Golden Gate Bridge in SanFrancisco. Here the parabola is formed when vertical supportsare placed on the catenary cable.

The Exploratorium in San Francisco has a hands on exhibit ofa catenary arch.

1 The equation of a catenary curve is: y=a cosh (x/a) where the x-axisis the directrix.


A chain hang-ing freely forms acurve called a cat-enaryl curve. Thecurve looks muchlike a parabola,and even Galileofirst believed it

I -

An old but frustratingpuzzle is to fit these fourpieces together so theyform a T. Good luck!


The TProblem

See the appendix for the solution to the T problem.

v


Thales & the Thales (640-546 B.C.)was known as one of

Great Pyramid the Seven Wise Menof ancient Greece.Called the father of

deductive reasoning, he introduced the study of geometryinto Greece. He was a mathematician, a teacher, a philosopher,an astronomer, a shrewd businessman and the first geometerto prove his theories by a step-by-step proof. Thales correctlypredicted the solar eclipse in 585 B.C., and astounded theEgyptians when he calculated the height of their Great Pyra-mid using shadows and similar triangles.

procedure:The above diagram shows the shadow cast by the pyramid. Arod of known length, I DC I, is placed perpendicularly at the tipof the shadow at C. The rod's shadow has measured lengthICE I. I AF I is the length of 1/2 the pyramid's side. Now the

height of the pyramid, x, can be easily calculated using similartriangles, AABC and ACDE.(x/ICDI)=(IACI / ICEI),thusx=(1CDI-IACI) / ICEI


One of the qualifica- Hotel Infinitytions for hotel clerk at Ho-tel Infinity is a workingknowledge of infinity. Paul applied, was interviewed, and start-ed work the following evening. Paul wondered why the hotelrequired that all its clerks know about infinity, infinite sets, andtransfinite numbers. He figured since it was an infinitely roomedhotel it would be no problem finding rooms for its guests. Afterhis first night on the job, he was glad he had that knowledge.

When Paul relieved the day clerk, she informed Paul that therewere an infinite number of rooms presently occupied. As sheleft, a new guest walked in with a reservation. Paul needed todecide which room to give the guest. He thought for a moment,then moved each occupant to a room with the next highest num-ber, and therefore Paul was able to vacate room #1. Paul feltgood about his solution, but just then an infinite bus load of newguests arrived. How would he give them their rooms?

1 9°°° I211111.1

-Irl r-rf Ell

The idea of a hotel infinity was first conceived by German mathematician David Hilbert(1862-1943).

See appendix for Paul's solution.


example, sodium chlorate

Crystals -Nature 'sPolyhedra

cubes and tetrahedra, while chrome alum crystals are in theform of octahedra. Equally fascinating are the appearance ofdecahedra and icosahedra crystals in the skeletal structures ofradiolaria - microscopic sea animals.

Polyhedra are solids whosefaces are in the shape ofpolygons. They are calledregular polyhedra if theirfaces are identical poly-gons and their angles areidentical throughout. Thus,a regular polyhedron hasall its faces identical, all its

From classical times polyhe-dra have appeared in mathe-matical literature, but theirorigin is much older, beinglinked with the origin of na-ture itself. Crystals grow inthe shapes of polyhedra. For

crystals appear in the shape of


edges identical, and all its corners identical. There are infinite-ly many types of polyhedra, but there are only five regularpolyhedra - called Platonic solids.1 They are named afterPlato, who discovered them independently about 400 B.C.Their existence was known before this time by thePythagoreans, and Egyptians had used some of them inarchitecture and other objects they designed.

1See section on Platonic Solids.

The five Platonic solids

tetrahedronoctahedron

hexahedron(cube)

dodecahedron icosahderon

I

A-71-17".

I


Pascal's triangle, Blaise Pascal(1623-1662) was

the Fib onacci a famous Frenchmathematician

sequence & who might have.. become one of

binomial formula the great math-ematicians if it

were not for his religious beliefs, poor health and unwillingnessto exhaust a mathematical topic. His father, fearing that hisson would share his keen interest in mathematics1 and wantinghim to develop a broader educational background, initally dis-couraged him from studying mathematics in order that hemight develop other interests. But by the age of twelve Pascalshowed such a gift for geometry that his mathematical incli-nation was thereafter encouraged. He was very talented,and at the age of sixteen he wrote an essay on conics thatsurprised and astounded mathematicians. In his work was thetheorem that came to be known a Pascal's theorem, whichstates in essence that opposite sides of a hexagon, which isinscribed in a conic, intersect in three collinear points. At theage of eighteen he invented one of the first calculatingmachines. At this time he suffered from poor health, and madea vow to God that he would give up his work in mathematics.But three years later he wrote his work on the Pascal triangleand its properties. On the night of November 23, 1654, Pascalhad a religious experience that prompted him to devote hislife to theology and abandon mathematics and science. Exceptfor one brief period (in 1658-1659), Pascal never studiedmathematics again.

Mathematics has a way of connecting ideas that appearunrelated on the surface. So it is with the Pascal triangle, the

1 Etienne Pascal, was very much interested in mathematics, and in factthe curve limagon of Pascal is named after him rather than his son.


Fibonacci sequence and Newton's binomial formula. The Pascaltriangle, the Fibonacci sequence, and the binomial formula areall interrelated. The design illustrates their relationships. Thesums of the numbers along the diagonal segments of the Pascaltriangle generate the Fibonacci sequence. Each row of the Pascaltriangle represents the coefficients of the binomial (a+b) raised toa particular power.

For example,

(a+b)0 = 1 1

(a+b)l = la + lb 1 1

(a+b)2 = la 2 + 2ab + lb 2 1 2 1

(a+b)3 = la 3 + 3a 2 b +3ab2 + lb 3 1 3 3 1

I1 Fibonacci's

2 3Sequenece3

Ca'58

1 \ a

6 4 1* 0 0 0 0

(a+ b= ( ) an+ (N ) ewlt b+ ( ) la-2 b

Newton 's binomial formula

Pascal'striangle


Mathematics W ho would believethat knowledge of math-

of the ematics can help one'sbilliard game? Given a

billiard table rectangular billiard tablewhose sides are in whole

number ratios, e.g. 7:5, a ball hit from a corner at a 45 degree an-gle will end up in one of the corners after a certain number of re-bounds. In fact, the number of rebounds is tied in with the di-mensions of the table. The number of rebounds before reaching acorner is given by the formula:

length + width -2.

STARTING POINT

ENDING POCKETafter 10th rebound

With the above table the rebounds total 10.7+5-2=10 rebounds.

Note the formation of isosceles right triangles in determiningthe path of the ball.


V arious geomet-

ric shapes appear inmany facets ofthe physical world.Many of theseshapes are not visi-ble to the nakedeye. In this partic-ular electron's paththe appearance ofpentagons is evi-dent.

The geometry ofan electron's

path


Topologists have creat-ed some fascinating ob-jects. The Moebius strip,created by the Germanmathematician Augustuslinhis /1'7nn 1040N toVneUsUIUc ob/7j-ec0t. Lb

one such object.

The above diagram illustrates a strip of paper that has beenglued to form a band. One side of the paper is on the inside ofthe band while the other side lies on the outside of the band. Ifa spider were to crawl along the outside of the band, the onlyway it could get to the inside section would be to somehowcross its edge.

This diagram illustrates the Moebius strip, formed by taking astrip of paper and twisting it once before gluing its endstogether. Now the paper strip in this form no longer has twosides. It is one sided. Suppose the spider were to begincrawling along the Moebius strip, it would be able to crawlover the entire strip without ever having to cross an edge. Toprove this, take a pen and continually draw a line. You will

The MoebiusStrip & TheKlein Bottle


have covered the entire strip and returned to your startingpoint without lifting your pen.

Another interesting property of the Moebius strip is todiscover what happens when it is cut along the center mark.Try it.

Moebius bands, as in fan belts for automobiles or belts formechanical devises, are of particular interest in industry, sincethey wear more uniformly than conventional belts.

Just as intriguing as the Moebius strip is the Klein bottle. FelixKlein, a German mathematician (1849-1925), devised atopological model of a special bottle which has only onesurface. The Klein bottle has an outside but no inside. It passesthrough itself. If water were poured into it, the water wouldjust come out of the same hole.


There is an interesting connection between the Moebius stripand the Klein bottle. If the Klein bottle were cut halfway alongits length, it would form two Moebius strips!


This puzzle was created A Sam Loydby the famous puzzlistSam Loyd. The object is to Puzzlework one's way out of thediamond. Start at thecenter, where the #3 appears. This number indicates you mustmove 3 squares or spaces right, left, up, down, or diagonally.Doing this, you land on another number which tells you howmany more square spaces you can move in one of the eightpossible directions.

Good luck!

a for 4 5 5 1 L p

.5 2 6 4 .5 85 -9 I*

IE]3 El 6 4 l

] 2.643 l 28t 4I Th3E !3

l 2 9 | +x7

Se th apeni fo th souint0a Ly'uze


Mathematics & Mostofusatonetime or another

paperfoldtng have folded a pieceof paper and tuckedit away, but fewer

of us have intentionally folded paper in order to study themathematical ideas it revealed. Paper folding can be botheducational and recreational. Even Lewis Carroll was a paperfolding enthusiast. Although paper folding transcends manycultures, it is the Japanese who are associated with developingand popularizing it into the art form called origami.

Some mathematical aspects of paperfoldingMany geometric concepts naturally appear when paper folding.Some of these are: square, rectangle, right triangle, congruent,diagonal, midpoint, inscribe, area, trapezoid, perpendicular bi-sector, the Pythagorean Theorem, geometric & algebraic ideas

Here are some examples of paperfolding that illustratethe use of these concepts.i) from a rectangular shaped paper form a square

[E -off this part

ii) with the square paper form 4 congruent righttriangles

iii) find the midpoint of a side of a square m

iv) inscribe a square in the paper square

v) studying the paper's creases, notice that theinscribed square is 1/2 the area of the large square


vi) form two congruent trapezoids by taking a squaresheet of paper and folding it along any edge so thecrease passes through the center e

vii) make the perpendicular bisector of a segment byfolding the square in half-the crease will be theperpendicular bisector to the side of the square

Liviii) Demonstrate the Pythagorean TheoremFold the square paper as shown in the diagram.

B N

c2 = area of square ABCDa2 = area of square FBIMb2= area of square AFNO

r

M

A

I

0K

By matching up congruent shapes,the area of square FBIM = area of A ABK

andthe area of AFNO = the area of BCDAK(the remaining area of square ABCD).

D

C

Thus, a2 + b2 =c2 F a B

b

A


ix) Demonstrate the theorem that the measure of the angles ofa triangle total 180 degrees, by taking any shaped triangle andfolding it along the dotted lines, as illustrated.

al + cQ + bQ = 1801 -they form a line

x) Construct a parabola by folding tangent lines.

procedure: Locate the focus point of the parabola a few inchesfrom the side of a sheet of paper. Crease the paper between 20and 30 times as illustrated in the diagram. These creases arethe tangents of the parabola and outline the curve.


Each term in the Fibonaccisequences is generated by addingthe two previous terms. Anysequence generated in this manner iscalled a Fibonacci-like sequence.

TheFibonacci

Trick

1, 1, 2, 3, 5, 8, 13, 21,34, 55, 89, 144, 233, 377,610, 987, 1597, 2584, 4181,6765, 10946, 17711, 28657,46368, 75025, 121393,196418, 317811, 514229, .0.

Select any two numbers and generate a Fibonacci-like sequencestarting with the two numbers you selected. The sum of the firstten numbers in your sequence will automatically be eleven timesthe seventh term.

Can you prove this for any two starting numbers?

See appendix for proof of Fibonacci trick.

1See section on The Fibonacci sequence for more information.

The EvolutionofMathematicalSymbolsclarity and certainly to save time, effort and space.

During the fifteenth century, some of the first symbols used forplus and minus were p and m, respectively. Germanmerchants, for example, were using + and - signs to denotedefective weights. At this time + and - signs were adopted bymathematicians, and after 1481 they began to appear inmanuscripts. The symbol for times , x, is attributed to WilliamOughtred (1574-1660), but it met opposition from somemathematicians because they contended it would be confusedwith the letter x.

There were often as many symbols for a concept as there weremathematicians. For example, in the sixteenth century FrancoisVieta first used the word aequalis and later the symbol, -, todenote equality. Descartes preferred oc for equality, but it wasRobert Recorde's (1557) symbol, =, which ultimately wasadopted. For him, parallel lines were objects that were mostalike and denoted equality.

Although letters had been used for unknowns by the ancientGreek mathematicians Euclid and Aristotle, this was not acommon practice. In the 16th century, words such as radix(Latin for root), res (Latin for thing), cosa (Italian for thing), coss(German for thing) were used to refer to unknowns. Between theyears 1584-1589 when lawyer Francois Vieta was between termsin the Brittany Parliament, he made an extensive study of theworks of many mathematicians. He developed the notion of


From the early Baby-lonian scribe poking hiscuneiform into a claytablet to take the spaceallotted for zero, mathe-maticians have devisedsymbols for conceptsand operations - for


using letters for positive known and unknown quantities.Descartes modified his idea and introduced the notion of usingthe first letters of the alphabet for known quantities and the last

This symbol was first used byItalian mathematician Fibonacciin 1220. It denotes a/and probablywas derived from the Latin wordradix meaning root. The symbolwe use today, F';is from 16thcentury Germany.

17th century Germanmathematician, ThisLeibniz, chose this usedsymbol for divismultiplication. Fren

I.E.C170(

This symbol was used foraddition by Tartaglia,Renaissance mathematician.It is derived from the Italianword piu (more).

The German mathematician,Christoff Rudolff used thissymzbolfor cube root in 1525.1/ is from France in the 17thcentury.

In 1859 Benjamin Peirce,reversed D was a Harvard professor, usedto denote this symbol for pi. ir

;ion by originated in England inchman the 18th century.Gallimard in the1S.

lXThe ancient Greekmathematician, Diophantusused this symbol forsubtraction.

letters for unknown quantities. And finally, in 1657, letters wereused for both positive and negative numbers by John Hudde.


00 had been used by the Romans to denote 1000, and later anyvery large number. In 1655, John Wallis, an Oxford professor,used the symbol, -o, for infinity for the first time. But the symbolwas not generally employed until 1713 when it was used byBernoulli.

Other symbols that evolved were the use of parentheses in 1544,square root brackets and braces in 1593 and the square rootradical, which was devised by Descartes (he used X- for cuberoot).

It is hard to imagine working mathematical problems without a+ sign or symbol for 0, or any of the other mathematicalnotations we take for granted. It is also difficult to realize it hastaken centuries for them to evolve and be universally accepted.

The chart compares mathematical symbols andphrases used in the past and present.

.......... A.T.......

p +

m

V used under the radical

Cardan(1501-1576) ..v.7.p:.14 ....

Chuquet 1484 123 +120 +71m 12x3 +12+7x

BombelliHi3

Stevin 1585 14+3+ 6® +(® 1+3x+6x2 +x 3

1+3x+6xx+x3 1+3x+6x2+X3Descartes


This sketch by Some geometricLeonardo da Vincishows his use of designs ofregular polygons inthe design of a Leonardochurch. Leonardo's Vi cinterest and study in da Vincigeometric structuresand his knowledge of symmetry were his tools for creatingarchitectural plans for adding chapels to churches withoutupsetting the design and symmetry of the main building.

A~

V r

* ~

Ite r@W A} 4 ;

, affrIf AX}Y lSi~ X -

Written below in dif-ferent number systemsare ten historical dates.Write each in base ten,

and identify what historical event took place in that year.

11-

eee.

R?~to

>tt~ IVnlILXVI

/AYNE

MDCCLXX

1LLELLOLOELThis chart of various number system may help you in deciphering the numbers.

Today |12 1 13 14 15I 61 71I 01 1 1311111 1l

Babylonian (1500 B.C.) I y PT IfTf Wlas rIV17PT}IRl( I1$j4 tv"' ? 1 ? ?Chinese (500 B.C.) | -| - |-- I -t |* III 'l I -|- I | ? ? ?Greek (400 B.C.) I A 53 ir 1A IJE If IZ Ili le I I IjA|ie|irl? I ? JrEgyptian (300 B.C.) illi j)I i t 5i"P', , wi's,4,:1 n Ilnilnw4t ? IRoman (200 B.C.) J=~I1 I'm IM IyflIjIIXIffInIlII? =?Maya (300 A.D.) | * I,,.,^ u1IL ' I" ' l" l?

Hindu (11th century) I 2 1 l 18 lI alcO lE1Ci2oS1215 ?' 1? 1?Base Two (computers) II 1101111oIliWIpgI 1OgiIwIucdI;III ? I? I?

See appendix for solution.


Ten HistoricalDates


"The advancement andperfection of mathematics areintimately connected with theprosperity of the State. "

- Niasnolonn I

Napoleon Bonaparte (1769-1821) had a special respect for mathe-matics and mathematicians, and enjoyed the subject himself. Infact, he is attributed with the following theorem.

If three equilateral triangles are constructed off the sides of anytriangle, then the centers of the circles which circumscribe eachequilateral triangle are vertices of another equilateral triangle.

Napoleon'sTheorem


Lewis Carroll Charles LutwidgeTh Dodgson(1832-1898),

T ean English mathema-tician and logician, isMathematician better known by hispseudonym Lewis

Carroll and as the author of Alice in Wonderland and Alicethrough the Looking Glass. In addition, he published many textsdealing with various fields of mathematics. His book PillowProblems - 72 problems which were nearly all written andsolved while he lay awake at night - deals with arithmetic,algebra, geometry, trigonometry, analytical geometry, calculusand transcendental probability.

'Contrariwise," continued Tweedledee"if it was so, it might be; and if Itwere so, It would be: but as it isn't,it ain't. That's logic.' -Lewis Carroll

A Tangled Tale was originally printed as a monthly magazinearticle, and later compiled into a delightful story containingmathematical puzzles in its ten chapters. It is said that QueenVictoria was so taken by Carroll's Alice books that she sent forevery book he had written. Imagine her surprise at the stack ofmath books she received.


problem # 8 from Pillow Problems"Some men sat in a circle, so that each had 2 neighbours; andeach had a certain number of shillings. The first had oneshilling more than the second, who had one shilling more thanthe third, and so on. The first gave one shilling to the second,who gave two shillings to the third, and so on, each givingone shilling more than he received, as long as possible. Therewere then 2 neighbours, one of whom had 4 times as much asthe other. How many men were there? And how much had thepoorest man at first? "

See the appendix for the solution.

This maze was drawn by Lewis Carroll when he was in his twenties.He made the maze with paths that cross over and under each other. Theobject is to find a path out of the center.


Counting onf Because paper writingmaterials were at a premi-Fingers urn during the MiddleAges, counting and com-municating results by fin-

ger signs were often used. As the diagram illustrates the fingersystem was able to represent both small and large numbers.


The figures belowinvolve the use of theMoebius strip. If youmake paper models ofthese topological models,and cut them aiong the

A Twist tothe Moebius

Strip


Heron s |M any have learned ingeometry how to compute thei neorem area of a triangle using the

length of its altitude and of itsrelated base. But without Heron's Theorem, calculating its areaknowing only the lengths of its three sides would requireknowledge of trigonometry.

C

Heron is best known in the history of mathematics for theformula,

area of triangle= -J/s(s - a)( s - b)(s - c)

where a, b, and c are the lengths of the sides of the triangle and sis half the sum of the lengths of the sides of the triangle.

Heron's formula was known earlier to Archimedes, whoprobably had proven it, but the earliest written record we haveis from Heron's writing, Metrica. Heron is best described as anon-classical type of mathematician. He was more concernedabout the practicality of mathematics than about theory and thetreatment of mathematics as a science and an art. As a result, heis also remembered as the inventor of a primitive type of steamengine, various toys, a fire engine that pumped water, an altarfire that lit as soon as the temple doors opened, a wind organ,and various other mechanical devices based on the properties offluids and the laws of simple mechanics.

These rare Gothic plansshow the use of geometryand symmetry in the archi-tecture of the Dome of Mi-lan. These plans were pub-lished in 1521 by CaesarCaesariano, master archi-tect of the Dome of Milan.


A look atGothic

architecture& geometry


N apier s W orking with complicatedfigures had become more andBones more tedious, especially for sci-entists doing astronomical cal-

culations, seamen needing to perform practical navigationproblems and merchants rendering their accounts. Then, in the17th century, John Napier (1550-1617), a famous Scot-tish mathematician, revolutionalized computation with his dis-covery of logarithms (a method of using exponents to perform com-plicated multiplication and division by converting it to addition andsubtraction). 1 Napier's method of computation with logarithmsand the tables he developed simplified difficult calculationsinvolving multiplication, division, exponents, and finding roots.Although the theory of logarithmic and exponential functions isan essential part of mathematics, with the introduction of mod-em electronic calculators and computers, the logarithmic tablesand their uses have become as obsolete as the slide rule. But thedevelopment of these tables and shortcut methods of computa-tion were a marvelous computing method widely used forcenturies by mathematicians, accountants, navigators, astrono-mers and scientists.

