X e V e X

Editors

Michael RosenbergBenjamin Kaplan

Faculty Adviser

Rabbi Ely Stern

Contributors

Gabrielle Amar-OuimetSarah Ascherman

Jacob BermanHadassah Brenner

Ma hew HirschfeldBenjamin Kaplan

Henry KofflerMax Koffler

Jasmine LevineTyler Mandelbaum

DJ PresserMichael Rosenberg

Tess Solomon

Eratosthenes: Jack of all TradesJacob Berman '16

Eratosthenes of Cyrene, an ancient Greek mathe-matician, is best known for being the first person tomeasure the earth’s circumference. As well as be-ing a great mathematician, Eratosthenes was alsoa well known geographer, astronomer and histo-rian. Despite being very knowledgeable, his Greekcolleagues scorned him and called him Beta, rep-resenting the second letter of the Greek Alphabet.This meant to say that althoughEratosthenes was involved in ev-erything, he always was the sec-ond best. Even ChristopherColumbus studied Eratosthenes’smeasurements before he set offon his voyage to the West. Eratos-thenes, a man well ahead of histime, laid a strong foundation inmathematics, astronomy, history,and geography.

Eratosthenes was born inCyrene, Greece, which is nowpartof Libya, in 276 bce As a youngman he studied in Athens, and soon made a namefor himself in many fields. Eventually, he madesuch a name for himself that he caught the atten-tion of the ruler of Egypt, Ptolemy III. PtolemyIII invited Eratosthenes to Alexandria to both tu-tor his son and become the head librarian of thegreat Alexandrian University. Eratosthenes tookhis chance and, in Alexandria, he learned and dis-cussed with many other great scholars. Eratos-thenes died in 195 bce in Alexandria by suppos-edly starving himself due to the fact that he wasblind and could no longer work.

In both mathematics and astronomy, Eratos-thenes had his most lasting discoveries. Eratos-thenes is best known for accurately measuring thecircumference of the world. He did this by ob-serving that the sun shone directly down at a wellin Syene at the same time that it cast a shadow

in Alexandria, directly south of the well. By un-derstanding this, Eratosthenes measured the an-gle of shadow to earth and then was able to cal-culate the circumference of earth. He found thecircumference to be nearly 250,000 stadia, whichis approximately 25,000 miles. All of these com-plex calculations left him with only an astonishing.16% error! Eratosthenes went on to draw 675 star

diagrams, and was even believedto have created a calendar includ-ing leap years.

Eratosthenes also made greatstrides as both a historian and ageographer. As a historian, Er-atosthenes worked to better or-ganize the library of Alexandria.He was the founder of scientificchronology, which helped figureout the exact time that the con-quest of Troy took place. As a ge-ographer, he created the first mapof the world. Using the primitive

geographical knowledge of the era, he made a mapthat include parallels and meridians. This map wasthe foundation for many of the terms in mappingthat we use today.

Eratosthenes, although not so well known, hada huge impact on the world today. Seventeen cen-turies after Eratosthenes had died, a Spanish ex-plorer by the name of Christopher Columbus wasstudying Eratosthenes’s calculations about the cir-cumference of the world. If Columbus had notdisregarded Eratosthenes’s exact calculations, hewould have realized that he had not reached India,but rather was in a new continent. Eratosthenes isunlike other famous mathematicians, because un-like the rest, he isn’t overwhelmingly famous forone specific discovery, but rather he was known asa jack of all trades.

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My Respective Numbers of the MoonHadassah Brenner '18

I study by night.The moon in its perfection.My respective numbers of the moon.Are amongst my observation.

Four times totalIt waxes and it wane.sAppearing and diminishingInto the dark’s shadowy vein.s

Twice the whole of itShines bright in the sky.Gleaming in its splendorBy the by.

Dividing itselfIn the first and third quarter.The poor moon grievesAt the severing of its grandeur.

Twelve cyclesIt performs per year.Repeating the processIt holds so dear.

On 354.37 nightsI gape at its lure.Sometimes crisp and clearSometimes glowing murky and im-pure.

On quite a few occasionsThree to be exact.The moon blotted out its advisoryThe solar light doused and compact.

And so here I recordNever did it vanish completely.I studied it night by nightAs it embarked upon its journey.

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Biography on Grigori PerelmanMax Koffler '16

Griogori Perelman was born to Jewish parentsin Leningrad, Soviet Union, in the midst of theCold War in 1966. His mathematical brilliancebecame apparent at age 10, when he was en-rolled in an after-school math program ran bygenius Sergei Rukshin. Instead of teaching inRussian, as all patriotic teachers should, Ruk-shin taught Perelman only in English so thathe would be able to participate in internationalcompetitions that were conducted in English.This gave Perelman a sig-nificant advantage over hispeers. The small group ofRukshin’s students, followingtheir mentor’s advice, attendedLeningrad’s Special Mathemat-ics and Physics School Number239, where they were all placedin the same class. After gradua-tion, many of them participatedin the 1982 InternationalMathematical Olympiad,but only onemember of the Soviet teamwouldwinthe first-place goldmedal—Grigori Perelman, witha perfect score. His gold medal automatically gavehim entry into a top Soviet university, and he choseLeningrad State University—perhaps the Harvardequivalence in the USSR. In his undergrad yearshe had remarkable success and publishedmany pa-pers, including Realization of abstract k-skeletonsas k-skeletons of intersections of convex polyhedra inR2k−1. Despite his achievements, he was unableto immediately enroll in a graduate program at theSteklov Mathematics Institute, as the Institute hada strict no-Jew policy. Aleksandr Danilovic Alek-sadrov, an old comrade and mentor of Perelman,wrote to the director requesting that Perelman be

allowed to undertake graduate work under his su-pervision. The request, highly unusual comingfrom someone of Aleksandrov’s high standing, wasgranted. Perelman successfully defended his thesis,Saddle Surfaces in Euclidean Spaces in 1990.

After the fall of the iron curtain, Perelman trav-eled to the US and lectured at prestigious uni-verses, among them Princeton, MIT, and Har-vard. Perelman was best known for his work incomparison theorems in Riemannian geometry.

Among his notable achieve-ments was a short and elegantproof of the “soul conjecture”.In 1999, the Clay Mathemat-ics Institute announced it wouldaward $1,000,000 to any personwho could prove the “Poincaréconjecture.” Perelman spentmany years working on this, andpublished his proof in 2003. In

May 2006, a committee of nine mathematiciansvoted to award Perelman a Fields Medal for hiswork on the Poincaré conjecture. However, Perel-man declined to accept the prize, making him thefirst and only person to ever decline this esteemedaward. He also declined to accept the million dol-lar prize. Perelman was quoted as saying that“theprize is completely irrelevant for me. Everybodyunderstood