Using logarithms, Napier also invented numbered rods, calledNapier's bones, to help merchants' accounting. Merchantswould carry a set of rods, made of ivory or wood, with them toperform multiplication, division, square roots and cube roots.Each rod was a multiplication table for the digit at its top. Tomultiply 298 by 7 one would line up the 2, 9, and 8 rods and thencount down to the seventh row, and sum the two numbers asillustrated.

1 For example, to perform the operation-3600/.072, one wouldconvert these numbers to their exponent (i.e. logarithmic) form, using logtables. To perform division of numbers written in exponent form of thesame base, is a simple matter of subtracting their exponents. So the twologarithmic numbers for 3600 and .072 are subtracted, and the result isconverted back by using log tables to a number in base ten.


51~

,.o75

34

6

18;Z4

X6

-44,-

I16-4>

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A rt & Consciously or unconsciouslyP . . mathematics has influenced art

Protective and artists over the centuries.Projective geometry; the gold-G eom etry en mean; proportion; ratios; op-tical illusions; symmetries; geo-

metric shapes; designs and patterns; limits and infinity; andcomputer science are some of the areas of mathematics whichhave affected various facets and periods of art - be it primitive,classical, Renaissance, modem, pop or art deco.

Pt

The superimposed lines illustrate the use of projective geometry by Leonardo daVinci in his masterpiece, The Last Supper.

: arm - -


An artist painting a three dimensional scene on a twodimensional canvas must decide how things change whenviewed from different distances and positions. It is in this areathat projective geometry developed and played a major part inRenaissance art. Projective geometry is a field of mathematics thatdeals with properties and spatial relations of the figures as they areprojected-and therefore with problems of perspective. To createtheirrealisticthree dimensional paintings, Renaissance artistsused the now established concepts of projective geometry -

point of projection, parallel converging lines, vanishing point.Projective geometry was one of the first non-Euclideangeometries to surface. These artists wanted to depict realism.They reasoned that if they perceived a scene outside through awindow, it would be possible to project what they saw onto thewindow as a collection of points of vision if they kept their eye ata single focal point. Thus, the window could act as their canvas.Various devices were created to enable the artist to actuallytransform the window into a canvas. These wood cuts byAlbrecht Durer illustrate two such devices. Note that the artist'seye is at a fixed point.


Infinity & Every circle has a fixedcircumference - a length of a

the Circle finite amount. One method forderiving the formula for thecircumference of a circle is to use

the concept of infinity. Study the sequence of perimeters ofinscribed regular polygons. (A regular polygon has all sides thesame size and all angles the same measure.) By calculating andstudying the perimeter of each inscribed regular polygon, onenotices as the sides of the polygon increase its perimeter comescloser and closer to the circumference of the circle. In fact, thelimit of the perimeters as the number of sides approach infinityis the circumference of the circle. The diagram illustrates that themore sides a polygon has the closer its sides are to the circle, andthe more it resembles a circle.


Take any size circular The Amazingtrack made up of any two ranconcentric circles, as illus- Tracktrated below. Can youshow that the area of thetrack is equal to the area of a circle whose diameter is a chord ofthe larger circle but tangent to the smaller circle?

See appendix for proof.


This 17th centuryPersian drawing ingeni-ously pictures fourhorses. Can you locatethe four horses?

WI

..t i,

,.11 I

Iu . "I%

See appendix for solution.

The PersianHorses & SamLoyd's Puzzle


This drawing may have been the inspiration for the jockey anddonkey puzzle by master puzzlist Sam Loyd (1841-1911).

I-----------------------------------

4I

------- S---- -------------

Loyd's original version was created around 1858, when he was ateenager.

The problem is to cut the picture into threerectangles along the dotted lines, and rearrange therectangles without folding them, to show twojockeys riding two galloping donkeys.

This puzzle was an instant success. In fact, it was so popular thatSam Loyd reportedly earned $10,000 in a matter of weeks.

See appendix for solution to Sam Loyd's puzzle.

i i ,,q i

I

I

i

it& I 6I I 'r I


Lunes |The word lune has its origin from theLatin word lunar - moon shaped.Lunes are plane regions bounded by

arcs of different circles (see shaped crescents in the diagram).Hippocrates of Chios (460-380 B.C.), not to be confused with thephysician of Cos (author of the Hippocratic oath), did extensivestudy with lunes. He probably believed that they couldsomehow be used to solve the problem of squaring a circle.'

A C

He found and proved that:

The sum of the areas of two lunes constructedon two sides of a triangle which is inscribed

in a semicircle, equals the area of that triangle.

1 See section on the Impossible Trio.


If ABC, AEB, BFC are semicircles, thenthe area of lune(l) +area of lune(2)=area of triangle ABC.

proof

Although Hippocrates was not successful in his effort to square acircle, his quest helped him discover many new important math-ematical ideas.

area of semin AEB/areaofsemi))ABC=6r 1 AB 1 2/8)/(in 1 AC 1 2)/8= I AB 12/, AC 2-

area of semiE)AEB = area of semi) ABC( I AB 12/ 1 AC 12)-line (a)

similarlyarea of seminOBFC = areaof semi®)ABC (I BC 12/ 1 AC 1 2) -line(b)

Now adding like terms and factoring lines (a) and (b) -area of semiOAEB + area of semi(DBFC =

areaof semiEABC ((I AB 12 + I BC 12 )/ 1 AC 12 -line (c)

AABC is a right triangle because it is inscribed on a semi.Thus,

I AB 12 + I BC 12 = I AC 12 -by the Pythagorean theorem

Substituting this in line (c) and simplying we get -area of semi®)AEB + area of semiOrFC = area of semiOABC

subtracting area (3) + area (4) =area (3) + area (4)QED area of lune(l) + area of lune (2) = area A ABC


Hexagons in Many of nature's crea-tions are beautiful models

N ature for such mathematical ob-jects as squares and circles.The regular hexagon is oneof the geometric figures

found in nature. A hexagon is a 6-sided figure. It is a regularhexagon if all its sides are the same length and its angles are thesame size.

Mathematicians have shown that only regular hexagons, squaresand equilateral triangles can be fitted together (tessellated) sothere is no wasted space.

Of the three, the hexagon has the smallest perimeter when theirareas are equal. This means, that in forming a hexagonal cell tobuild the honeycomb, the bee uses less wax and does less workfor the same space. The hexagon shape is found in honeycombs,snowflakes, molecules, crystals, sealife and other forms.

Walking in a snow flurry you are in the midst of some wondrousgeometric shapes. The snowflake is one of the most exciting

A //'


examples of hexagonal symmetry in nature. One can spot the

hexagons in the formation of each snowflake. The infinitenumber of combinations of hexagonal patterns, explains the

common belief held that no two snowflakes are alike.1

Nancy C. Knight of the National Center for Atmospheric Researchin Boulder Colorado, may have discovered the first set of identicalsnowflakes. They were collected on November 1, 1986.


Googol 6 A googol is 1 followedby a hundred zeros, 10100.

Go ogo ip lex The name googol was coinedby mathematics author Dr.Edward Kasner's nine year

old nephew. The nephew also suggested another number muchlarger than a googol called a googolplex, which he described as 1followed by as many zeros as you could write before your handgot tired. The mathematical definition for a googolplex is 1followed by a googol of zeros, 10googO91

10 000000000000 000000 000 000 000 000 000000 000 000000 000 000000 000 000 000 000 000000 000 000000 000 000

000 000 000 000Using large numbers:1) If the entire universe were filled with protons and electrons so thatno vacant space remained, the total number would be 10110. Thisnumber is larger than a googol but much less than a googolplex.

2) The number of grains of sand on Coney Island is about 1020.

3) The number of words printed since the Guttenberg Bible (1456)until the 1940's is about 1016.


This 3 by 3 magiccube is an arrangementof the first 27 naturalnumbers so that the sumof each row or columnof three numbers is 42.

1 0_26 6

8-

22

2-

1

/

/

- 3

19

1227 -

/

/1

7/18

7

4

1.j


Fracta is- For many centuries the

Ae 1 or objects and concepts ofreal or Euclidean geometry (such aspoint, line, plane, space,im aginary.? squares, circles, ...) were

considered to be the thingsthat describe the world in which we live. The discovery ofnon-Euclidean geometries has introduced new objects thatdepict the phenomena of the universe. Fractals are such objects.These shapes are now considered to portray objectsand phenomena of nature. The idea of fractals originated in thework of mathematicians from 1875 to 1925. These objects werelabeled monsters, and believed to have little scientific value.Today they are known as fractals - coined by Benoit Mandelbrotin 1975, who made extensive discoveries in this area.Technically, a fractal is an object whose detail is not lost as it ismagnified - in fact the structure looks the same as the original.

In contrast, a circle appears tobecome straight as a portion of it ismagnified. However, there are twocategories of fractals - the geometricfractals which continuously repeatan identical pattern and the randomfractals. Computers and computergraphics are responsible for bringing

these "monsters" back to lite byThe snowflake curvel is an ex- almost instantaneously generatingample of a fractal generated byequilateral triangles being add- the fractals on the computer screened to sides of existing equilateral and thereby displaying their bizarretriangle. shapes, artistic designs or detailed

landscapes and scenes.

It had been felt that the orderly shapes of Euclidean geometrywere the only ones applicable to science, but with these new

1See section on snowflake curve for additional information.


forms nature can be viewed from a different perspective.

Fractals form a new field of mathematics - sometimes referred

to as the geometry of nature because these strange and chaotic

shapes describe natural phenomena such as earthquakes, trees,

bark, ginger root, coastlines, and have applications in astrono-

my, economics, meteorology and cinematography.

a random Jractal

The Peano curve isan example of a frac-tal and also a space-filling curve. In aspace filling curve,every point within agiven area is tracedand gradually black-ens the space. Thisexample is partiallytraced.

Cesaro curve - a fractal


Nanoseconds - It takes an electricimpulse one-billionth

measuring time of a second to travel 8inches. One-billionth of

on computers a second has come to becalled a nanosecond.Light travels one foot in

one nanosecond. Computers today are built to perform millionsof operations per second. To get a feeling for how fast a largecomputer can work, consider half of a second. Within a half asecond the computer would have performed the following tasks:

1) debited 200 checks to 300 different bank accounts;2) examined the electrocardiograms of 100 patients;3) scored 150,000 answers on 3000 exams, and evaluated the

validity of each question;4) figured the payroll for a company with one-thousand

employees;5) plus time left over for some other tasks.

It's mind boggling to tryto imagine how fast acomputer would be if itcould be powered bylight rather than electrici-ty. What kind of numbersystem would be neededto harness the use oflight? Would it be basedon the number of colorsin the spectrum of light?Or perhaps some otherproperty of light?


Leonardo da Vinci Geodesic Domewas intrigued bymany fields of study of Leonardo daand their interrelation.Mathematics was one V inciof them, and he usedmany concepts ofmathematics in his art, his architectural designs and his inven-tions. Here is a rendition of a geodesic dome he sketched.


M agic M agic squares have intriguedpeople for centuries. From ancient

Squares times they were connected with thesupernatural and the magical

world. Archeological excavations have turned them up inancient Asian cities. In fact, the earliest record of the appearanceof a magic square is about 2200 B.C. in China. It was calledlo-shu. Legend has it that this magic square was first seen byEmperor Yu on the back of a divine tortoise on the bank of theV-11-i,, T2,.-

The black knots repre-sent even numbers andthe white knots are oddnumbers. In this magicsquare, its magic num-ber (the total of anyrow, column or diago-nal) is fifteen.

leIn the Western world magic squares were first mentioned in 130A.D. in the work of Theon of Smyrna. In the 9th century, magicsquares crept into the world of astrology with Arab astrologersusing them in horoscope calculations. Finally, with the works ofthe Greek mathematician Moschopoulos in 1300 A.D., magicsquares and their properties spread to the western hemisphere(especially during the Renaissance period).

SOME PROPERTIES OFMAGIC SQUARES:The order of a magic square is defined bythe number of rows or columns. Forexample, this magic square has order 3because it has 3 rows.

16

6

8

2

10

12

14

4


The magic of a magic square arises from all the fascinatingproperties it possesses. Some of the properties are:

1) Each row, column, and diagonal total the same number. This magicconstant can be obtained in one of the following ways:

a) Take the magic square's order, n, and find the value of1/2(n(n2 +1)) where the magic square is composed ofthe natural numbers 1,2,3,. . ,n2. 15

8 1 6 order 3.magic number 4 A\'

1 - - ¶1/2(3(32+1))=15

b) Take any size magic square and, starting from the lefthand corner, place the numbers sequentially alongeach row. The sum of the numbers in either diagonalwill be the magic constant.

2) Any two numbers (in a row, column, or diagonal) that areequidistant from the center are complements. Numbers of a magicsquare are complements if their sum is the same as the sum ofthe smallest and largest numbers of that magic square.

8 [1 6 This magic square has complements-[3 1517 8&2 6&4 3&7 1&91419 12 1

WAYS TO TRANSFORM AN EXISTING MAGIC SQUARE TO

ANOTHER MAGIC SQUARE:

3) Any number may be either added or multiplied to every number of amagic square and it will remain a magic square.

4) If two rows and two columns, equidistant from the center, areinterchanged the resulting square is a magic square.

5) a) Interchanging quadrants in an even order magic square results ina magic square. at is one of the fourIt 9 71 ections of a magic square


b) Interchanging partial quadrants and rows in an odd order magicsquare results in a magic square.

More has been written on magic squares thanany other mathematical recreation. Benjamin Franklin spentconsiderable time devising methods of forming magic squares.It is quite challenging to take the first 25 counting numbers andarrange them in a 5x5 square so that each row, column, and

r - -r - -r -,

THE STAIRCASE METHOD

for a 3x3 magic squareSTEP 3

I2i

STEP 1 STEP 2

l11i1r3 5

41 I2

STEP 4 STEP 5

8 1 6 iX- +-+-A

35 7 iii

STEP 7

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I I I- ..L - .L. - r- -

I I I

r --- - .L -n

8 STEP 6

F- - - -STEP 8

COMPLETED

8 1 6


diagonal total the same amount. This would be referred to as amagic square of order 5. Any magic square with an odd numberof rows or columns is an odd order magic square. If there are aneven number, it is an even order magic square. A generalmethod for making any size even order magic square is stillbeing explored. On the other hand, for the odd order ones thereare a number of general methods that can be used to make anysize odd order magic square. The staircase method, invented byLa Loubere is well known among magic square enthusiasts.The diagrams illustrate a 3x3 magic square being made usingthis method.

THE STAIRCASE METHOD

1) Start with the number 1 in the middle box of the top row.

2) The next number is placed diagonally upward in the next box,unless it is occupied. If it lands in a box in an imaginary squareoutside your magic square, find its location in your magic squareby matching the location its holds in the imaginary square toyour magic square.

3) If in your magic square the diagonally upward box isoccupied, then place the number in the box immediately belowthe original number, e.g. for numbers 4 and 7.

4) Continue following steps (2) and (3) to obtain the locations ofthe remaining numbers for the magic square.

Now try making the 5x5 magic square using the first 25 countingnumbers and the staircase method. Test some of the methods foraltering magic squares on it to see how they work.

Using any of the magic squares you have constructed, multiplyeach of its numbers by a constant you choose. Is the resultingsquare a magic square?


For the even order magic squares various methods have beendevised for constructing a particular even order magic square.

example: The diagonal method applies only to a 4x4 magicsquare.

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5

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2

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14

3

71

15

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procedure:Begin by sequentially placing the numbers in the rows of themagic square. If a number lands on one of the diagonals, itslocation must be switched with its complement.

With a 4x4 magic square, either rows or columns may beinterchanged, and it will remain a magic square. Also ifquadrants are interchanged, it results in a magic square.

See if you can devise your own methods for making magicsquares of other even orders, or discover a general method for alleven order magic squarest You may also want to research othermethods for making any odd order magic squares that havebeen devised.

1 Many have devoted much time and effort to discovering a generalmethod of even order magic squares. Hyman Sirchuck of Howell, NewJersey claims to have devised a method for making even order magicsquares.


The Fibonacci The-"S ecialquence is 1,1,2,3,5,8,13,... IT e Specialin which each term is the Magic Squaresum of the two termspreceding it. When theFibonacci numbers 3, 5,8, 13, 21, 34, 55, 89, 144 are matched up with counting numbers1, 2, 3, 4, 5, 6, 7, 8, 9, a new square is formed. It does not have thetraditional properties of a magic square, but the sum of theproducts of the 3 rows (9078+9240+9360 =27,678) equals thesum of the products of the 3 columns (9256+9072+9350=27,678).

8 1 6

3 5 7

4 9 2


Chinese M athematics is universal.History shows its uses and

Tri angle discoveries are not unique to oneTra g region as illustrated by this

Chinese version of the Pascaltriangle. Although Pascal made some significant discoveries inhis triangle of numbers, it appeared in a book printed in around1303, three-hundred and two years before Pascal was born!1

11 1

1 2 11 3 3 1

1 4 6 4 1

lSee section on Pascal's triangle.


A rchimedes of Syra-cuse (287 B.C. to 212 B.C.)was a leading mathemati-cian of the HellenisticAge.

| The Death ofArchimedes

I >I,)"#r 1.

During the Second Punic War, Syracuse was besieged by theRomans from 214 B.C. to 212 B.C. At this time Archimedesinvented ingenious defense weapons - catapults, pulleys andhooks to raise and smash Roman ships, parabolic mirrors to setfire to ships - which were responsible for holding the Romansback for nearly three years. Syracuse eventually fell to the Ro-mans, however. Marcus Claudius Marcellus, commander, gaveexpress orders that Archimedes was not to be harmed. A Romansoldier entered the home of Archimedes and found him workingon a math problem, totally oblivious to his presence. The soldierordered him to stop, but Archimedes did not pay any attention.Angered, the soldier killed Archimedes with his sword.


A Non-Euclidean The 19th centu-ry was a period

W world of revolutionaryideas in politics,art, and science,

as well as inmathematics with the development of its non-Euclidean geom-etries. Their discoveries marked the beginning of modernmathematics in the same way impressionistic painting markedthe beginning of modem art.

During this period, hyperbolic geometry (one of thenon-Euclidean geometries) was discovered independently byRussian mathematician Nicolai Lobachevsky (1793-1856) andHungarian mathematician Johann Bolyai (1802-1860).

An abstract design of Poincar6 hyperbolic geometric model.

We find that hyperbolic geometry, like other non-Euclideangeometries, describes properties that are alien, since we areconditioned to think of geometry in Euclidean terms. For


example, in hyperbolic geometry, line does not imply straightand parallel lines do not remain equidistant though they do notintersect because they are asymptotic. When non-Euclideangeometries are studied in detail one finds that they may indeedgive a more accurate description of phenomena in our universe.As a result, different worlds have been described in which thesegeometries could exist.

One such world is the model created by the French mathema-tician Henri Poincare (1854 - 1912). His imaginary universe isbounded by a circle (visualize a sphere for a 3-dimensionalmodel) whose temperature at the center is absolute 0. As onetravels from the center the temperature around rises. Assumethe objects and the inhabitants in this universe are unaware ofchanges in temperature, but the size of everything changes as itmoves about. In fact, every object and living thing enlarges as itapproaches the center and shrinks proportionally as itapproaches the boundary. Since everything changes size, onewould be unawareand unable to de-tect size changes.This means one'ssteps would be-come smaller asone moves towardthe boundary, andindeed, it wouldappear that one isno nearer theboundary than be-fore approachingit. This phenome-non makes thisworld appear infi-nite, and here the shortest distance between two points is acurved line because to get from A to B it would take less steps


(since they are larger) if we move toward the center in an arcshape. Here is a world in which a triangle's sides would bemade up of arcs, as triangle ABC in the diagram. And evenparallel lines have a new look. Line DCE is parallel to line ABsince they would never intersect.

Actually Poincar6's universe may describe the world in whichwe live. If we look at our position in the universe and if we wereable to travel distances measured in light years, then perhaps wewould discover changes in our physical size. In fact in Einstein'stheory of relativity the length of a ruler shortens as it approachesthe speed of light!

Poincar6 was an original thinker.The diversity of subjects he lec-tured on while a professor at theSorbonne in Paris (from 1881-1912) illustrate this point. Hiswork and his ideas cover suchsubjects as electricity, potentialtheory, hydrodynamics, ther-modynamics, probabilty, celes-tial mechanics, divergent series,asymptotic expansions, integralinvariants, stability of orbits,shapes of celestial bodies, andthe list goes on. His work can

definitely be said to have stimulated mathematical thought ofthe twentieth century.

Square numbers, Cannopyramidal numbersand their summationscan be used in deter-mining the number ofcannon balls in a square based pyramid.

odd numbersI

square numbers

pyramidal numbers

lp.4

3 5 7 9iee 0eese esseegs eseseesse

1 4 9 16 25=1+3 =4+5 =9+7 =16+9

* SO 005 0eeS *

seesee 500000 00 sees

1 5 14 30=1÷4 =5+9 =14÷16

55=30+25

41N ~,

Study the patterns of these numbers.How many cannon balls are there in the diagram below?


n Balls &Pyramids

.


Conch oid of Very often, searching for. solutions to certain mathe-N icom edes matical problems gives rise

to new concepts and dis-coveries. The three famous construction problems of antiquity- trisecting an angle (dividing an angle into three congruentangles), duplicating a cube (constructing a cube with twice thevolume of a given cube), and squaring a circle (constructing asquare with the same area as a given circle) - stimulatedmathematical thought, and as a result many ideas werediscovered in an attempt to solve these. Although it has beenshown that these three ancient problems are impossible toconstruct using only a straightedge and compass, they were solvedusing other means - one of which was the conchoid.

The conchoid is one of the curves of antiquity which wasinvented by Nicomedes (circa 200 B.C.) and used to duplicate acube and trisect an angle.

L

To form a conchoid begin with a line L and point P. Then draw raysthrough P which intersect L. Mark off a fixed distance, a, on each ofthese rays. The locus of these points form the conchoid. The curvatureof the conchoid depends on the relationship between a and b, i.e. a=b,a<b, or a>b. The polar equation of the conchoid is r=a+bsece .


To trisect <P, make <P one of the acute angles of right AQPR.Draw a conchoid with 3,P and with QR as its fixed line, L. Forthe fixed distance from QR use 2h, twice the hypotenuse I PR I.At R construct RS I QR intersecting the conchoid at S. Now<QPT can be shown to be one-third <QPR.

Q

S

R

h

proof

Locate M, the midpoint of TS, then I RM I =h, since ASRT is aright triangle and the midpoint of the hypotenuse is equidistantfrom its vertices.

Now m<1=m<2=kQ because I MS I = I MR I =h. And m<3=2kQ,since <3 is an exterior angle for ASMR. Also m<3=m<4=2kQbecause I MR I = I PR I =h.

m<5=m<2=k2, since PQ I I RS, because segments PQ and SR arecoplanar and perpendicular to line QR.

Thus, m<QPR=3kQ and 1/3(m<QPR)=k2 =m<5.

Therefore, <QPR is trisected.

.


Tie Trefoil Tieing knots has been aroutine process for most

K not of us from the time wemastered tieing our shoe-

laces. Of course tieing knots can be an art, especially when onesees a sailor rigging a boat. But the subject of knots is also amathematical idea in the field of topology. Knots form a relative-ly new area. The most important idea that has been provenabout them thus far is that a knot cannot exist in more than threedimensions.

Making a trefoil knotTo form the trefoil knot, illustrated below, take a long strip ofpaper and give it 3 half twists. Now join its ends together withtape. Using a pair of scissors cut midway between its edgesalong the entire strip. You will end up with one band with atrefoil knot in it.


Benjamin Franklin's magic ihe Iagicsquare features a variety ofnumerical oddities in addition Square ofto the regular properties of amagic square.1 Each row totals Benjamin260. Halfway totals 130. Ashaded diagonal up four num-Franklin

bers and down four totals 260.The sum of any four numbers equidistant from the center is 130.The four corners and the four center numbers total 260. The sumof the four numbers forming a little square (2x2) is 130.

52 el 4 13 229 36 45..... .... . ... .-

14 $ 62 51 46 3, 3G0 19

560 5 i 12 21.28.. 3 4.

117 i6 i59 54 4330 27220

2558 710 2:2 3942,.I--- i - -

9 8 7564 4 2524.

5 0.6a3 ; ;2 i:1 5 18 31 3eWi4 7.1 . .. -33-

161 844.9"48 33,~l

1 See section on Magic Squares.

Irrational Numbers&the PythagoreanTheorem

Examples are:Xi'-, I~; T5i 7c; -48; e; 235; A;(p

When one attempts to write an irrational number as a decimal itturns out to be a never ending non-repeating decimal.

examples:

nJ2 -1.41421356. .

q235 15.3297097...

7i 3.141592653.

e 2.71828182..

p -1.61803398... -the golden ratio

For thousands of years mathematicians have been devisingmethods to obtain more accurate decimal approximations ofirrational numbers. With the use of high powered computers

and infinite number series, their approximations can be carriedout to any desired decimal accuracy. Considering the time andeffort that has been expended in devising these methods, it is

amazing that the exact location for many irrational numbers canbe found using the marvelous Pythagorean theorem. AncientGreek mathematicians had proven the Pythagorean theorem1 ,

and had used it in constructing exact irrational numberlengths.


Irrationalnumbers arenumbers thatcannot be

expressed byan ending or

a repeating

1See section on Pythagorean Theorem. Note that 7c and e cannot beconstructed using a straight edge and compass because besides beingirrational numbers they are also transcendental numbers.


To locate the 42, , 4- 4i A F7 48, . . . on the number line,construct right triangles with hypotenuses these lengths. Thenuse a compass to measure and to arc its location on the numberline as illustrated.

0

As the diagramillustrates, to constructVCi one can use thelength for V51 and 1, ordevise another construc-tion using other lengths,such as 7 and vA7.

N'5_1

,T5-2

1ao THEJOY OF MATH-EMATICS

Prim' e A number is prime if it is acounting number greater thanNlum bers one whose only factors are itselfand one.

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1067416061773610046717601660 06667166 .....636l166 I260 1201 I'll,520371 5555 1:.019502633062,936656319

56013016233631 666611 5111 .... 667 1163670651611662606677561 3411619216520.6.66973.01900206195656669 I 27 12 0595136

At9 I~.3 onOtoe0, .7 the abov number~101 was discoveretob h ags nwnpienme so tha date..... , After 061800 ,Ihour of ompuer tme, aur Nicke and Curt~ Nol(ayad

CA, high.schoo stdets fon. teprm.nmer..7¶Contnuig OidepedenlyCur Nol isoerd lrerprm

nubr,....1afwmots ae....n May of 1979 Harry11.2 .Neso.o.heLiemoe a dscvre apim nalytwc the1 II 6

size of11 Noll-, naelyI

Alhog.cmutr.hv .be programmed.tday tolocatprime numbers, the Greek mathematician 2.3.Eratosthenes (121575-.600.

19 ..)ivntdtIstecniqu of. a1 numerical sieve to discover1,:1


the prime numbers smaller than some given number. Thediagram has the prime numbers circled that are less than 100.

The Sieve of Eratosthenes

procedure:

1) 1 is crossed out since it is not classified as prime.

2) Circle 2, the smallest positive even prime. Now cross out every 2nd

number, i.e. all multiples of 2.

3) Circle 3, the next prime. Now cross out every 3rd number, i.e. all

multiples of 3. Some may already be crossed out since they are also

multiples of 2.

4) Circle the next open number, namely 5. Now cross out every 5th

number.

5) Continue the process until all numbers up through 100 are either

circled or crossed out.

® t. @it M >wi'

(i) X X 34 t '3 (i 3 a

>~(~ XK (i7+W 5} )0.

()5(ij)t At8(!

X ,> '94 VJ486 '8 ) '9Q

'fl .b W e HiXb


The Golden The golden rectangle isa very beautiful and excit-

Rectangle ing mathematical object,which extends beyond

the mathematical realm. Found in art, architecture, nature, andeven advertising, its popularity is not an accident. Psychologicaltests have shown the golden rectangle to be one of the rectanglesmost pleasing to the human eye.

Ancient Greek architects of the 5th century B.C. were aware ofits harmonious influence. The Parthenon is an example of theearly architectural use of the golden rectangle. The ancientGreeks had knowledge of the golden mean, how to construct it,how to approxi-mate it, and howto use it to con-struct the goldenrectangle. Thegolden mean, ,0(phi), was not co-incidentally thefirst three lettersof Phidias, the fa-mous Greeksculptor. Phidiaswas believed to The Partenon in Athens, Greece.

have used thegolden mean and the golden rectangle in his works. The societyof Pythagoreans may have chosen the pentagram as a symbol oftheir order because of its relation to the golden mean.

Besides influencing architecture, the golden rectangle alsoappears in art. In the 1509 treatise De Divina Proportione by LucaPacioli, Leonardo da Vinci illustrated the golden mean in themake up of the human body. The use of the golden mean in arthas come to be labeled as the technique of dynamic symmetry.


Albrecht Diirer, George Seurat, Pietter Mondrian, Leonardo daVinci, Salvador Dali, George Bellows all used the golden rectan-gle in some of their works to create dynamic symmetry.

Bathers (1859-1891) by French impressionist George Seurat. There are three goldenrectangles shown.

When the geometric mean is located on a given segment, AC, the

golden mean1 is formed,so that

(I AC I / I AB I=I AB I / I BC I),then I AB I is the golden mean,also known as the golden section,the golden ratio, or the golden proportion.

A B C

1To determine the value of the golden ratio, one must solve theequation (1/x) = (x/(1-x)), where x= I AB I, I AC I =1, and I BC I =(1-x).The golden ratio, I AC I/ IAB I or I AB I / I BC I comes out to be -[(1+ 3)/21-1 .6.


Once a segment has been divided into a golden mean, the goldenrectangle can easily be constructed as follows:

D 'E 'F

lIj

I 1 I

A L-- -- -- - -- J- - - - CB

1) Given any segment AC, with B dividing the segmentinto a golden mean, construct square ABED.

2) Construct CF perpendicular to AC.

3) Extend ray DE so that line DE intersects line CF atpoint F. Then ADFC is a golden rectangle.

A golden rectangle can also be constructed without alreadyhaving the golden mean, as follows:

D M C

A FN B

1) Construct any square, ABCD.

2) Bisect the square with segment MN

3) Using a compass, make arc EC using center N andradius I CN I.

4) Extend ray AB until it intersects the arc at point E.

5) Extend ray DC.

6) Construct segment EF perpendicular to segment AE,and ray DC intersects ray EF at point F. Then ADFE isa golden rectangle.

II III ,1I1:

I1.I N


The golden rectangle is also self-generating. Starting with goldenrectangle ABCD below, golden rectangle ECDF is easily made bydrawing square ABEF. Then golden rectangle DGHF is easilyformed by drawing square ECGH. This process can be continuedindefinitely.

B E c B E C B E G

4 ~G G

A F D A F D A F D

Using the final product of these infinitely many golden rectan-gles nestled in one another the equiangular spiral (also called thelogarithmic spiral) can be made. Using a compass and the squaresof these golden rectangles, make arcs which are quarter circles ofthese squares. These arcs form the equiangular spiral.

NOTE:The golden rectangle continually gener-ates other golden rectangles and thusoutlines the equiangular spiral. The in-tersection of the diagonals pictured isthe pole or center of the spiral.

O is the center of the spiral.

A radius of the spiral is a segment withendpoints the center 0 and any pointof the spiral.

Notice that each tangent to the point ofthe spiral forms an angle with thatpoint's radius, e.g. TiPF O. The spiral isan equiangular spiral if all such anglesare congruent.

This is also called a logarithmic spiralbecause it increases at a geometric rate,.i.e. a power of some number and apower or exponent is another name for K

logarithm.

The equiangular spiral is the only type of spiral that does not alter itsshape as it grows.


Nature has many forms of packaging - squares, hexagons, cir-cles, triangles. The golden rectangle and the equiangular spiralare two of the most aesthetically pleasing forms. Evidence of theequiangular spiral and the golden rectangle are found in starfish,shells, ammonites, the chambered nautilus, seedhead arrange-ment, pine cones, pineapples, and even the shape of an egg.

Equally exciting is how the golden ratio is linked to the Fibonac-ci sequence. The limit of the sequence of ratios of consecutiveterms of the Fibonacci sequence - ( 1, 1, 2, 3, 5, 8, 13,....[Fn i+Fn 2],. .) - is the golden mean, cp.

1 2 3 5 8 13 21 Fn+11/' 1 ' 2 ' 3 ' 5 ' 8 ' 13 F n

1, 2, 1.5, L 6,1.625,L 6153,1.619,

Up = V - 1.62

Besides appearing in art, architecture and nature, the goldenrectangle is even used today in advertising and merchandising.Many containers are shaped as golden rectangles to possiblyappeal to the public's aesthetic point of view. In fact, thestandard credit card is nearly a golden rectangle.

Yet the golden rectangle interrelates with other mathematicalideas. Some of these are: infinite series, algebra, an inscribedregular decagon, Platonic solids, equiangular and logarithmicspirals, limits, the golden triangle, and the pentagram.


In a broad sense flexagons can beconsidered a type of topologicalpuzzle. They are figures made

from a sheet of paper, but end up

having a varying number of faceswhich are brought to view by a--- - - - - -- -- --- - - - . -

series of flexings.

The object below is a called a tri-tetra flexagon. Tri stands for

the number of faces and tetra for the number of sides of the

object. F FONT

I I

fold here

BACK

IL.

2

2

3

1

step 1.

33F~.Nfold here

a #3 isconcealed here

4 a #I'

concealed

s2 2 2 herestep 3. step 2.

2

2

LBJWNow on the front all 2's are shown and on theback all Vs.To make the 3's appear flex along the verticalcrease.

Making aT- *TetraFlexagon

2

2

108 THE IOY OF MATHEMATICS

Finding infinity Can you imagine.1 what infinity is? In-

in sm all places finity is a never end-ing amount. The con-cept of infinity is

hard to grasp. While we can easily grasp that the number 7 candescribe 7 apples, and the number 1 billion (written1,000,000,000) can describe the number of grains of sand in a jar.But an infinite amount is endless. A very exciting way to get aphysical feeling of infinity is to hold up a mirror directly in frontof another larger one. What happens is that you will see amirror inside a mirror inside a mirror inside a mirror.... neverending.

Many people think that an infinite amount must take up a greatdeal of space, but on this small line segment AB, A B,there are an infinite number of points.

To prove this, we use the idea that between any two points anotherpoint can be found. So if point A and point B are on a line segment,then point C can be found between them. Then between pair A and Canother point can be found, as well as between the pair C and B another


point can be found. This process of finding another point between anytwo continues forever, so there are an infinite number of points onsegment AB.

Another way to describe an infinite amount is by using the fleastory.

Half-flea wants to jump across the room. His friend tells him that hewould never reach the other side if he promised to always jump inleaps that are only 1/2 the remaining distance. Half-flea says he wouldhave no trouble reaching the other side. He first jumps 1/2 way across,then 1/2 way across the remaining distance, then 1/2 way across thenext distance, and so on. Even though he is so very close to the otherside he has to follow the rule: each jump must be 1/2 the remain-ing distance. Half- flea finally realizes that there will always be aremaining distance he would have to jump 1/2 way, and this would goon forever unless he gives up.

Thus, even though infinity is an endless amount that cannot beidentified by a number, we have discovered that it can fit in avery small space as well as a very large space.

110 THE IQY OF MATHEMATICS

The FivePlatonicSolids

Platonic solids are convex solidswhose edges form congruent regularplane polygons. Only five such solidsexist.

Tho -nwrrdcnl; ,-ns- .nv-TV VJ.L OVIIU 'At lasy

3-dimensional object, such as a rock, a bean, a sphere, a pyra-mid, a box, a cube. There is a very special group of solids calledregular solids that were discovered in ancient times by the Greekphilosopher, Plato. A solid is regular if each ofits faces is the same size and shape. So a cube isa regular solid because all its faces are the samesize squares, but this box, on the right, is not aregular solid because its faces are not all thesame size rectangles. Plato proved that there were onlyfive possible regular convex solids. These are the tetrahedron, thecube or hexahedron, the octahedron, the dodecahedron, and theicosahedron.

exahedron or cube

octahedron

dodecahedron

-


Here are patterns for making all five regular solids. Why notcopy them, cut them out and try to fold them into their3-dimensional forms?

Ktetrahedron

'lihexahedron or cube

octahedron

icosahedron

dodecahedron

-


1

ifo

lhe Pyramidmethod9r makingiagic squares

Procedure:1) Sequentially place the numbers 1 to 25 of the magic squarealong diagonal boxes, as shown;2) Relocate any number landing outside the magic square inan imaginary square in their respective positions in the magicsquare (hollowed numbers were relocated).

3 16 9 22 15

220 8 21 14 2 20H

1 725 13 11 19 25- -2 - - --- -[

6 24112 5 18 6 2: - E- -7S -- - -11 17 10 23

15 22

The pyramid methodis one of the methodsused for constructingodd order magicsquares. The examplebelow illustrates how tomake a 5x5 magicsquare.


Although Plato is at- The Kepler -tributed with the discov- Seryof the five Platonic Poinsot Solidssolids (tetrahedron, hex-ahedron or the cube, octahedron, dodecahedron, icosahedron)and Archimedes is attributed with the Archimedean solids, thesefour non-convex solids were unknown to the ancient world.Kepler discovered his two in the early 1600's and Louis Poinsot(1777-1859) rediscovered these two and discovered two more in1809. Their shapes are often used today as light fixtures andlamp shades.

small stellated dodecahedron great stellated dodecahedron

great dodecahedron great icosahedron


The False This diagram ap-' pears to be a spiral,Spiral Optical but closer examina-

.1 tion shows it is com-IL lusi on posed of concentric

circles. The unit of di-rection was discovered by Dr. James Fraser, and it was firstdescribed in the British Journal of Psychology in January 1908. It isalso often called the twisted cord effect. Two cords of contrastingcolors are twisted to form a single cord. It is then superimposedon different backgrounds. The illusion created is so convincingthat even tracing the concentric circles to destroy the illusiion ofa spiral or a helix is a difficult task.


The golden rectan- The icosahedrongle appears in somany facets of our & the goldenlives - architecture,

art, nature and sci- rectangleences, as well asmathematics. Luca Paoioli's book, De Divina Proportione (illus-trated by Leonardo da Vinci in 1509), presents fascinating exam-ples of the golden mean in plane and solid geometry. Thedrawing below is such an example. Here, three golden rectanglesintersect each other symmetrically and each perpendicularly tothe other two. The corners of these rectangles coincide with thetwelve corners of a regular icosahedron.


Zeno s Paradox Paradoxes are inter-esting, entertaining-A chiles & and a very importantpart of mathematics.

the tortoiseThey emphasize howimportant it is to state

and prove ideas carefully so there are no loopholes. In mathe-matics, we try to make mathematical ideas cover as many facetsas possible, i.e. we try to generalize a concept and thereby makeit apply to more objects. It is important to generalize, but it canbe dangerous. One must proceed cautiously. Some paradoxesillustrate this danger.

- - w

__1 AW


In the 5th century B.C., Zeno, using his knowledge of infinity,sequences and partial sums, developed this famous paradox. Heproposed that in a race with Achilles, the tortoise be given ahead start of 1000 meters. Assume Achilles could run 10 timesfaster than the tortoise. When the race started and Achilles hadgone 1000 meters, the tortoise would still be 100 meters ahead.When Achilles had gone the next 100 meters the tortoise wouldbe 10 meters ahead.

Zeno argued that Achilles would continually gain on thetortoise, but he would never reach him. Was his reasoningcorrect? If Achilles were to pass the tortoise, at what point of therace would it be?

See the appendix for the solution to Achilles & the tortoise.

Eublides' & Zeno 's paradoxesThe Greek philosopher Eublides argued that one could neverhave a pile of sand. He proposed that certainly one grain of sanddoes not constitute a pile of sand. And if one adds another grainof sand to the one, they do not make a pile. He said if you do nothave a pile of sand and if by adding one to what you have youstill do not have a pile, then you will never have a pile of sand.

Along the same line of reasoning, Zeno looked at the points on aline segment. He argued if a point has no size, then addinganother point to it will still have no size. Thus one will never getan object of any size by joining points. But he further argued thatif a point does have size then a line segment must be infinitelylong because it is composed of an infinite number of points.


(1623-1662) when hemystic hexagram.

was

The MysticHexagram

I

3

z

I

If a hexagon is inscribed in a conic, then the intersectionof the three pairs of opposite sides are collinear.

- Mathematics is an unend-ing treasure of intriguingideas. This particular theorem

- was proven by the Frenchmathematician Blaise Pascal

een years old. He named it the

Turn the triangle of penniesupside down by sliding onepenny at a time to a new loca-tion so that it touches two otherpennies.


The PennyPuzzle

The minimum number of moves needed is three.


Tessellations Tessellations of a planesimply means being ableto cover the plane with flat

tiles so that no gaps and no tiles overlap. Given certain shapesone can use mathematics to decide ahead of time whether itwould be possible without actually laying out the tiles. To figurethis out one must know the mathematical fact that a circle has360.

Equipped with this tool and some geometry let's considercovering (tessellating) a floor with regular pentagons. A regularpentagon has five congruent sides and five congruent angles. Tofigure out the measure of the angles of a pentagon, divide thepentagon into triangles, as illustrated. For any triangle, its threeangles total 180. All of the five triangles composing thepentagon are congruent, since their corresponding sides andangles are congruent. Wecan now determine themeasure of the pentagon'sangles to be 108Q. Thuswhen we try to put identicalregular pentagons side byside we find we will have agap because the pentagonscannot complete a circle or360Q (1089 + 1089 + 1089=324").


Now let's try to tessellate a floor using equilateral triangles.Each angle in a equilateral triangle is 60Q. We see we can lay sixidentical equilateral triangles together, and they will completethe circle.

What about using squares, hexagons, octagons, or a combina-tion? Here are some tessellations of planes.

.. . W.. -.-.


In a similar way, space can be tessellated - tiled with three-dimensional solids. Pictured below are truncated octahedra.They are the only Archimedean polyhedra that can be used to fillspace while leaving no gaps and using no other shaped solids.

The renowned Dutch artist M.C. Escher used many mathemati-cal concepts in his work - the Moebius strip, geodesics,projective geometry, optical illusions, the tribar, trefoil and tes-sellations - to name some. A number of his famous works useexciting tessellations which he created, for example, Metamorpho-sis, Horseman, Smaller and Smaller, Square Limit, Circle Limit. Inaddition to art, the study and applications of tessellating space isof special interest to the field of architecture, interior design,commercial packaging.


Diophantus is often called D iophantus'the father of algebra. Little isknown about his life except Riddlethat he lived between 100 to400 A.D. However, his age at death is known because one of hisadmirers described his life in an algebraic riddle.

Diophantus' youth lasted 1/6 of his life. He grew a beardafter 1/12 more of his life. After 1/7 more of his life,Diophantus married. Five years later he had a son. The sonlived exactly 1/2 as long as his father, and Diophantus diedjust four years after his son's death. All of this totals theyears Diophantus lived.

See appendix for solution to Diophantus'Riddle.

T he KonigsbergBridge Problem&Topology

K6nigsbergl is a city on the Preger River that contains twoislands and is joined by seven bridges. The river flows aroundthe two islands of the town. The bridges run from the banks of

4

Diagram of the Kbnigsberg Bridge Problem.

the river to the two islands in the river with a bridge connectingthe islands. It became a town tradition to take a Sunday walk,and try to cross each of the seven bridges only once. No one hadsolved the problem until it came to the attention of the Swissmathematician Leonhard Euler (1707-1783). At that time, Euler


Topology originat-ed with the solutionin 1736 of a famousproblem -

the Konigsberg BridgeProblem.

-- I -- S-�--- I

WgQg-.--


was serving the Russian empress Catherine the Great in St.Petersburg. In the process of solving this problem, Eulerinvented the branch of mathematics known as topology. Hesolved the Kdnigsberg Bridge Problem by using an area oftopology today called networks. Using networks, he proved thatthe problem of crossing each bridge of Konigsberg only oncewas not possible.

This problem and Euler'ssolution launched the studyof topology. Topology is arelatively new field. Themathematicians of the 19thcentury began delving intotopology along with theirstudies of other non-Euclid-ean geometries. The first trea-tise of topology was writtenin 1847.

1 In the 18th century Konigsberg was a German city. Today it isRussian.


Networks A network is basically a diagramof a problem. The network for theKonigsberg Bridge Problem isillustrated below.

C

A

D DThe network for the Konisgberg Bridge Problem.

A network consists of vertices and arcs. A network is traveled ortraced by passing through all the arcs exactly once. A vertex maybe crossed any number of times. The above diagram shows thevertices for the Kbnigsberg Bridge Problem as A, B, C, D. Notethe number of arcs passing through each vertex-A has 3, B has5, C has 3, and D has 3. Since these are all odd numbers, thesevertices are called odd vertices. An even vertex would have aneven number of arcs passing through it. Euler discovered manyproperties about the number of odd and even vertices a networkcould have and still be traceable. Specifically, Euler noted thatfor a vertex to be odd, one would have to begin or end thejourney at that vertex. With this in mind, he reasoned that since anetwork can have only one beginning and only one ending, thata traceable network could have only two odd vertices. Thus,since in the K6nigsberg Bridge Problem there are four oddvertices, it cannot be traced.


Which of the above networks is transversible (can be traced overwithout doubling back)?

H~I~rCan you find the path that passes through each door only once withoutlifting your pencil? Try to prove your solution by drawing a network.


A ztec One of the earliest and mostimportant calculating devices was

Calendars the calendar - a system ofmeasuring and recording the

passage of time. Realizing nature furnishes a regular sequenceof seasons which control food supplies, early people tried tocorrelate the solar day, the solar year and the lunar month. Sincethe lunar month is about 29.5 days while the solar year is 365days 5 hours and 48 minutes and 46 seconds, it was not possibleto get a whole number quantity of lunar months for a solar year.This is the major problem in developing consistent calendars.Even our present calendar is not consistent because everycentury year that is not divisible by 400 (e.g. 1700, 1800, 1900)must lose its extra leap day even though it is a leap year.

The Aztecs had two calendars - the religious calendar bore norelationship to the lunar month and the solar year. This calendarwas important for ceremonial purposes, and Aztecs wouldinclude their birth dates as part of their names. The calendarconsisted of twenty signs and thirteen numbers that followed ina fixed cycle of 260 days. Their second calendar wasagriculturally oriented and contained 365 days.1 Cyclicalmovements of heavenly bodies, enabled the Aztecs to regulatetheir calendar and accurately predict events such as eclipses.

1 The Aztec borrowed many elements from the Toltec and Mayacivilizations, including many parts of their calendar.


In 1790, the Aztec sunstone or stone calendar was discovered whilerepair work was being done on a cathedral in Mexico City. Thecathedral had been built on the site of an ancient Tenochtitlanpyramid-temple. The circular disc is 12 feet in diameter and weighs 26tons. It recorded the history of the world according to the Azteccosmology.

At the center, the sungod (Tonatiuh) is sculptured. Around thesungod the four suns or cosmogonic worlds ( Tiger, Water, Wind andRain of Fire) appear indicating the time before the Aztecs. Here thesymbols of movement also appear. Another band of twenty glyphs -alligator, wind, house, lizard, serpent, death, deer, rabbit, water, dog,monkey, grass, reed, jaguar, eagle , vulture, earthquake, flint, rain,flower-represent the twenty days of the Aztec month.


The The beauty of a mathemati-cal problem does not lie in its

Im possible answer but in the methods ofIi its solution. There exist prob-

Trio lems in which the solution isfinally determined to be no

solution. Somehow no solution seems like a frustrating answer,but the thought process used to arrive at this conclusion is oftenmost fascinating, and exciting discoveries of new ideas are madeen route. So it was with the threefamous problems of antiquity.

trisecting an angle -dividing an angle into threecongruent angles

duplicating a cube - constructing a cube with twice thevolume of a given cube

squaring a circle -constructing a square with the samearea as a given circle

These problems stimulated mathematical thought anddiscoveries for over 2000 years until it was determined in the19th century that the three construction problems were not

posibl ii-ina nnlv a sfrai-chfPdcrP and,a straight- rl-v-v- T --- a Odoes not have compass. It was deduced that anarks on it, straightedge can be used to construct-~der has.

line segments whose equations arelinear (first degree equations), e.g.

.- ) A A __ Lu- shUe .n L-..AhyJA`-'. P1 Lucopascs~, oln LIt: uLone IMdILU,

can construct circles and arcs whoseequations are second degree, e.g. x2 + y2=25. When theseequations are solved simultaneously by using linear combinationsthey produce at most second degree equations. But the equationsthat are derived in solving the three construction problems byalgebraic means are not first or second degree equations butrather cubic (a 3rd degree equation) or involve transcendentalnumbers. Therefore, a compass and a straightedge alone cannotbe used to derive these types of equations or numbers.


Trisecting an angleCertain specific angles such as 135Q or 90Q can be trisected usingonly a compass and straightedge. But given any angle, it isimpossible to trisect it with only the use of a straightedge andcompass because the equation used to solve this problem can beshown to be a cubic in the form a3 - 3a -2b = 0 .

Duplicating a cubeIn an attempt to duplicate a cube - that is double its volume -one might try to double the length of its sides. But this actuallycreates a cube whose volume is 8 times the given cube.

The volume of the cubethat is to be duplicated = a3

a

To duplicate this cube we want a cube whose volume is doubleor 2a3.

q j-x' = 2a' which meansx = aV2

Again we end up with a cubic wecannot construct with only acompass and straightedge.

x

x

x

a


Squaring a circleGiven a circle with radius r, its area is nr2.So we want to construct a square whose area is irr2.x2 =7tr2, which means x =r'47. Since 7t is a transcendentalnumber it cannot be expressed in a finite number of rationaloperations and real roots, thus the circle cannot be squared usingonly a compass and straightedge.

x

x x

x

Although we see these three problems of antiquity are impossi-ble to construct with only a compass and straightedge, ingeniousmethods and devises were created to solve them. And equallyimportant, these problems stimulated the evolution ofmathematical thought over the centuries. The conchoid ofNicomedes, the spiral of Archimedes, the quadratrix of Hippias,conic sections, cubic and quartic curves and severaltranscendental curves are some of the ideas which sprang fromthese three problems of antiquity.


A 3 by 3 magic square Ancientis at the center of this an-cient Tibetan seal, another Tibetanexample illustrating thatmathematical ideas are not Magic Squarerestricted to countries orby borders. The numeralsappearing in the magic square are:

4 9 23 5 78 1 6


Perimeter, The diagram belowrepresents an infinite

A rea & the number of triangles.*.Each inscribed triangle is

Infinite Series formed from the mid-points of the sides of the

triangle that circumscribes it. To determine the sum of the pe-rimeters of these triangles, we must look at the following series:

'U,,

1/2 + 1/4 +1/8 +1/16 +1/32 +1/64 +1/128 +...

.L r- < All


By looking at a number line one can determine the sum of thesefractions.

1/2 1/4 1/8

0 1/2 1

We notice that each successive fraction added to this seriesbrings the sum closer and closer to 1, yet it will never go beyond1. We conclude that this series has as its sum 1.

Now you may wonder how this information will help usdetermine the sum of the triangles' perimeters. First let's list theperimeters of each of the triangles:

30,15, 15/2, 15/4, 15/8, 15/32, 15/64, 15/128,...'

Now we sum this series to determine the sum of the perimetersof the triangles.

30+15 +15/2 +15/4 +15/8 +15/16 +15/32 +15/64 +15/128 +...

simplifying the above we get

45 +15(1/2 + 1/4 + 1/8 + 1/16 +1/32 +1/64 + 1/128 +...)

now replacing 1 for the value of the sum of the series we get

45 +15(1) = 45 +15 = 60 as the perimeter.

To determine the sum of the areas of the triangles is anotherchallenge. You may have to do some research on the sum of anew infinite series.

1 These values are determined using a theorem from geometry thatstates: A segment joining the midpoints of two sides of a triangle is half thelength of the side opposite it.

If two opposite cornersof a checkerboard are re-moved, can the checker-board be covered by domi-noes?

Assume that each domino is the size of two adjacent squares ofthe checkerboard. The dominoes cannot be placed on top of eachother and must lie flat.

7

I

I

II

See the appendix for the solution to the checkerboard problem.


TheCheckerboardProblem

K-K

I

I

L


Blaise Pascal (1623-1662) is Pascal'sknown as one of France's famousmathematicians and scientists. He Calculatoris credited with making many theo-rectical mathematical and scientific discoveries, such as thetheory of probability, the theory of liquids and hydraulicpressures. In addition, he invented this calculating machine atthe age of eighteen. With the machine, one could add longcolumns of figures. Pascal's invention helped introducefundamental principles that are the basis for modern calculators.


Isaac Newton Isaac Newton (1642-1727) was one of the in-& Ca ICU lusventors of calculus and ofthe theory of gravitation.

Although Newton was a mathematical genius, he devoted muchof his life to the study of theology. In 1665 the university he wasattending at Cambridge was closed because of bubonic plague.He stayed home during this time, and developed his form ofcalculus, formulated the theory of gravitation and worked onother physical questions. Unfortunately, 39 years elapsed beforehis work was published.

This is one of Newton 's sketches showingthe effects of gravity on an elliptical orbit.


It is important to always bear in Japanesemind that mathematics was devel-oping simultaneously in various Calculuscultures in all parts of the world. IFor example, Seki Kowa, a 17th cen-tury Japanese mathematician, is credited with the developmentof Japanese calculus. It was called yenri (circle principle). Theillustration below was drawn in 1670 by a pupil of Seki Kowa. Itmeasures the circle's area by summing up a series of rectangles.


The art of reasoning touchesmany aspects of our lives, be itdeciding what to eat, using amaDn hiuino' a gift or nrnuinic a

geometric theorem. All sorts of skills and techniques enter intoproblem solving. A single flaw in our reasoning can create somerather strange or ridiculous results. For example, if you are acomputer programmer, you dread overlooking a step that mightlead to an infinite loop. Which of us has been certain of ourexplanation, solution or proof only to discover a mistake? Inmathematics, dividing by zero is a common error which can causeoutlandish results as illustrated by the proof of 1=2. Can youfind where the error is?

1=2?I)2)3)4)5)6)7)8)9)

If a = b & b, a) 0, then 1 = 2.

Proofa, b > 0 given

a = b givenab = b x of= step 2ab - a' = b2 - a' - of step 3a(b - a) = (b + a) (b - a) factoring, step 4a = (b + a) ,' of =, step 5a = a + a substitution, steps 2 & 6a = 2a addition of like terms, step 71 = 2 f of =, step 8

See the appendix for the flaw in the proof of 1=2.


Patterns and symmetry The Symmetryof natural phenomenaabound. In 1912, physi- of Crystalscist Max Von Laue passedX-rays through a spheri-cal crystal onto a photographic plate. Dark points appeared thatwere arranged in perfect symmetry, which were later joined toform this design.


The M music and mathematicshave been linked from an-

M a them a tics cient times. During the0 .medieval period the educa-

of M usic tional curriculum groupedarithmetic, geometry, as-

tronomy and music together. Today's modern computers areperpetuating that tie.

Score writing is the first obvious area where mathematics revealsits influence on music. In the musical script, we find tempo (4:4time, 3:4 time, and so forth), beats per measure, whole notes, halfnotes, quarter notes, eighth notes, sixteenth notes, and so on.Writing music to fit x-number of notes per measure resemblesthe process of finding a common denominator-the differentlength notes must be made to fit a particular measure at a certaintempo. And yet the composer creates music that fits sobeautifully and effortlessly together in the rigid structure of thewritten score. When a completed work is analyzed, everymeasure has the prescribed number of beats using the variousdesired lengths of notes.

In addition to the obvious connectionof mathematics to the musical score,music is linked to ratios, exponentialcurves, periodic functions and comput-er science. With ratios, the Pythagore-ans (585-400 B.C.) were the first toassociate music and mathematics. Theydiscovered the connection betweenmusical harmony and whole numbersby recognizing that the sound caused

by a plucked string depended upon the length of the string.They also found that harmonious sounds were given off byequally taut strings whose lengths were in whole number ratios- in fact every harmonious combination of plucked strings


could be expressed as a ratio of whole numbers.the length of the string of whole numbers ratios,could be produced. For example.starting with a string that producesthe note C, then 16/15 of C's lengthgives B, 6/5 of C's length gives A,4/3 of C's gives G, 3/2 of C's gives F,8/5 of C's gives E, 16/9 of C's givesD, and 2/1 of C's length gives low C.

Have you every wondered why agrand piano is shaped the way it is?Actually there are many instrumentswhose shapes and structures arelinked to various mathematical con-

By increasingan entire scale

cepts. Exponential functions andcurves are such concepts. An exponential curve is one described

4

2I

by an equation of theform y=kX, where k>0.An example is y=2X.And its graph has theshape illustrated.

X1 2

Musical instruments that are either string instruments or formedfrom columns of air, reflect the shape of an exponential curve intheir structure.

The study of the nature of musical sounds reached its climaxwith the work of the 19th century mathematician John Fourier.He proved that all musical sounds - instrumental and vocal -

could be described by mathematical expressions, which were the


sums of simple periodic sine functions. Every sound has threequalities - pitch, loudness and quality - which distinguishes itfrom other musical sounds.

I I 11111111111111 IIIIIIIIIIIiIIIIII liii 111111

Exponential curves are outlined by the strings of this grand piano and the pipes ofthis organ.

Fourier's discovery makes it possible for these three properties ofa sound to be graphically represented and distinct. Pitch isrelated to the frequency of the curve, loudness to the amplitudeand quality to the shape of the periodic function.

Without an understanding of the mathematics of music,headway in using computers in musical composition and the


design of instruments would not have been possible.Mathematical discoveries, namely periodic functions, wereessential in the modem design of musical instruments and in thedesign of voice activated computers. Many instrumentmanufacturers compare the periodic sound graphs of theirproducts to ideal graphs for these instruments. The fidelity ofelectronic musical reproduction is also closely tied to periodicgraphs. Musicians and mathematicians will continue to playequally important roles in the production and reproduction ofmusic.

The diagram shows a string vibrating in sections and as a whole. The longestvibration determines the pitch and the smaller vibrations produce harmonics.


A palindrome is a word,verse, number, etc. thatreads the same backwardsas forward.

Examples are:(1) madam, I'm Adam(2) dad(3) 10,233,201(4) "Able was I ere I saw Elba"

An interesting number curiosity is the following:

Start with any whole number. Add to it the number formed by revers-ing its digits. To this sum add the number formed by reversing thesum's digits. Continue this process, and see if eventually you will endup with a numerical palindrome.

Will you always end up with a numerical palindrome?

1284+ 4821

6105+ 5016

11121+ 12111

23232 a numerical palindrome

NumericalPalindromes


A teacher announces thata test will be given on one ofthe five week days of nextweek, but tells the class,"You will not know whichday it is until you areinformed at 8a.m. of your1p.m. test that day.-

Why isn't the test going to be gi

See the appendix for the solution for the unexpected examparadox.

The Paradoxof the

UnexpectedExam

.


Babylonian The Babylonians proba-bly adopted clay tablets

Cuneiform and cuneiform (wedged-shaped) writing of the

Text Mesopotamians becausewriting material, such as

papyrus, was not available. Their number system was a base 60positional number system which used the two symbols, T forI and < for 10. '(4 = 60x10=600. Their clay tablet recordsshowed evidence of sophisticated computation which they wereable to carry out with their number system. This particularBabylonian problem and its solution were written during thereign of Hammurabi (1700's B.C.). The problem deals withlengths, widths and areas.

S spirals are forms whichappear in many facets ofnature, such as vines,shells, tornados, hurri-canes, pine cones, theMilky Way, whirlpools.


The Spiral ofArchimedes

The Archimedean spiral is a two dimensional spiral. One way toenvision it is to consider a bug crawling along a line whichpasses through the spiral's center, the pole. The bug crawls at auniform rate while the line uniformly rotates about the pole. Thepath traced by the bug will be an Archimedean spiral.


Evolution of Pour les astres en gin-.ral at pour les grand

M athem atical Cometes en particulier,trois mile ans ne sont

Ideas pas grand 'chose: dans lecalendrier de 1 'eternitMc'est moins qu'une sec-

onde. Mais pour 1'homme vous saves comme moi, mathematician lec-teurer, que trois mile ans c'est boucoup beaucoup!

-Flammaron, 1892

It is easy to lose perspective of the fact that mathematics is an ev-olution of ideas beginning with the earliest discoveries given byprehistoric people dividing up their food and discovering theconcept of number. Each contribution no matter how small isimportant to the development of mathematical thought. Somemathematicians have spent a lifetime on a single idea whileothers have varied their studies. For example, look at acondensed view of the development of Euclidean geometry.Geometric ideas were being discovered by a variety of peoplesduring ancient times. Thales (640 -546 B.C.) was considered thefirst to take a logical approach to geometric ideas. Othersfollowed over a period of about 300 years, discovering much ofthe geometry studied in high school. In about 300 B.C., Euclidcollected and organized the geometric ideas that had beencreated. It was an enormous task. He compiled all thisinformation into a mathematical system which has come to beknown as Euclidean geometry.. In his book, The Elements, hearranged the information so that it followed a logicalprogression. The Elements, written more than two thousandyears ago, is far from a perfect mathematical system whenscrutinized by today's mathematicians; but it remains aphenomenal piece of work.


)4)

Apollonius, inspired by Euclid's work, made historical contribu-tions to mathematics in the field of conics, astronomy andballistics. The diagram above illustrates one of his intriguingproblems:

Given three fixed circles,find a circle tangent to all three.

The eight solutions are illustrated.

The Four ColorMap Problem -Topology turns thetables on map coloring

only four differentcolors to differentiate the various countries. In 1976, the famousfour color map problem was set to rest with a computer proof byK. Appel and W. Haken of the University of Illinois, but theircomputer proof continues to be challenged.

The problem was to prove that every map drawn on aplane can be colored with only four colors so thatadjacent territories have different colors.

For an added twist, now consider map coloring on differenttopological models. Topologists study very unusually shapedsurfaces - doughnuts, pretzles, Moebius shaped surfaces, andto them a sphere can be distorted into a plane by puncturing a


It had alwaysbeen an unprovenrule among mapmakers that a mapdrawn on a flatsurface or on asphere required


hole in it and stretching and flattening it out. So in essence thenumber of colors required to color a plane and a sphere are thesame. Topology is the study of properties of objects that remainunchanged when the objects are distorted - stretched or shrunkas on a rubber sheet. What type of properties remain unchangedunder these conditions? Since distortion is allowed, we realizethat topology cannot deal with size, shape or rigid objects. Someof the characteristics that topologists look for are the position orlocation of points inside or outside a curve, the number ofsurfaces of an object, whether an object is a simple closed curve,the number of inside and outside regions it has. So with thesetopological objects, map coloring is an entirely new problemsince the four color map solution does not apply to them.

1 ry vanous map colon---on a strip of paper. Thentwist it into a Moebius strip(give it a half-twist and jointhe ends together). Will fourcolors always be enough?1K- 1:1- -. . XA1- :- .1Not likely. what is me Mcbius strip.minimum number of colorsyou can come up with that would work on any map. Try thesame on a torus ( which has a doughnut shape). The easiest wayto experiment with this object is to envision the formation of a

doughnut shape from a flatcodl~ll a5 nearr nt. (lnw Sn

on one side of the paper.Roll it into a cylinder. Thenimagine bending the ends ofthe cylinder together so asto form a doughnut. Canyou determine the mini-mum number of colors

torus needed for a torus?


A rt & There are many shapes innature that are symmetrical -

D ynam ic leaves, butterflies, the humanbody, snowflakes. Yet, there are

Sym m etrmany natural shapes that arenot symmetrical. Consider the

shape of an egg, of one wing of a butterfly, of the chamberednautilus, of the calico surfperch. These asymmetrical forms alsopossess a beautiful balance in their forms which has come to be

Composition with Yellow, 1936 by Mondrian. Mondrian is said to have approachedevery canvas in terms of the golden rectangle.


known as dynamic symmetry. The shape of the goldenrectangles or the proportion of the golden mean can be found inall shapes with dynamic symmetry.

The use of the golden mean and the golden rectangle in art is thetechnique of dynamic symmetry. Albrecht Durer, George Seurat,Pietter Mondrian, Leonardo da Vinci, Salvador Dali, and GeorgeBellows all used the golden rectangle in some of their works tocreate dynamic symmetry.

Illustrations show dynamic symmetry for the chambered nautilus, an egg, thewing of a butterfly, and the calico surfperch.

1See section The Golden Rectangle.


Transfinite How many elements wouldNe you say are in the sets below:

Numbers(a, b, c} ?(-1, 5, 6, 4, 1/2) ?( I

If you answered 3, 5, and 0 you are actually describing thecardinality of these sets.

Now, how many elements would you say are in the set,(1, 2,3,4,5, ... ) ?

If you answered an infinite amount, you are not specific enoughbecause there are various infinite sets. In fact, there is an infiniteset of infinite cardinal numbers called transfinite numbers.

As the name implies, transfinite (to cross over the finite) cardinalis a "number" that describes an infinite amount. No finitenumber can adequately describe an infinite set. Two sets can berepresented by the same cardinal number if the elements of oneset can be paired off with the elements of the other set so noelements are left over in either set.

for example,(a, b, c, d}| I I I have cardinality 4, i.e. 4 elements in each set.

(1,2,3,4)

set A=(1, 23 4 ' n...

set B=1 2, 22, 32, 4,52,. .. ,n 2,...}

Set A and set B have the same cardinality because the elementsof each set can be paired off as shown. Yet this seems


paradoxical, since set A has numbers which are not perfectsquares, but no elements are left out in the pairing off process.

19th century German mathematician George Cantor solved thisparadox by creating a new number system - one that deals withinfinite sets. He adopted the symbol, t (aleph - the first letterof the Hebrew alphabet), as the "number" of elements in infinitesets. Specifically, K 0 (aleph null), is the smallest of transfinitecardinals.

o 0 describes the number of elements in:

positive integers=(1, 2, 3, 4, 5,. . .,n,. .) nwhole numbers= (0, 1, 2, 3, 4, 5,.. .,n-1,. . representspositive integers= (+1,+2,+3,+4,+5,. . ..n. } a positivenegative integers= (-1,-2,-3,-4,-5,.. .,-n,. ... integerintegers= (. .. ,-3, -2, -1, 0, 1, 2, 3,. .rational numbers.

All of these sets and any others that can be paired off with thepositive integers are considered to have cardinality X 0

The examples below show methods for making one-to-onecorrespondences between the positive integers.

72,32 4,5. . .7 positive integers

(0, 1, 2,3,4,. . .,n-1,.. .} whole numbers

1, 2, 3, 4, 5, 6, 7, 8, 9....) positive\ \ \ \ \ \ \ integers

(1/1, 2/1, 1/2, 1/3, 2/2, 3/1, 4/1, 3/2, 2/3,...)

rational numbers


The chart below shows the order of how the rational numbersare placed in the preceding set.

1/1 1/2 1/3 1/4 1/5 ...

2/1 2/2'2/32/4V2/5 ...

3/1 3/2 3/3 3/4 3/5 ...

1 /2 /4/1 4/2 4/3 4/4 4/5 ...

5/1 5/2 5/3 5/4 5/5 ...

Cantor invented thismethod for arrangingthe rational numbersin some order so thatevery rational numberwould appear some-where in this ar-rangement.

Cantor also developed an entire arithmetic system for operatingwith these transfinite numbers,

0' O 1' 2' 3'- -. n,

He also proved that

x o< 1 < 2 < 3 <. . < K noc -

and that K 1 describes the cardinality of real numbers, pointson a line, points in a plane, and points in any portion of a higherdimension.


This logic problemdates back to eighthcentury writings.

A farmer needs to take his goat, wolf and cabbage across the river. Hisboat can only accommodate him and either his goat, wolf, or cabbage. Ifhe takes the wolf with him, the goat will eat the cabbage. If he takes thecabbage, the wolf will eat the goat. Only when the man is present arethe cabbage and goat safe from their respective predators.

How does he get everything across the river?

See appendix for the solution of the logic problem.

The snowflake curve 1 derivesits name from the snowflake-like shapes it assumes as it isgenerated. To generate a snow-flake curve begin with an equi-

lateral triangle, figure 1. Then trisect each side of the triangle.Now on each of the middle thirds make an equilateral trianglepointing outward, but deleting the base of each new triangle thatlies on the old triangle, figure 2. Continue this procedure witheach equilateral triangular point - trisecting the sides anddrawing new points, figure 3. Thus the snowflake curve isgenerated by repeating this process.

lSee section on Fractals for additional information.


TheSnowflakeCurve


figure 1

figure 3

The astonishing characteristic of the snowflake curve isthat its area is finite while its perimeter is infinite.

The perimeter is continually increasing without bound. Thiscurve can be drawn on a tiny piece of paper because its area isfinite, namely 1 3/5 times the area of the original triangle.


The number zero isindispensable to ournumber system. Butwhen number systemswere being invented,

they did not automatically include zero. In fact, the Egyptiannumber system did not have or require zero. Around 1700 B.C.

MAYA ZERO

BABYLONIAN ZERO

rr. rr==722 = 2(60)2 + 0(60) + 2

ZERO ON THE ABACUS

the positional number system of base 60 evolved. TheBabylonians used it in conjunction with their 360 day calendar.They performed sophisticated mathematical computations withit, but no symbol for zero was devised. An empty space was leftin the number, standing for zero. About 300 B.C. Babyloniansused this symbol for zero,O . The Maya and Hindu numbersystems evolved after the Babylonian. Their systems were thefirst to use a symbol for zero that functioned both as a placeholder and as the number zero.


Pappus Theorem:If A, B, C, are pointson line li and D, E, F

are points on line 12;

then P, Q, and R arecollinear.

Apply Pappus' theorem to solve the Nine Coin Puzzle.

Nine Coin Puzzle: Rearrange these nine coins, which are 8 rows of 3,into 10 rows of 3.See appendix for solution to the nine coin puzzle.

Pappus' theorem& the nine coin

puzzle

1',


A Japanese This Japanese magic cir-cle is from the work of Seki

Magic Circle Kowa. He was a 17th cen-tury Japanese mathemati-cian who is credited withdiscovering a form of cal-

culus and matrix operations used to solve systems of equations.

ALi,C+ Bf 4A

-t It6 A

) t OS *AA

go 4 -rI 5 '

6

In the magic circle each diameter of numbers totals the sameamount. The method used to form this magic circle seemssimilar to the method Gauss used to sum the first 100 naturalnumbers.

As the story goes, when Gauss was in primary school, his teacher gavethe class the assignment to total the first 100 natural numbers. All theother students in the class busily began adding columns of numbers inthe traditional manner of approaching this problem. Gauss sat at hisdesk thinking. Apparently his teacher thought he was daydreaming andasked him to get busy. Gauss replied that he had already solved theproblem. The teacher asked for the solution, and Gauss illustrated hissolution.

1+2+3+4+5+.. .+50+51+. . .+96+97+98+99+100

L I IGauss paired the numbers so the sum of each pair was 101, and hefigured there were fifty pairs. Thus the sum came to 5050 = (50)(101).


There are a great Sphericalmany uses and applica-tions of various geomet- Dome 6 Waterric shapes to everydayneeds and situations. D titillationOne unusual example ison the island of Symi, off Greece, where a solar distillation unitin the shape of a hemisphere provides each of the island's 4,000inhabitants with approximately one gallon of water per day.

The sun's heat evaporates water from the central supply ofseawater. Then the fresh water condenses on the underside of aclear spherical dome, and drops of water slide down the sidesand are collected along the edge of the dome.

SUNLIGHT

II I ICLEAR DOME

/ A FRESH\/ ~WATER\

el1=S E a WAT ER S--


The helix is a fascinatingmathematical object whichtouches many areas of ourlives, such as geneticmake-up, growth patterns,GAG At2 -~ 1 YE ^o- I .A- --- -

motion, tne natural worlaand the manufacturedworld.

To understand the helix it is important to look at its formation.When a group of congruent rectangular blocks is joined length-wise, a long rectangular column is formed. Perform the sameprocess with rectangular blocks in which one face from eachblock is beveled. Then the resulting column curves aroundforming a circle. But if one face of each rectangular blockwere cut diagonally, the column would curve around on itselfand form a 3-dimensional helix. Deoxyribonucleicacid, DNA-

DNA double helix Rectangular blocks cutdiagonally form a 3-D helix.

the inheritance chromosomes, are composed from two such3-dimensional helices. DNA has two columns of sugarphosphate molecules that twist around joining skewedmolecular units, as the above modified rectangular blocks.

The HelixMathematics& Genetics


There are different types of helices. In fact, the straightrectangular column and the circle column can be consideredspecial cases of a helix. Helices can twist clockwise (right-handed) or counter-clockwise (left-handed). A clockwise helix,such as a corkscrew, when reflected in a mirror, appears to be acounterclockwise helix.

- - ^ A - 1 A l- - - L - - - A. - - - - I I - - A -

examples or different types or nelcesappear in many facets of our world.Circular staircases, cables, screws,bolts, thermostat springs, nuts, ropes,and candy canes can be bothright-handed and left-handed helices.Helices which spiral around cones arecalled conical helices, as seen inscrews, bedsprings, and the spiral framp designed by Frank Lloyd formation of prochorite crystal

Wright in the Guggenhein Museum inNew York.

In nature we also find many forms of helices - the horns ofantelopes, rams, narwhal whales and other mammals, viruses,some shells of snails and mollusks, the plant structure of stalks,of stems (e.g. peas), of flowers, of cones, of leaves and others.The human umbilical cord is a triple helix formed from one veinand two arteries that coil to the left.

It is not uncommon for right and left helices to intertwine.One botanical couple is the honeysuckle (left-handed) and thebindweed (right- handed which includes the morning glory in itsfamily). They were immortalized by Shakespeare in A Midsum-mer Night's Dream. Queen Titania says to Bottom, "Sleep thou,and I will wind thee in my arm...So doth the woodbine (acommon term for bindweed) the sweet honeysuckle gentlytwist."


Motion is yet another area where helices appear. Examples ofhelical paths are found in - tornadoswhirlpools, draining water,a squirrel's path up or down a tree, and the clockwise spiral ofthe Mexican free-tailed bats of the Carlsbad caverns in NewMexico.

honeysuckle

a screw

a spring tecoma

With the discovery that the helix is linked to the DNA molecule,it is not surprising to find the appearance of helices in so manyareas. The varying forms of helices and their growth patterns innature are themselves governed by a genetic code, and thus arecontinually generated by nature.


in the nineteen hundreds A/Magic "Line"Claude F. Bradon discoveredhow magic squares could beused to form artistically pleasing patterns. He discovered that ifthe numbers in a magic square were connected consecutively,they formed interesting patterns that have come to be known asmagic lines. Actually a magic line is not a line, but a patternthat is formed. When these patterns are filled in with alternatingshades, some very unique designs are created. As an architect,Bragdon used magic lines in architectural ornaments andgraphic designs of books and textiles.

8 3- - -4The magic line for lo shu,

1 4,5 If9 the earliest known magicsquare. It is from China

6-- -- 7 2 in 2200 B.C.

165\5\ \

1

3--2 13/Y N I

/- -:~

156-- -7'/ t\12

I'---4 I

The magic line for Albrecht Darer'smagic square of 1514.


Mathematics & We are all familiarwith many of the

A architecture mathematical formsused in architecture,such as the square,

rectangle, pyramid andsphere. But there are some architectural structures which havebeen designed in less recognizable shapes. A striking example isthe hyperbolic paraboloid used in the design of St. Mary'sCathedral in San Francisco. The Cathedral was designed by PaulA. Ryan and John Lee and engineering consultants Pier LuigiNervi of Rome and Pietro Bellaschi of M.I.T.

St. Mary's Cathedral


At the unveiling when asked what Michelangelo would havethought of the Cathedral, Nervi replied: "He could not havethought of it. This design comes from geometric theories notthen proven."

The top of the structure is a 2,135 cubic foot hyperbolicparaboloid cupola with walls rising 200 feet above the floor andsupported by 4 massive concrete pylons which extend 94 feetinto the ground. Each pylon carries a weight of nine millionpounds. The walls are made from 1,680 prepounded concretecoffers involving 128 different sizes. The dimensions of thesquare foundation measure 255' by 255'.

A hyperbolic paraboloid combines a paraboloid (a parabola revolvedabout its axis of symmetry) and a three dimensional hyperbola.

hyperbolic paraboloid equation:Y2 -x2 = z

b2 a2 c ab>O, c#0


H istory of in the second half of the 19thO century, there was a great surge

Op ti cal of interest in the field of opticalillusions. During this period

IllUStOnS nearly two hundred paperswere written by physicists and

psychologists describing optical illusions and why they occur.

Optical illusions are created by our eyes' structure, our mind, ora combination of both. What we see is not always what exists. Itis important not to base conclusions strictly on what is per-ceived, but rather to verify by actual measurement.

4.-

.40

04-

/0

ee00,1-0�00100oe01.1000000,

N

N 1

N*

N 00

%N "

N" .

N"'0

N,"00 '00N

Zollner's illusion

It was this illusion that triggered the study of optical illusions inthe 19th century. Johann Zollner (1834-1882), an astrophysicistand professor of astronomy (who made many contributions tothe study of comets, the sun and the planets and was theinventor of the photometer) happened upon a piece of fabricwith a design similar to the above drawing. The vertical lines

II


are actually parallel, but certainly do not appear so. Somepossible explanations for this optical illusion are:

1) the difference between the acute angles which are set indifferent directions on the parallel segments.

2) the curvature of the eye's retina3) superimposed segments cause our eyes to converge and

diverge which makes the parallel segments curve.

It was discovered that this illusion is most intense when thediagonal segments form angles of 45Q with the parallel segments.

This famous optical illusion was created by cartoonist W.E. Hill andpublished in 1915. It is classified as an oscillation illusion because oureyes shift between two figures - an old woman and a young woman.

Can you make the blackfaces become tops and albottoms of the cubes? I

I I


Trisecting & Geometry has awealth of ideas, con-

the equilateral cepts and theorems.It can be very interest-

triangle ing to discover aproperty which ap-plies to a geometric

object. For example, take any triangle and trisect its three angles.Then study the figure formed by the trisectors. What do you no-tice?l

1It can be proven these trisectors always formn an equilateral triangleregardless of the shape of the original triangle.


From each home three

separate paths must be

made - one to the well, oneto the grain mill, and one to

the woodshed. None of

these paths cross each other.

vWater, C

Pro1

Food,rrainPlem

Can you solve this

problem?

I t

"ate'r

/ f

wood

grain

See appendix for the solution to the Wood, Water, GrainProblem.

CharlesBabbage-the Leonardoda Vinci ofmoderncomputers

This illustration is a portion of Charles Babbage's difference en-gine which he started to build in 1823 and abandoned in 1842.


Considered the Leonar-do da Vinci of moderncomputers, Charles Bab-bage (1792-1871) was anEnglish mathematician,engineer, and inventor.Besides inventing the firstspeedometer, various pre-cision machinery, codesand a method for identify-ing lighthouses from theirbeams, he spent most of


his time making a machine that would perform mathematicaloperations and calculate tables.

Babbage's original model of his difference engine was made withtoothed wheels on shafts that were turned by a crank and couldproduce a table of squares up to 5 decimal places. Later Babbagedesigned a much grander machine with a 20 decimal place ca-pacity that would stamp the answer on a copper engraver'splate. In the process of making parts, he became an expert tech-nician, developed superior tools and techniques foreshadowingmodern methods. He was continually perfecting parts anddesigns and scrapping old work. His perfectionism and the levelof technology at that time kept him from completing the endproduct. When he abandoned the difference engine, he conceivedthe idea for the analytical engine 1 - capable of doing anymathematical operation, having a memory capacity of 1000 fiftydigit numbers, using tables from its own library, comparinganswers and making judgements not in the advance instructions.The machine's performance relied on mechanical parts andpunched cards. Although his idea was never realized, thelogical structure of his analytical enginee is used in today'scomputers.

The analytical engine actually represents a class of machines, inthe same sense as today's computer is. It is quite astonishingBabbage singlemindedly pioneered this contemporary idea,engineered the machine, developed the tools to construct it,designed its various stages, and developed the mathematicsneeded to program it. A phenomenal task! A working model ofthe analytical engine was built by IBM as a tribute to CharlesBabbage.

lAda Lovelace, daughter of Lord Byron, encouraged and worked withCharles Babbage on his analytical engine. Besides monetarily contribut-ing to his work, her keen knowledge of mathematics was invaluable inthe computer programming aspect for the analytical engine. And equallyimportant was her overall enthusiasm for his project.


Mathematics Since the representa-tion of the human body& Moslem Art was forbidden to Mo-hammedans, their artform was channeled

into other areas. It was confined to ornament and mosaic, andconcentrated on geometric designs. As a result, there is adefinite connection between their art and mathematics.

The wealth of patterns that were created display:* symmetries* tessellations, reflections, rotations, translations of

geometric forms* congruences between dark and light patterns

This design shows translations This design illustrates tessellationsof geometric forms. reflections, rotations, and symmetric

Congruences between light and da:forms are also highlighted by this design.

1 A tessellation of a plane is the covering of the plane with aparticular shaped tile so that no gaps are left and they do not overlap.


The Chinese magicsquare, illustrated below,is about 400 years old.In base ten numerals it

translates to:

27 29 2 4 13 369 11 20 22 31 18

32 25 7 3 21 2314 16 34 30 12 528 6 15 17 26 191 24 33 35 8 10

A ChineseMagic Square


Infinity The diagram below illustratescircumscribing regular polygons.

& Lim its The number of sides of theI polygon successively increases. It

would seem that the radii wouldgrow without bound, but in fact the increasing radii approach alimit that is about 12 times that of the original circle.


There are ten stacks of Counterfeitten silver dollars. You aregiven the weight of a real Coin Puzzlesilver dollar, and are told I

each counterfeit coinweighs one gram more than a real silver dollar. You also knowthat one of the stacks is completely counterfeit, and you can usea scale that weighs by grams. What is the minimum number ofweighings needed to determine the counterfeit stack?

"' I yF>II,'eL 2c

Kr";* ,

GirlF.>UIC

See appendix for counterfeit coin puzzle solution.


The Parthenon Ancient Greek archi-& tects of the 5th century

-An Optical & B.C. were masters inthe use of optical illu-

Ma thematical sions and the goldenmean in the design of

D esign their buildings. These-architects found thatstructures built pre-

cisely straight did not end up appearing straight to oureyes. This distortion is due to our retinas' curvature, causingstraight lines that fall at particular angles to appear to curvewhen our eyes view them. The Parthenon is one of the most

The Parthenon

famous examples illustrating how these ancient architects com-pensated for the distortion caused by our eyes. As a result,the columns of the Parthenon actually curve outward, as do thesides of the rectangular base of the Parthenon. Diagram Iillustrates how the Parthenon would have appeared if thearchitects had not made these adjustments.


Thus, by making these compensations, the structure and thecolumns appear straight and aesthetically pleasing.

Ancient Greek architects and artists also felt the golden meanand golden rectangles enhanced the aesthetic appeal of struc-tures and sculptures. They had knowledge of the golden mean- how to construct it, how to approximate it, and how to use itto construct the golden rectangle. The Parthenon illustrates thearchitectural use of the golden rectangle. Diagram 2 belowshows how its dimensions fit almost exactly into the goldenrectangle.

,- - - - - - - e - - - - -- a.-

l-.- -.-. ..- .-.-.-. -.. . .. . . m -

diagram 2

1 See section on Golden Rectangle for additional information.

';-.- I

l


Probabilityand Pascal'sTriangle

.0000

0 00

1

the hexagonal obstacles and collect below. At each hexagonthere is an equal chance for a ball to roll to the right or to the left.As illustrated the balls distribute themselves according to thenumbers of the Pascal triangle. The balls collect at the bottomand produce the bell shaped normal distribution curve. Thiscurve is used - by insurance companies to set rates, in science

The triangle belowformed from hexagonalblocks, has a unique way ofgenerating the Pascal trian-gle. Balls from a reservoiron top pass down through

Am.

0 ..b 4Iff", 11�41� '.. �-- . �


to study the behavior of molecules, and in the study ofpopulation distribution.

Pierre Simon Laplace (1749-1827) defined the probability of anevent to be the ratio of the number of ways in which the eventcan happen to the total possible number of events. Thereforewhen flipping a coin, the probability of getting a head is

I - number of heads on a coin2 - number of possible events

(head and tail)

The Pascal triangle can be used to calculate differentcombinations and the total possible combinations. For example,when four coins are tossed in the air the possible combinationsof heads and tails are:

4 heads - HHHH=13 heads & 1 tail - HHHT, HHTH, HTHH, THHH = 42 heads & 2 tails -HHTT, HTHT, THHT, HTTH, THTH,

TTHH = 61 head & 3 tails -HTTT, THTT, TTHT, TTTH = 44 tails - TTTT =1

The fourth row from the top of the Pascal triangle indicates thesepossible outcomes - 1 4 6 4 1. The sum of these numbersrepresents the total possible outcomes = 1+4+6+4+1=16. Thusthe probability of tossing 3 heads and 1 tail is:

4 - possible combination of 3 heads and 1 tail is16 - total possible combinations.

For large combinations when the Pascal triangle would betedious to extend, Newton's binomial formula can be used. Eachrow of the Pascal triangle contains the coefficients in theexpansion of the binomial (a+b)n. For example, to find the


coefficients for (a+b)3 , one looks at the 3rd row from the top row(which is considered the zero row, i.e. (a+b)0 =1). At row three,one finds 1 3 3 1, which are the resulting coefficients,

la 3 + 3a2b +3ab2+1b3 =(a+b)3.

The binomial formula can be used for the nth row of the Pascaltriangle.

binomial formula:

(a+ b)n = an + n (an-lb')+ n(n ~l)(an-2b )+...+b

the rth coefficient is r! (n - r)!

the number of combinations of n objects taken r at a time is

C(nr)= n!r!(n-r)!

10 objects taken 3 at a time would be

C(10,3) = 10!3!(10-3)!

= 10 9 8 7 6 5 4 3 2 1 = 120321 7654321

120 possible combinations for 10 objects taken 3 at a time, orcheck the 10th row of the Pascal triangle.


As a rope is wound or Theunwound around anothercurve (here a circle), it Involutedescribes an involute curve.Many examples of involutes appear in nature as found on thetips of a hanging palm leaf, the beak of an eagle, the dorsal fin ofa shark.

The pentagon,the pentagram& the goldentriangle

The golden triangle is an isosceles triangle with a vertex angle36Q and base angles 72Q each. Its legs are in a golden ratio to itsbase. When a base angle is bisected, the angle bisector dividesthe opposite side in a golden ratio and forms two smallerisosceles triangles.


Starting with a regularpentagon, the penta-gram is generated bydrawing in the diago-nals of the pentagon.Within the shape of thepentagram exist goldentriangles. These goldentriangles divide thesides of the pentagraminto golden ratios.


One of these triangles is similar to the original triangle. The othertriangle serves as the generator of the spiral's curve.

Continuing the process of bisecting a base angle of the newgolden triangle generates a series of golden triangles and theformation of an equiangular spiral.2

FBI7 = golden ratio, (= 2 ) - t 6180339...

1 See footnote on page 32 for additional information on the goldenratio.

2 See The Golden Rectangle for information on the equiangularspiral.


Three men Three men are placedin a line perpendicularfacing the wall to a wall. They areblindfolded. Then threehats are taken from a

bin of three tan hats and 2 black hats. The men are given thatinformation. Then the blindfolds are removed. Each man isasked to determine what color hat he is wearing. The man

6* &

farthest from the wall who sees the two men and their hats infront of him says, "I do not know which color hat I am wearing."The second man from the wall who heard the reply and sees theman and the hat ahead of him says the same thing. The thirdman who sees only the wall, but has heard the two replies says,

"I know which color hat I am wearing. "

Which color is he wearing and how did he determine it?

See the appendix for The Three Men Facing the Wall solution.


If a square is formed, usingthe sum of any two consecu-tive Fibonacci numbers asthe length of its sides, an in-teresting geometric fallacyoccurs.

example:1) Use the consecutive Fibonacci numbers 5 and 8.2) Form a 13x13 square.3) Cut it up as illustrated. Now calculate the square's area and

the rectangle's area. The square's area is 1 unit greater than thatof the rectangle.4) Do the same process for the Fibonacci numbers 21 and 34. Inthis case, the area of the rectangle is 1 unit greater than thesquare.

The I unit discrepancy will alternate between the square and the

rectangle, depending on which consecutive Fibonacci numbersare used.1

lThe sequence of ratios formed from consecutive Fibonaccinumbers 1/1, 2/1, 3/2, 5/3, 8/5, 13/8, ...Fn+i/Fn alternates in valueabove and below the golden ratio. The limit of this sequence is the valueof the golden ratio (1+45)/2. For additional information see the sectionon The Golden Rectangle.

Geometricfallacy & the

Fibonaccisequence

_'1WLj I 11 I I I I 11 I I q I I I i�H I I 11 I- I 11 I

IC

h il; r- - I 11 I I


M azes Today we think of a maze as a recrea-tional puzzle, but in earlier times mazesconjured up mystery, danger and confu-

sion. One could surely get lost in the intricately winding pathsor perhaps encounter monsters harbored in the maze's interior.In ancient times, labyrinths were often constructed to defend for-tresses. Invaders would be forced to travel a great distance in amaze, and thus be exposed to attack.

Mazes have appeared in various parts of the world over thecenturies.

* stone carvings in Rock Valley, Ireland - circa 2000 B.C.* Minoan labyrinth in Crete - circa 1600 B.C.* Italian Alps, Pompeii, Scandinavia* turf mazes of Wales and England* mosaic mazes set in floors of European churches* African textile mazes* Hopi Indian rock carvings of Arizona

Today, mazes are of special interest in the field of psychologyand computer design. For decades psychologists have usedmazes to study the learning behavior of animals and humans.Computerized robots have been designed to problem solvemazes as an initial step in the design of more advanced learningmachines.

The garden at Hampton Court.


Topology is the field of mathematics that studies mazes as abranch of networks (solving problems by a method ofdiagramming). A Jordan curve is often mistaken for a maze. Intopology we learn that a Jordan curve is a circle that has beentwisted or bent inward and around without crossing itself. It hasan inside and outside as does a circle, unlike a maze. Thus, theonly way of getting from the inside to the outside is to cross thecurve.

Since robots are used to solve mazes, systematic methods had tobe devised to solve mazes.

methods for solving mazes(1) For a single maze, shade blind alleys and loops you see. The

remaining routes will be to the goal. Then select the mostdirect path. If the maze is complicated this method would behard to apply.

(2) Go through the maze always keeping one hand (either theright or the left) in contact with the wall. This method iseasy, but does not work for all mazes. Exceptions are (a)those mazes with two entrances and a path connecting themthat does not pass through the goal, (b) mazes with pathsthat loop or surround the goal.

(3) French mathematician M. Tremaux has devised a generalmethod to solve any maze.

procedure:(a) As you go through the maze, constantly draw a line to the

right.(b) Whenever you come to a new juncture take any path you

like.(c) If on your new path, you come to an old juncture or dead

end, turn and go back the way you came.(d) If returning on an old path you come to an old juncture, take


This maze appears on a Navajo blanket.

anv npw nath- if there is

one. Otherwise take anold path.Never enter a pathmarked on both sides.s method is supposed tofool proof - but may takerme time.

Leather with a life sizele or with a pencil inid, mazes continue toMllenge and intrigue the

mind and provide enjoy-ment.

This maze of London appeared in The Strand Magazine in April 1908 with thefollowing instructions: 'The traveller is supposed to enter the Waterloo Road, and hisobject is to reach St. Paul's Cathedral without passing any of the barriers which are placedacross those streets supposed to be under repair. "


In this illustration Chinesea checker boardpattern of the Checkerboards"Chinese countingboard is shown. The Chinese were the first to evolve a system ofset rules for solving systems of simultaneous equations.

They would place components in a checkerboard lay out, andthen apply the rules which were based on matrices to solve theproblem.

� I W PI it 0 i � Ila


Conic It is often baffling to some peoplethat mathematicians will pursue a

Sections problem or an idea simply because itis interesting or curious. Looking

back at the ancient Greek thinkers, we find them studying ideasintensely without regard to their immediate usefulness, butbecause they were exciting, challenging or interesting. And so itwas with their study of conic sections.

Their primary interest in these curves was to use them to helpsolve the three ancient construction problems - squar-ing a circle, doubling a square, and trisecting an angle. Theseproblems had no practical value at the time, but they werechallenging and stimulated mathematical thought. More oftenthan not the practical use of a particular mathematical ideadoes not manifest itself for years. Conic sections created duringthe third century B.C., gave 17th century mathematicians foun-dations to begin to formulate various theories involving curves.For example, Kepler used ellipses to describe the paths ofplanets, and Galileo found the parabola to fit the motions oftrajectiles on the earth.

The diagram on page 197 illustrates how a plane intersecting thedouble cone produces the circle, ellipse, parabola, andhyperbola.

question: How should a plane intersect the cone to produce astraight line, two intersecting lines, or a point?

There are many examples in the universe that form these curves.One contemporary and exciting example is Halley's Comet.

In 1704 Edmund Halley worked on the orbits of various cometsfor which data was available. He concluded that the comets of1682, 1607, 1531, 1456 were a single comet orbiting the sunelliptically about every 76 years. He successfully predicted its


return in 1758, and as a result it became known as Halley'sComet. Recent research suggests that Halley's Comet may havebeen recorded by the Chinese as early as 240 B.C.

J

examples of these curves in the universeparabola-* arc of spouting water* shape of flashlight's light on a flat surface

ellipse-* orbital paths of some planets and some comets

hyperbola-*paths of some comets and other astronomical objects

circle-* ripples on a pond* circular orbits* the wheel* objects in nature


The Archimedean screw,when submerged in waterand rotated, was able topump water uphill.

It is still used for irrigation in various parts of the world.

Archimedes (287-212 B.C.) was a Greek mathematician andinventor. He discovered the laws of lever and pulley. Hisdiscoveries led to the invention of machines capable of movingheavy loads easily. He is also credited with discovering amethod to compare volumes by submerging objects in water-hydrostatics - buoyancy - the use of calculus ideas -inventing the catapult - inventing contoured mirrors toconcentrate the sun's rays.

The Screw ofArchimedes


Optical illusions are Irradiationcreated by our minds, oureyes' structure or both. For Opticalexample, when we view a O tcregion that has both light illusionand dark objects, the fluidsin our eyes are not perfect-ly clear and light scatters while passing to the retina at the rearof the eye (this is where the eye detects light). As a result, brightlight, or light areas spill over onto dark areas of the image that ison the retina. Thus a light region will appear larger than a darkone of equal size, as in the diagram below. This explains whydark clothing, especially black, makes you appear more slenderthan if you were wearing light, or white clothing of the samedesign. This illusion is called irradiation, and was discovered byHerman L. F. von Helmholtz in the 19th century.


ThePythagoreanTheoremand PresidentGarfield

ISee section on Pythagorean Theorem.

President James AbramGarfield (1831-1881), thetwentieth president of theUnited States, was quite in-terested in mathematics. In1876, while serving as amember of the House of Rep-resentatives, he discoveredan interesting proof of thePythagorean theorem1 . TheNew England Journal of Educa-tion published the proof.

Db


The proof uses two methods to calculate the area of a trapezoid.

method (1): area of a trapezoid = 1/2(sum of the bases)(altitude)

method (2): dividing trapezoid into 3 right triangle and calculating

the area of these 3 right triangles.

proof

Construct trapezoid ABCD with AB I I DC, angles C and B rightangles, and the indicated lengths, a, b and c.

Calculate the area of the trapezoid using the two methods listedabove.

area method (1) = area method (2)

1/2(a+b)(a+b) =1 /2(ab)+1 /2(ab)+1 /2(c c)

(a+b) (a+b) = ab + ab + c2

a2 +2ab + b2 =2ab+c2

QED a2 + P = c2


The W heel Two concentric circlesare shown on this wheel.

Paradox of The wheel moves from Ato B as it rotates once.

Ari s to tle Notice I AB I correspondsto the circumference of the

large circle. Since the small circle also rotated once and traveledthe distance I AB I, is not its circumference I AB I ?

Galileo's explanation of Aristotle's wheel paradoxGalileo analyzed the problem by considering two concentricsquares on a square "wheel". As the box flips 4 times,(transversing the perimeter of the square wheel, IABI), wenotice the small box is carried along 3 jumps. This illustrateshow the small circle is also carried along the distance I AB I, andthat I AB I does not represent its circumference.

A B

A B


On the Salisbury Plain in StonehengeEngland stands the awesomestone structure known asStonehenge. It was started around 2700 B.C., and the last of thethree stages was completed about 2000 B.C.

What was the purpose of Stonehenge? What meaning did ithave for the different groups of people who used and developedit? Was it:

* a religious temple?* a lunar and solar observatory for the

winter solstice sunset and the summersolstice sunrise?

* a lunar calendar?* a primitive computer for predicting lunar

and solar eclipses?

Since there is no written documentation by the builders or usersof Stonehenge, there is no way to know its true purpose. Fromthe shreds of evidence, all theories remain speculation. What isevident though, is that the builders had knowledge of a form ofmeasurement and geometry.


Art as varied as the earlyHow many cave drawings, the Byzan-tine icon, Renaissancedim ensi o nspainters, and the Impres-there? sionists depict subjectsthat existeither in secondor third dimension. Art-

ists, scientists, mathematicians, and architects have developedtheir own renditions of how they believe certain objects wouldappear in the 4th-dimension. One example is this 4th-dimen-sional drawing of a cube, called a hypercube, by architect ClaudeBragdon in 1913. Bragdon incorporated his hypercube drawingand other fourth dimensional designs into his works. Anexample is his Rochester Chamber of Commerce Building.

The hypercube by Claude I

The existence of other dimensions beyond the third dimensionhas always been an intriguing notion. From a mathematicalpoint of view, the likelihood of higher dimensions follows a logi-cal progression of thought.

For example, begin with a zero-dimensional object, a point. Nowmove the point one unit to either the right or the left, and a linesegment is formed. The line segment is a one-dimenional object.


Move the line segment one unit upward or downward, and asquare is formed. The square is a two-dimensional object. Pro-ceeding in the same manner, take the square and move it oneunit outward or inward, and a cube is formed, which is a three-dimensional object. The next step is to somehow visualize movingthe cube one unit in the direction of the 4th dimension, and pro-ducing the hypercube, also called tesseract. In a similar wayone can arrive at the hypersphere, a 4-D sphere. But mathema-tics does not stop with the 4th dimension, but considers the nth-dimension. Startling mathematical patterns appear when datareferring to vertices, edges, and faces of different dimensionalobjects is collected and organized.

sseract.

The possible existence of a 4th-dimension has intrigued manypeople. Artists and mathematicians have tried to image anddraw how certain objects would appear in the 4th dimension.The tesseract and the hypercube are 4th-dimensional representa-tions of a cube. A cube drawn on paper is sketched in perspec-tive to imply its 3-dimensional characteristic. Thus a tesseractdrawn on paper is a perspective of a perspective.


Computers & Humans, being 3-D crea-tures, can easily visualize

D im ensions and understand the first_ three dimensions. Although

dimensions exist mathemat-ically beyond the third, it is difficult to accept something wecannot see or imagine. Computers are beginning to be used tohelp in the visualization of higher dimensions. For example,Thomas Bancroft (a mathematician) and Charles Strauss (a com-puter scientist) at Brown University have used a computer togenerate motion pictures of a hypercube moving in and outof a 3-D space, thereby capturing various images at differentangles of the hypercube in a 3-D world. It is analogous to a cube(i.e. 3-D object) passing through a plane (2-D world) at differentangles and recording the cross-section impressions it makes onthe plane. A collection of these impressions will help give abetter picture to a 2-D creature of a 3-D object.

This illustration showsdifferent impressions asphere would leave asit passes or intersects aplanes, i.e. the 2nddimension. It isanalogous to ahypercube passingthrough space, i.e. the3rd dimension.

We now have 2-D holograms picturing 3-D objects. Hologramsare now being commercially used in advertising and graphics.Perhaps in the future, 3-D holograms will be developed andused to picture 4-D objects.

Have you considered that even your best friend could be a 4th-dimensional creature who appears to you as 3-D creature?


Topology studies the The Doubleproperties of an objectthat remain unchanged M oebius Stripwhen the object is dis-torted (stretched orshrunk). Unlike Euclidean geometry, topology does not dealwith size, shape, or rigid objects. Basically, it studies elastic ob-jects, which is why it has come to be referred to as rubber sheet ge-ometry. The Moebius strip was created in the 17th century bythe German mathematician Augustus Moebius, and is one of theobjects studied in topology. By taking a strip of paper, giving ita half twist and gluing its ends together, a Moebius strip isformed. It is fascinating because it has only one side. A pencilcan be used to trace its entire surface without lifting the pencil.

Now let's consider the double Moebius strip. It is formed by takingtwo strips of paper, simultaneously giving them a half twist

and gluing the ends together. The strips appear to be two nes-tled Moebius strips. But are they?

Make the illustrated model and test it. Run your finger betweenthe two strips to see if they seem nestled. Take a pencil andbegin tracing along one of them, until you arrive back at yourstarting place. What happened?

What happens if you try to unnestle them?

ParadoxicalCurve -space-fillingcurve

~~-- -_v. ...... "- I -. ,- -_I -

fill a given space. Euclidean curves are planar, flat. Mathemati-cians of the Euclidean era had not yet considered that curvescould generate themselves in the manner below.

t- -I - --

The example above shows the stages of the space-filling curvewhich will envelop the space of an entire cube by continuallygenerating itself in the special way illustrated.


A curve is usually con-sidered one-dimensional,and composed of pointswhich are considered tohave zero or no dimen-sion. With these notionsit somehow seems contra-diictorv that a cuirve can


Often called an ancient The A bacuscomputer, the abacus is one ofthe oldest accounting devicesknown. This ancient calculator was and is used in China andother Asian countries to add, subtract, multiply, divide andcalculate square and cube roots. There are many types of abaci.For example, the Arabic abacus had ten balls on each wire withno center bar. History shows that even the ancient Greeks andRomans used abaci.

The Chinese abacus consists of thirteen columns of beadsdivided by a crossbar. Each column has five beads below thecrossbar and two beads above it. Each bead above the crossbar isequal to five beads below the cross bar in its column. Forexample, a bead above the crossbar in the ten's column is worth5x1O or 50.

This a lu is b

This abacus has value 1,986 written in its beads.

210 THE JOY OFMATHEMATICS

Mathematics How can mathemat-ical objects be present

& Weaving in woven fabrics?

Does a weaver consciously analyze the design from a mathemat-ical point of view.

Studying these pieces of fabric one finds many mathematicalconcepts present in them:

* lines of symmetry* tessellations* geometric forms

p Droportional objects* reflected patterns

Shoshone Indian designOjibwa Indian design

Potawatomi Indian designdesign from the Congo

Can you find the mathematical ideas listed above in thesefabrics' designs? Can you find other mathematical ideas present?

I I


In the 17th century, a 69 M ersenne 'sdigit number was pro-posed prime by French N um bermathematician Marin Mer-senne. In February of 1984, a team of mathematicians successful-ly programmed their Cray computer (able to sample whole clus-ters of numbers simultaneously) to solve this three century oldpuzzle. After 32 hours and 12 minutes of computer time, thethree factors (listed below) of the Mersenne number werediscovered. This feat now has cryptographers worried, sincemany cryptographic systems use multidigit numbers, which aredifficult to factor, in order to encode secrets and keep themsecure.

MERSENNE'S NUMBER132686104398972053177608575506090561429353935989033525802891469459697factors:178230287214063289511 and 61676882198695257501367 and12070396178249893039969681

Factoring a number involves breaking it down into a product ofsmaller prime numbers. The task is simple with small numbers,and can be done by trying all primes numbers smaller than thenumber as divisors. Large numbers require other mathematicalmethods. The number of computations needed to factor a num-ber grows exponentially as the numbers grow. Even a computerperforming a billion operations per second would take severalthousand years to factor a 60-digit number by the above method.

In 1985-86, Robert Silverman (of Mitre Corp. Bedford, MA) andPeter Montgomery (of System Development Corp. Santa Monica,CA) developed a method of factoring using microcomputers,rather than special computers developed expressly for factoringor the costly Cray computer. Their method is fast and veryinexpensive. One of their recent accomplishments was factoringan 81-digit number using eight microcomputers which ran for150 hours.


TangramPuzzle

Using the seven pieces of atangram, determine how the fig-ures below are formed.


This illustration shows Infinite vshow a set of points of a . 0semicircle, which is finite in Finitelength, is matched (put into aone-to-one correspondence)with a set of points of a line, which is infinite in length. Thissemicircle has a perimeter of 57r. The line tangent to thesemicircle is infinite in length. A ray with endpoint P (center ofthe semicircle) is drawn so that it intersects the line and thesemicircle. The points of intersection of the semicircle and theline are thus matched up by the ray in a one-to-one correspon-dence. As the ray moves along the semicircle and approachesray PQ it intersects the line further and further out.

What would happen when the ray becomes ray PQ?

5 P Q

1The ray would be parallel to the line.


Triangular, There are manydifferent labels given

Square to numbers. Some ofS uar their names were de-&Pentagonal rived because of the

1 shape of geometric ob-N um bers jects they form. Be-

low we see that oddnumbers form triangular shapes and hence they are also calledtriangular numbers. Perfect square numbers , i.e. 12=1, 22=4, 32=9,..., form squares.

Each group of numbers has a pattern associated with it. Try toform other sequences of numbers that are linked to geometricobjects, and determine their particular pattern.

triangular numbers

* **-- -- ' etc.

l 3 6)

l0- --- 4- - -.

square numbers

I I I*- a--- -- etc.

4 9

pentagonal numbers

* e t' , \ ,

* -4 - >- .- '- ---. etc.I 5 12


In 200 B.C., Eratos-thenes devised this inge-nious method for measur-ing the distance aroundthe Earth.

To measure the circumference of the Earth, Eratosthenes usedhis knowledge of geometry, particularly the theorem:

Parallel lines cut by a transversalform congruent alternate interior angles.

He was aware that at noon during the summer solstice in the cityof Syene (in Egypt) a vertical rod did not cast a shadow, while inAlexandria (5000 stadia - 500 miles away) the vertical rod cast ashadow which formed a 72 12' angle. With this information hewas able to calculate the circumference of the Earth to within 2%of its actual value.

ri's raysII

L:III

III

II

procedure:Since light rays travel parallel to each other, <CAB and <B in theabove diagram are congruent alternate interior angles.Thus, the distance between Syene and Alexandria is a fraction ofthe distance around the Earth. This fraction is 7Q 12'/ 360Q =1/50. Therefore, the distance around the Earth is (500 miles) x 50= 25,000 miles.

Eratosthenesmeasures the

Earth


Projective Using techniques ofprojective geometry

Geometry & and solving systems ofequations, Narendra

Linear Karmarkar, while aBell Laboratory mathe-

Programming matician, discovered amethod to drasticallycut down the time

needed to solve very cumbersome linear programming prob-lems, such as allocating time on communication satellites, sched-uling flight crews, and routing millions of telephone calls overlong distances.

An artists rendition of a geometric solid and its many facets.

Until recently, the simplex method, developed by mathematicianGeorge B. Danzig in 1947, had been used. It requires largeamounts of computer time and is not feasible for enormous


problems. Mathematicians visualize such problems as complexgeometric solids with millions or billions of facets. Each cornerof each facet represents a possible solution. The task of thealgorithms is to find the best solution without having tocalculate every one. The simplex method by Danzig runs alongthe edges of the solid, checking one corner after another butalways heading for the best solution. In most problems itmanages to get there efficiently enough, as long as the number ofvariables (unknowns) is no more than 15,000 to 20,000. TheKarmarker algorithm takes a short-cut by going through themiddle of the solid. After selecting an arbitrary interior point,the algorithm warps the entire structure, i.e. reshaping theproblem, in a way designed to bring the chosen point exactlyinto the center. The next step is to find a new point in thedirection of the best solution and to warp the structure again,and bring the new point into the center. Unless the warping isdone, the direction that appears to give the best improvementeach time will be an illusion. The repeated transformations,based on concepts of projective geometry, lead rapidly to thebest solution.

lAn algorithm is a procedure of computation for arriving at asolution. For example, the process and steps of long division is analgorithm. In long division we mentally take short-cuts in the divisionprocess-if 29 is to be divided into 658, one thinks of 29 as being close to30 and figures how many 30's are in 65 rather than trying to figure outhow many 29's in 658 all at once. The Karmarker algorithm also hasspecific short-cuts built into it, as its transformation/warping process.


The spider & Henry Ernest Dudeneywas a renowned 19th

the fly century English puzzlecreator. Most of today's

problem puzzle books have manyof his gems, but very often

they are not credited to him. In the 1890's he and Sam Loyd, thefamous American puzzlist, collaborated on a series of puzzlearticles.

Dudeney's first book, The Canterbury Puzzles, was published in1907. Five additional books were later published and theyremain a treasury of mathematical teasers.

12'

30'

The spider and the fly, which first appeared in an Englishnewspaper in 1903, is one of his best known puzzles.

In a rectangular room, 30'x12'x12', a spider is at the middleof an end wall, one foot from the ceiling.

The fly is at the middle of the opposite end wall, one foot abovethe floor. The fly is so frightened it can't move.

What is the shortest distance the spider must crawl in order tocapture the fly? (Hint: it is less than 42')

See appendix for the solution to the spider & the fly problem.


What kind of mathe- Mathematicsmatical concepts wouldbe connected to soap &bubbles? The shapes thatsoap film forms are gov- soap bubbleserned by surface tension.

The surface tensiondiminishes the surface area as much as possible. Consequently,each soap bubble encloses a certain amount of air in such a waythat the surface area for the given amount of air is minimized.This explains why a single soap bubble becomes a sphere, whilea cluster of bubbles, as in foam, has a different formation. Infoam, the edges of the soap bubbles meet at angles of 120Q, calledtriple junctions. A triple junction is essentially the point wherethree line segments meet, and the angles at the intersection are120 each. Many other natural phenomena (a few examples arethe scales of a fish, the interior of a banana, formation of kernelsof corn, the plates on a turtle's shell) conform to a triple junction,which is an equilibrium point of nature.


T e coin The top coin has been movedhalfway around the coin below it

paradox ending up in the same positionin which it originally started.Since it traveled half of its cir-

cumference, one would have expected it to end upside down.Take two coins and study the movement. Can you explain whythis did not happen?


An hexamino is a flat Hexaminoesobject composed of sixsquare units. Startingwith a cube of volume 1 cubic unit, cut along seven of its edgesso it falls apart and lies flat. The resulting figure is a hexamino.Depending along which edges the cube is cut, different shapedhexaminoes result. Some hexaminoes are pictured below.

How many different hexaminoes are there?

The occurrence of theFibonacci sequence innature is so frequentthat one is convinced itcannot be accidental.

a) Consider the list of the following flowers with a Fibonaccinumber of petals: trillium, wildrose, bloodroot, cosmos,buttercup, columbine, lily blossom, iris.

b) Consider these flowers with a Fibonacci number of petal-like parts:aster, cosmos, daisy, gaillardia.

these Fibonacci numbers are frequentlyassociated with the petals of:

3 ....... lilies and irises5 ....... columbines, buttercups, and larkspur8 ....... delphiniums13 ...... corn marigolds21 ...... asters34,55, 84 ... daisies

BLOODROOT

COSMOS

TRILLIUM

WILD ROSE


The Fibonaccisequence &Nature


c) Fibonacci numbers are also found in the arrangement ofleaves, twigs, and stems. For example, select a leaf on a stemand assign it the number 0, then count the number of leaves(assuming none have been broken-off) until you reach onedirectly in line with the 0-leaf. The total number of leaves willmost likely be a Fibonacci number, and the number of turns inthe spiral before reaching the leaf directly above should also bea Fibonacci number. The ratio of leaves to turns in the spiral iscalled a phyllotactic (from the Greek word meaning leaf-arrangement ) ratio. Most phyllotactic ratios happen to beFibonacci ratios.

PEAR

CHE

ELM

leaf 2

e As


d) Fibonacci numbers have sometimes been called the pine conenumbers because consecutive Fibonacci numbers have a tenden-cy to appear as left and right sided spirals of a pine cone. This isalso true for a sunflower seedhead. In addition, you may findsome that are consecutive Lucas numbers.1

8 spirals to the right and 13 spirals to the left

sunflower seedhead

1 Lucas numbers form a Fibonacci-like sequence which starts with thenumbers 1 and 3; then consecutive numbers are obtained by adding theprevious two numbers. Thus, the Lucas sequence is 1,3,4,7,11, . . .It isnamed after Edouard Lucas, the 19th century mathematician who gavethe Fibonacci sequence its name and who studied recurrent sequences.Another way in which the Lucas sequence is related to the Fibonaccisequence is the following:

0, 1, 1, 2, 3, 5, 8, 13, .

Y372W


e) The pineapple is another plant to check for Fibonacci numbers.For the pineapple, count the number of spirals formed by thehexagonal shaped scales on the pineapple.

Fibonacci sequence & the golden ratioThe sequence of consecutive Fibonacci ratios,

1 2 3 5 8 Fn+1' 1' 2' 3' 5 F

1, 2 ,1. 5, 1. 6, 1. 625, 1. 6153, 1. 619, ..

alternates above and below the value of the golden ratio, vp. Thelimit of this sequence is vP. This connection implies that wherever(particularly in natural phenomena) the golden ratio, the goldenrectangle, or the equiangular spiral appear the Fibonaccisequence is present and vice versa.


The m onkey Three shipwrecked sailorsand a monkey found them-

and the selves on an island wherethe only food was coconuts.

coconuts They gathered coconuts allday, and decided to go tosleep and divide them up

the next day. During the night one of the sailors woke up anddecided to take his share of the coconuts rather than wait untilmorning. He divided the coconuts into three piles, but there wasone coconut left over which he gave to their monkey. He hid hispile and went back to sleep. Later, another sailor got up and didthe same thing as the first sailor, giving the leftover coconut tothe monkey. And later on the third sailor woke up andproceeded to divide the coconuts as the other two sailors haddone, also giving the left over coconut to the monkey. In themorning, when the three sailors got up, they divided the pile ofcoconuts into 3 shares with one left over for the monkey.

--i An&" I .

''.7z^771 7


What is the least number of coconuts the sailors collected?

Try the same problem with four sailors, and then with five sailors.

The equations used to solve these problems are called Diophantine, afterthe Greek mathematician Diophantus who first used these types ofequations for certain types of problem solving.

See the appendix for the solution to the monkey & the coconutsproblem.


Spider 1 Four spiders start crawling. from four corners of a 6x6 meter

Spirals square. Each spider is crawlingtoward the one on its right,moving toward the center at a

constant rate of 1 centimeter per second. Thus, the spiders arealways located at four corners of a square.

How many minutes will it take to meet at the center?

The curves formed by the spiders' paths are equiangular spirals.1

Try reworking the problem with other shaped regular polygons.

See appendix for the solution to the spider & spiral.

lFor additional information on the equiangular spiral, see the section onThe Golden Rectangle.

pA, I Af!%]Nf m= "ll - '19-

CPW2 - - --.- . -- - - I- -

i Fa r e- -9 t I I t -9 II -S -0 I t I t I -0 In -9 " A a - IV II-- -L 0

169

Si I i Hm al 1.

91133zm

6a m mo 461 M61

st I 1, 1 1

e- )I M-S 1-Qi Al .11 fil A Q

i R-P i -1 R-P n s e answers e i 4 p im

I F 6-a x x A Q1


SOLUTIONSpage 9-triangle to a square

page 17-the wheat and the chessboard

1 +(2) +(2)2 +(2)3 + (2)4+ + (2)631+2+4+8 + 16 +...

page 35-the Tproblem

page 37-He decided to shift each occupant to a room that had a number twice aslarge, so the guest with room #1 went to #2, #2 went to #4, #3 went to#6, etc. This opened up all the odd numbered rooms for the infinite busload of guests.

page 47-a Sam Loyd puzzleStarting at the center move the appropriate squares in thefollowing direction patterns: SW, SW, NE, NE, NE, SW, SW, SW,NW.

page 51-Fibonacci trickIf a and b represent the first two terms, then the following termsgenerated would be: a+b; a+2b; 2a+3b; 3a+5b; 5a+8b; 8a+13b; 13a+21b;21a+34b. Now taking the sum of the first ten terms gives 55a +88b,which is 11 times the seventh term, 5a+8b.

page 56-ten histroical dates:1879-birth of Einstein; 1066-Battle of Hastings; 476-Fall of Rome;1215-Magna Carta; 1455-Gutenberg Bible; 563-Birth of Buddha;1770-birth of Beethoven; 1969-walk on the moon; 1948-Ghandiassasinated; 1776-U.S.Declaration of Independence.


SOLUTIONSpage 59-problem #8 from Pillow ProblemsLewis Carroll's explanation-

m=# of menk=# of shillings of the last man (the poorest man)

After once around, each man is one shilling poorer , and the movingpile of shillings contains m shillings. After k times around each is kshillings poorer, with the last man having no shillings and the heapcontaining mk shillings. The process ends when the last man must passon the pile, which then contains (mk+m-i) shillings. The next to the lastman now has nothing, and the first man has (m-2) shillings.

The first and the last man are the only two neighbors whose shillingscan be in a ratio of 4 to 1. Thus either

mk+m-1=4(in-2) or else4(mk+m-1)=m-2.

The first equation gives mk=3m-7, i.e. k=3-(7/m), which gives nointegral values other than m=7, and k=2.The second equation gives 4mk=2-3m, which gives no positive integralvalues.Thus the answer is 7 men and 2 shillings.

page 69Proof for the amazing track:

The area of the race track is - 7tR 2 - nr2 .This is the difference of the large circle's area minus the small circle'sarea.

From the diagram on p. 69 we see that the chord's length is2l(R - rT).

Thus a circle with this diameter would have area 7i (R2 - r2), which sim-plifies to 7rR 2 - itr2 .

page 70-Persian HorsesThere are two horses horizontally placed belly to belly, and two horsesvertically placed belly to belly.

page 71-Sam Loyd's hors

60

W

ba

232 THE IOY OF MATHEMATICS

SOLUTIONSpage 116-Achilles and the tortoiseAchilles would reach the tortoise at one-thousand one hundred andeleven and one-ninth meters. If the race track is shorter than this, thetortoise would win. If it were exactly this size, it would be a tie.Otherwise, Achilles will pass the tortoise.

page 123-Diophantus' riddlen represents the number of years Diophantus lived.(1/6)n+ (1/12)n + (1/7)n +5 + (1/2)n +4 =nsimplifying: (3/28)n = 9

n = 84 years old

page 136-the checkerboard problemIt is not possible to cover the altered checkerboard with dominoes. Adomino must occupy a red and black square. Since both comersremoved were the same color, there will not be a compatible number ofred and black squares left.

page 140-The proof of 1=2?Division by zero occurs at step 6. The number zero is camouflaged bybeing expressed as b-a. b-a equals zero because a=b in the premise.

page 147-the unexpected exam paradoxThe test cannot be on Friday because it is the last day it could be givenand you could deduce this on Thursday, if you had not had it yet. Thecondition was that you would not know which day until the morning ofthe test. So if it were not Friday, that would make Thursday the lastpossible day. But it cannot be on Thursday because by Wednesday youwould know there are Thursday and Friday left. Since Friday was out,on Wednesday you would know ahead of time it would be Thursday,which you are not supposed to know ahead of time. Now this leavesWednesday as the last possible day, but Wednesday is out because ifyou did not have it by Tuesday, you would know by Tuesday youwould have it on Wednesday. Continuing with this reasoning, each dayof the week is eliminated.

page 159-farner, wolf, goat & cabbageHe first takes the goat across. He then returns and picks up the wolf. Heleaves the wolf off, and takes the goat back. He then leaves the goat atthe starting place, and takes the cabbage over to where the wolf is. Hethen returns, and picks up the goat, and goes where the wolf and thecabbage are waiting.

page 163-nine coin puzzle

*..v A ...


SOLUTIONSpage 175-wood,

The wood, water and grain problem has no solution so long as the pathsmust remain on the (Euclidean) plane. But if the houses are situated ona torus or doughnut surface (as illustrated) the solution is simple.

page 181-counterfeit coin puzzleOne weighing!Place on the scale one coin from the first stack, two from the second,three from the third, and so forth. You know how much the coins on thescale should weigh if none of them were counterfeit. Therefore todetermine which stack is counterfeit, look at the weight and figure outhow much heavier it weighs. This number will correspond to the stackfrom which that group of coins came. For example, if it's four gramsheavier, the forth stack is counterfeit because you placed four of itscoins on the scale.

page 190-three men facing the wallThe man furthest from the wall either sees 2 tan hats or a black and tanhat. If he saw 2 blacks hats, he would have known he had to have a tanhat. The middle man sees a tan hat because if he saw a black hat, hewould known he must be wearing a tan hat from the first response.Therefore the man facing the wall concludes he can only be wearing thetan hat the middle man sees.

page 218-the spider and the fly

FfiY

4oI-

3 - - - - - - - -32

| spider

J 2

:24


SOLUTIONSpage 227-the monkey and the coconuts

seventy-nine coconuts. Let n represent the amount of original coconuts.

# for monkey # each sailor hid for himself # left in pile1 (n-1)/3 (2n-2)/31 [(2n-5)/3]+3=(2n-5)/9 2(2n-5)/9=(4n-10)/91 [(4n-19)/9b-3=(4n-19)/27 2(4n-19)/27=(8n-38)/27

# each sailor took in morning1 [(8n-65)/27]+3=(8n-65)/81 0

Recall n=total original # of coconuts.(8n-65)/81=f, the number of coconuts each sailor received when thecoconuts were divided in the morning. Let f take one the varioussuccessive values of the counting numbers, starting with 1. Thesmallest counting number that gives a whole number value for n is f=7.With this value, n has value 79.

page 228-spider and spiralsNotice as the spiders move the size of the square they form shrinks, butit always remains a square. Each spider's path is perpendicular to thespider's path on its right. A spider will reach the spider on its right inthe same time it would take if the spider on the right had not moved.Each spider will have traveled 6 meters, which is 600 centimeters. 600centimeters will have taken 600 seconds or 10 minutes.


It would not be possible to form a complete bibliography forThe Joy of Mathematics or for More Joy of Mathematics. Even listingall the books in my library would not suffice, for many sectionswere written over a span of 25 years. Hence, this condensedbibliography-

Alic, Margaret. HYPATIA HERITAGE, Beacon Press, Boston, 1986.Asimov, Isaac. ASIMOV ON NUMBERS,242, Pocket Books, New York,1978.Bakst, Aaron. MATHEMATICS ITS MAGIC & MASTERY, D. Van Nostrand Co., New York, 1952.Ball W.W.Rouse and Coxeter, H.S.M. MATHEMATICAL RECREATIONS & ESSAYS, 13th ed., Dover

Publications, Inc., New York, 1973.Ball, W.W.Rouse. A SHORT ACCOUNT OF THE HISTORY OF MATHEMATICS, Dover Publications, Inc., New

York, 1960.Banchoff, Thomas F. BEYOND THE THIRD DIMENSION, Scientilc American Lbrary ,NewYork, 1990.Barnsaey,Michael. FRACTALSEVERYWHERE, AcademicPress,lnc., Boston, 1988.Beckman, Petr. A HISTORY OFi, St. Martirs Press, New York, 1971.Beier, Albert H. RECREATIONS IN THE THEORY OF NUMBERS, Dover Publications, Inc.,NewYork,1964.Bell, E.T. MATHEMATICSOUEEN & SERVANTOFSCIENCE,McGraw-HilI Book Co., Inc., NewYork,1951.BeI, E.T. MEN OF MATHEMATICS, Simon & Schuster, New York, 1965.Bell, R.C. BOARD AND TABLE GAMES FROM MANY CIVILIZATIONS Dover Publications, Inc.,

NewYork, 1979.Bell, R.C. OLD BOARD GAMES, Shire Publications Ltd., Bucks, U.K., 1980.Benjamin Bold. FAMOUS PROBLEMS OF GEOMETRY & HOW TO SOLVE THEM, Dover, Publications, Inc,

New York, 1969.Bergamini, David. MATHEMATICS, Time Inc., New York, 1963.Boyer, Carl B. A HISTORY OF MATHEMATICS, Princeton University Press, Princeton, 1985.Brooke, Maxey. COIN GAMES & PUZZLES, Dover Publications, Inc., New York, 1963.Bunch, Bryan H. MATHEMATICAL FALLACIES & PARADOXES, Van Nostrand Reinhold Co.,

New York, 1982.Campbell, Douglas and Higgins, John C. MATHEMATICS-PEOPLE, PROBLEMS, RESULTS,

3 volumes Wadsworth International, Belmont 1964Chadwick, John. READING THE PAST-LINEAR B AND RELATED SCRIPTS, University of Callomia

Press, Berkeley, 1987.Clark, Frank. CONTEMPORARY MATH, Frarklin Watts, Inc., New York, 1964.Cook, Theodore Andrea. THE CURVES OF LIFE, Dover Publications, Inc., New York,1979.Davis, Philip J. and Hersh, Reuben. THE MATHEMATICAL EXPERIENCE, Houghton Miflin Co., Boston,

1981.Detft, Peter van and Botermrans, Jack. CREATIVE PUZZLES OF THE WORLD, Harry N. Abrams,

Inc., NewYork, 1978.Doczi, Gy6rgy. THE POWER OF LIMITS, Shambhala Publications, Boulder, CO, 1981.Edwards, Edward B. PATrERN AND DESIGN WITH DYNAMIC SYMMETRY Dover Publications,

Inc., NewYork, 1967.Ellis,Keih. NUMBERPOWER, St.Martin'sPress NewYork, 1978.Emmet,E.R. PUZZLESFORPLEASURE, BellPublishingCo., NewYork, 1972.Engel,Peter. FOLDINGTHEUNIVERSE, Vintage Books, New York, 1989.Emst, Bruno. THE MAGIC MIRROR OF M.C.ESCHER, Ballantine Books, New York,1976.Eves, Howard W. IN MATHEMATICAL CIRCLES, two volumes, Prinde, Weber & Schmidt, Inc.,

Boston 1969.Filoiak, ANthony S. MATHEMATICAL PUZZLES Bell Publishing co., New York, 1978.Fixx, James. GAMES FOR THE SUPER INTELLIGENT, Doubeday & Co., Inc., New York, 1972.Fixx, James. MORE GAMES FOR THE SUPER-INTELLIGENT, Doubleday & Co., Inc.,

New York, 1972.Gamow, George. ONE, TWO, THREE ... INFINITY, Vking Press, New York, 1947.Gardner, Martin. PERPLEXING PUZZLES & TANTALIZING TEASERS Dover Publications, Inc.,

NewYork, 1977.


Gardner, Martin. THE UNEXPECTED HANGING, Simon & Schuster, Inc., New York,1969.Gardner, Martin. CODES, CIPHERS & SECRET WRITING, Dover Publications, Inc., New York, 1972.Gardner, Martin. MATHEMATICS MAGIC & MYSTERY, Dover Publications, Inc., New York, 1956.Gardner, Martin. NEW MATHEMATICAL DIVERSIONS FROM SCIENTIFIC AMERICAN, Simon &

Schuster, New York, 1966.Gardner, Main. MATHEMATICALCARNIVAL AlfredA.Knopf, NewYork, 1975.Gardner, Martin. MATHEMATICAL MAGIC SHOW Alfred A. Knopf, New York, 1977.Gardner, Martin. MATHEMATICAL CIRCUS, Alfred A. Knopf, New York, 1979.Gardner, Martin. MARTIN GARDNER'S SIXTH BOOK OF MATHEMATICAL DIVERSIONS FROM SCIENTIFIC

AMERICAN, UniversiyofChicagoPress, Chicago, 1983.Gardner,Martin THENEWAMBIDEXTROUSUNIVERSE, W.H.Freeman,NewYork, 1990GardnerManin. PENROSETILESTOTRAPDOORCIPHERS, W.H.Freeman&Co., New

York, 1988.Gardner, Martin KNOTTED DOUGHNUTS & OTHER MATHEMATICAL ENTERTAINMENTS, W.H.Freeman &

Co., New York, 1986.Gardner, Martin. TIME TRAVEL W.H.Freeman & Co., New York, 1987.Gardner, Martin. WHEELS, LIFE AND OTHER MATHEMATICAL AMUSEMENTS,

W.H.Freeman & Co., New York, 1983.Gardner, Martin. THE INCREDIBLE DR. MATRIX, Charles Scrbners Sons, New York, 1976.Gardner, Martin. THE 2ND SCIENTIFIC AMERICAN BOOK OF MATHEMATICAL PUZZLES & DIVERSIONS,

Simon & Schuster, New York, 1961.Gardner, Martin. THE SCIENTIFIC AMERICAN BOOK OF MATHEMATICAL PUZZLES & DIVERSIONS,

Simon & Schuster, New York, 1959.Ghyka, Mafila. THE GEOMETRY OF ART & LIFE, Dover Publications, Inc., New York, 1977.Gleick,James. CHAOS, PenquinBooks, NewYork, 1987.Glenn, William H. and Johnson, Donovan A. INVITATION TO MATHEMATICS, Doubleday & Co.,

Inc., Garden City, 1961.Glenn, William H. and Johnson, Donovan A EXPLORING MATHEMATICS ON YOUR OWN, Doubleday &

Co., Inc., GardenCity, 1949.Gobs, ElleryB. FOUNDATIONS OF EUCLIDEAN AND NON-EUCLIDEAN GEOMETRY, Holt, Rinehart and

Winston Inc., NewYork, 1968.Graham, LA. INGENIOUS MATHEMATICAL PROBLEMS & METHODS, Dover Publicatbons, Inc., New

York, 1959.Greenburg, Marvin Jay. EUCLIDEAN AND NON-EUCLIDEAN GEOMETRIES, W.H. Freeman & Co., New

York, 1973.GrOnbaum, Branko and Shephard, G.C. TILINGS AND PATTERNS, W.H. Freeman & Co., New

York, 1987.Gurfeld, Fredric V. GAMES OF THE WORLD, Hok, Rinehart & Winston, New York, 1975.Hambnridge, Jay. THE ELEMENTS OF DYNAMIC SYMMETRY, Dover Pubkcations, Inc., New York, 1953.Hawkins, Gerald S. MINDSTEPS TO THE COSMOS, Harper& Row, Publishers,New York, 1983.Heath, Royal Vale. MATH-E-MAGIC, Dover Publicatbons, Inc., New York, 1953.Hernick Richard, editor. THE LEWIS CARROLL BOOK, Tudor Putlishing, Co., New York,1944.Hoffman,Paul. ARCHIMEDESPREVENGE, W.W.Norton&Co., NewYork, 1988.Hoggatt, Vemer E., Jr. FIBONACCI & LUCAS NUMBERS, Houghton Mifflin Co. Boston, 1969.Hollingdale, 9uart. MAKERS OF MATHEMATICS, Penquin Books, London, 1989.Hunter, J.A.H. and Madachy, Joseph S. MATHEMATICAL DIVERSIONS, Dover Pubicatbons, Inc.,

New York, 1975.Huntley, H.E. THE DIVINE PROPORTION, Dover Publications, Inc., New York, 1970.Hyman, Anthony. CHARLES BABBAGE, Princeton University Press, Princeton, 1982.Ntrah,George. FROMONETOZERO, VikingPenquininc., NewYork, 1985.Ivins, William M. ART & GEOMETRY Dover Publications, Inc., New York, 1946.Jones, Madeline. THE MYSTERIOUS FLEXAGONS, Crown Pulishers, Inc., New York, 1966.Kaplan, Phil4. POSERS, Harper& Row, New York, 1963.Kaplan, Philp. MORE POSERS, Harper& Row, NewrYork, 1964.Kasner, Edward & Newman, James. MATHEMATICS AND THE IMAGINATION, Simon & Schuster, New

York, 1940.Kim,Scon. INVERSIONS, W.H.FreemanandCo., NewYork, 1981.


Kline, Morris. MATHEMATICS AND THE PHYSICAL WORLD, Thomas Y. Crowell Co., New Yotk, 1959.Kline, Morris. MATHEMATICS-THE LOSS OF CERTAINTY, Oxford Universfty Press, New York, 1980.Kline, Morris. MATHEMATICAL THOUGHT FROM ANCIENT TO MODERN TIMES, 3 volumes Oxford

University Press, New York, 1972.Kralchk, Maurice. MATHEMATICAL RECREATIONS, Dover Publicatons, Inc., New York, 1953.Lamb, Sydney. MATHEMATICAL GAMES PUZZLES & FALLACIES, Raco Publishing Co., Inc., New

York, 1977.Lang, Roberl. THE COMPLETE BOOK OF ORIGAMI, Dover Publirations, Inc., New York, 1988.Leapfrogs. CURVES, Leapfrogs, Cambridge, 1982.Lrnn, Charies F., edItor. THE AGES OF MATHEMATICS, 4 volumes, DouLbeday & Co., New York,1977.Locker, J.L, edhor. M.C.ESCHER, Harry N. Abrams, Inc., New York, 1982.Loyd, Sam. THE EIGHTH BOOK OF TAN, Dover Publications, Inc., New York, 1968.Loyd, Sam. CYCLOPEDIA OF PUZZLES, The Momingside Press, New York, 1914.Luckiesh,M. VISUALILLUSIONS, DoverPublications,Ic., New York, 1965.Madachy, Joseph S. MADACHY'S MATHEMATICAL RECREATIONS, Dover Publications, nc.,

New York, 1979..McLoughfin Bros. THE MAGIC MIRROR, Dover Publications, Inc., New York, 1979.Menninger, K.W. MATHEMATICS IN YOUR WORLD, Viking Press, New York, 1962.Montroll, John. ORIGAMI FOR THE ENTHUSIAST, Dover Publications, nc., New York, 1979.Moran, Jim. THE WONDEROUS WORLD OF MAGIC SQUARES, Vintage Books, New York, 1982.Neugebauer, O. THE EXACT SCIENCES IN ANTIQUITY, Dover Publications, Inc., New Yok, 1969.Newman, James. THE WORLD OF MATHEMATICS, 4 volumes, Simon & Schuster, New York, 1956.Ogilvy, C. Stanley and Anderson, John T. EXCURSIONS IN NUMBER THEORY, Dover Publications,

Inc., NewYork, 1966.Ogilvy, Stanley C. and Anderson, John T. EXCURSIONS IN NUMBER THEORY, Oxford Univershy

Press, New York 1966.Osen, Lynn, M. WOMEN IN MATHEMATICS, The MIT Press, Cambridge, 1984.Peat, F. David. SUPERSTRINGS AND THE SEARCH FOR THE THEORY OF EVERYTHING, Contemporary

Books, Chicago, 1988.Pedoe, Dan. GEOMETRY AND THE VSUAL ARTS, Dover Publications, Inc., New York, 1976.Peri, Teri. MATH EQUALS Addison-Wesley Publishing Co., Menlo Park, 1978.Peterson,tvars. THEMATHEMATICALTOURIST, W.H.Freeman&Co., New York, 1988.Pickover, Clifford, A. COMPUTERS, PATTERN, CHAOS & BEAUTYS Marin's Press, New York, 1990.Ransom, William R. FAMOUS GEOMETRIES, J. Weston Walch, Portland, 1959.Ransom, William R. CAN & CANT IN GEOMETRY, J. Weston Walch, Portland, 1960.Rodgers, James T. STORY OF MATHEMATICS FOR YOUNG PEOPLE, Pantheon Books, New York, 1966.Rosenberg, Nancy. HOW TO ENJOY MATHEMATICS WITH YOUR CHILD, Stein & Day, New York, 1970.Rucker, RudoR V. B. GEOMETRY, RELATIVITY &THE FOURTH DIMENSION, Dover Publications

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MATHEMATICS, W.H.Freeman&Co., NewYork, 1988.SteenLynnAuthur,ed. MATHEMATICSTODAY, VintageBooks, NewYork, 1980.Stevens, PeterS., PATTERNS IN NATURE, L111b, Brown and Co., Boston, 1974.Stokes, William T. NOTABLE NUMBERS, Stokes Publishing, Los Altos, 1986.Storme, Peter and Sryle, Paul. HOW TO TORTURE YOUR FRIENDS, Simon & Schuster, New York,1941.Struik, Dirk J. A CONCISE HISTORY OF MATHEMATICS, Dover Publications, Inc., New York, 1967.Warden,B.L.vander. SCIENCE AWAKENING, ScienceEditions, NewYork, 1963.Weyl, Hermann. SYMMETRY, Princeton University Press, Princeton, 1952.


INDEXAabacus, 209A-khowavizmi, 2anatomy and mathematics, 32-33Apollonius, 151Archimedes, 89, 198

Archimedean polyhedra, 122Arcrihimedean screw, 198

architecture and mathematics, 170-171Aristotle's wheel paradox, 202Aztec calendars, 128- 129

BBabbage, Charles, 176-177Babylonians, 2,4

cuneiform text, 148base ten, 2, 3, 24, 25billiards, 42binary base system, 24-25binomial formula, 40,41, 185, 186Bragdon, Claude F and magic lines,

169

Ccalculus, 138-139calligraphy and mathematics, 16Cantor, George, 157-158cardinality, 156-157Carroll, Lewis, 58-59

Alice in Wonderland, 58Alice Through the Looking Glass,

58A Tangled Tale, 58

catenary curve, 34Cesaro curve, 79circumference, 68computers, 24-25, 176-177, 206

Charles Babbage, 176-177nanoseconds, 80graphics, 5

conchoid of Nicomedes, 94-95conic sections, 196-197counting, 24-25

fingers, 60crystals, 38, 141cycloids, 6-8

curtate cycloid, 8

Dda Vinci, Leonardo, 32-33, 55, 81,

103decimal fraction, 2, 3dimensions, 206Diophantus, 123, 227Dodgson, Charles Lutwidge, 58-59Dudeney, Henry Ernest, 9, 218

duplicating a cube, 94, 130-132, 197Ddrer, Albrecht, 16, 67, 103dynamic symmetry, 103, 154-155

Eearthquakes and logarithms, 20-21electricity and mathematics, 24-25electron's path, 43Elements, The, 150equilateral triangle, trisecting, 174Eratosthenes, 215

earth's measurement, 215Escher, M.C., 122Euclid, 150Euclidean geometry, 150Euler, 124-125

FFibonacci (Leonardo da Pisa), 28-29Fibonacci sequence, 28-29,40, 41, 51,

87, 106, 191and a geometric fallacy, 191and the golden ratio, 106and nature, 222-225and the Pascal triangle and the

binomial formula, 40,41and a special magic square, 87and a trick, 51

finite, 213flexagons, tritetra, 107four color map problem, 152-153fractals, 78-79

GGalileo, 202game, mathematical topo, 26-27Gauss, Carl Friedrich, 164geodesic dome of Leorando da Vinci,

81geometry,

and designs of Leonard da Vinci, 63and electron's path, 43Euclidean, 150fallacy, 191and Gothic architecture, 63of nature, 78projective and art, 66-67

Ghobar numeral, 2,3Golden Gate Bridge and mathematics,

34golden rectangle, 29, 102-106, 115,

183golden ratio, mean, section,

proportion, 29, 32-33, 66, 102,183, 188-189, 225

golden triangle, 188-189


INDEXGothic architecture and mathematics,

63googol, 76googolplex, 76Great Pyramid and Thales, 36

HHalley's comet, 10-12,197Halley, Edmund, 10helix, 166-168Heron and Heron's Theorem, 62hexaminoes, 221hexagons in nature, 74-75hexagram, 118Hilbert, David, 9Hippocrates of Chios, 72Hindu numerals, 2, 3holograms, 206hyperbolic parabolid, 170-171hypercube, 204-206Hyzer's optical illusion, 13

Iicosahedrons, 115infinity, 37, 68, 108-109, 213

hotel infinity, 37perimeter and area 134-135and limits, 180

involute curve, 187irrational numbers, 98-99

phi, 0, 102-106pi, t, 18-19

KKepler's solids, 113Klein bottle, 45-46Klein Felix, 45Keenigsberg Bridge Problem, 124

LLaplace, Pierre Simon, 185Leonardo da Vinci, 32-33, 55, 81, 103Liber Abaci, 28-29linear programming, 216logarithms and earthquakes, 20-21Loyd, Sam, 47, 218

puzzle, 47jockey & horse puzzles, 71

lunes, 72-73

Mmagic circle, Japanese, 164magic cube, 77magic "line", 169magic squares, 82-87, 112, 169

Benjamin Franklin's, 97

ancient Tibetan, 133a Chinese magic square, 179

mathematical symbols, 52-54mathematics and

architecture, 170-171art, 66-67, 154-155calligraphy, 16evolution of mathematical

ideas,150-151genetics, 166-168Moslem art, 178music, 142-145nature, 74-75Parthenon, 182-183soap bubbles, 219typography, 16weaving, 210

mazes, 59, 192-194Mersenne's number, 211Mesopotamians, 2Moebius strip, 44-46, 61

Augustus Moebius, 44"double" Moebius strip, 207

Mondrian, Pietter, 154Moslem art and mathematics, 178

NNapier, John, 64

Napier's bones, 64-65nanoseconds, 80Napoleon, Bonaparte, 57networks, 126-127Newton, Isaac, 138non-Euclidean geometries, 67

fractals, 78-79hyperbolic, 90

Poincare's model, 90-92projective, 67, 216-217topology, 207

Koenigsberg Bridge Problem, 125numbers,

irrational, 98-99Mersenne's, 211odd, 93pentagonal, 214prime, 100-101pyramidal, 93square, 93, 214transfinite, 156-158triangular, 214

number systems, 56

0optical illusion, 5, 13, 66, 114, 199

history of, 172-173Parthenon, 182-183


INDEXPpalindromes (numerical), 146paperfolding, mathematics of, 48-50Pappus, 30

Pappus'Theorem, 163parabola, parabolic, 22-23, 34parabolic reflector, 22paradox,

coin paradox, 220paradoxical curve, 208unexpected exam, 147wheel paradox of Aristotle, 202Zeno's, 116-117

Pascal, Blaise, 40, 118, 137Pascal triangle, 29,40-41, 88, 184-185Peano curve, 79pentagon, 188-189pentagram, 188Penrose, Roger, 13pi, n, 18-19Plato, 39Platonic solids, 39, 110-111Poincard, Henri, 90Poinsot solids, 113Polyhedra 38-39

regular polyhedra, 39positional notation, base value sys-tem, 2Postscript, 16prime numbers, 100-101probability, 18-19

and Pascal's triangle, 184-185proofs,

1=2, 140puzzles,

amazing track, 69cannon balls and pyramids, 93checkerboard problem, 136counterfeit coin puzzle, 181Diophantus' riddle, 123Koenigsberg Bridge Problem, 124logic problem, 159monkey and coconuts, 226-227nine coins, 163penny puzzle, 119Persian horses, 70Sam Loyd's, 47, 71spider and spirals, 228spider and fly, 218tangram, 212three men facing the wall, 190T problem, 35triangle to square, 9wheat and chessboard, 17wood, water and grain, 175

Pythagorean Theorem, 4, 30, 49Pythagoras, 4and irrational numbers, 98-99Pythagoreans, 102, 142and President Garfield, 200-201

Qquipu, 14-15

Inca, 14-15

SShroeder's staircase, 5snowflake curve, 79, 160-161space-filling curve, 208spirals

equiangular, 105, 189, 228logarithmic, 105Archimedes', 149

spherical dome, 165squaring a circle, 72, 94, 130-132, 197Stonehenge, 203symmetry, 141, 155systems of equations, 216-217

Chinese "checkerboards", 195

Ttangram, 212tessellations, 120-122Thales, 36topo, 26-27topology, 31, 96, 107, 124-125, 152-

153, 207networks, 126-127four color map problem, 152-153

transfinite numbers, 156-157trefoil knot, 96tribar, 13trisecting and angle, 94, 130-132, 197twistors, 13typography and mathematics, 16

zzero, 2, 163Zeno's paradox, 116-117

About The Author

Mathematics teacher and consultant Theoni Pappas received herB.A. from the University of California at Berkeley in 1966 andher M.A. from Stanford University in 1967. She is committed todemystifying mathematics and to helping eliminate the elitismand fear that are often associated with it.

In addition to The Joy of Mathematics, her other innovativecreations include The Mathematics Calendar, The Children'sMathematics Calendar, The Mathematics Engagement Calendar, TheMath-T-Shirt, and WAhat Do You See?-an optical illusion slideshow with text. Pappas is also the author of the following books:More Joy of Mathematics, Math Talk, Mathematics Appreciation,Greek Cooking for Everyone, Fractals, Googols and OtherMathematical Tales, and The Magic of Mathematics.

Mathematics Titlesby Theoni Pappas

THE MAGIC OF MATHEMATICS$10.95 * 224 pageseavailable Spring'94

illustrated *ISBN:0-933174-99-3

FRACTALS, GOOGOL, and OtherMathematical Tales?

$9.95 - 64 pages * for all agesillustrated a ISBN:0-933174-89-6

THE JOY OF MATHEMATICS$10.95 * 256 pages

illustratedeISBN:0-933174-65-9

MORE JOY OF MATHEMATICS$10.95 * 306 pages

cross indexed with The Joy of Mathematicsillustrated * ISBN:0-933174-73-X

MATHEMATICS APPRECIATION$10.95 * 156 pages

illustrated * ISBN:0-933174-28-4

MATH TALKmathematical ideas in poems for two voices

$8.95 * 72 pagesillustrated - ISBN:0-933174-74-8

THE MATHEMATICS CALENDAR$8.95 * 32 pages * written annually

illustrated

THE CHILDREN'S MATHEMATICS CALENDAR$8.95 * 32pages * written annually

illustrated

WHAT DO YOU SEE?An Optical Illusion Slide Show with Text

$29.95 * 40 slides * 32 pagesillustrated * ISBN:0-933174-78-0

"The Joy of Mathematics is designed to help the reader become aware ofthe inseparable relationship of mathematics and the world by presentingglimpses and images of mathematics in the many facets of our lives."

-Science News

"Rarely does one come across books on mathematics with such breadth,beauty, and insight. Pappas' books always contain fascinating andvaluable information - not only for curious students and laypeople - butalso for the seasoned researcher. The eclectic Pappas is both a mathemati-cian and a poet, and with the cold logic of the one and the inspired vision ofthe other, she selects topics sure to stimulate readers' imagination andsense of wonder at the incredible vastness of our mathematical universe."

-Clifford A. Pickover, author of Computers & the Imagination

"The Joy of Mathematics presents the world of mathematics to readers ina fresh, original way. For any reader, mathematician or not... easilyreadable by all... readily usable for enrichment . ..I highly recommend it."

- The Mathematics Teacher

"The Joy of Mathematics is not just a mechanical reiteration of concepts:it explores the history of both concept and inventor, presents unusualexercises which challenge the mind to absorb not just the theory, but itspractical application; and peppers its presentation with an abundance ofblack and white drawings which explore math relationships to science andhuman endeavors. This approach goes far beyond the usual mechanicalpresentation, linking math to vital, everyday life and adding cultural andhistorical influences to connect math concepts to realistic challenges. Thetips, exercises and concepts are fun rather than tedious. "

-The Bookwatch

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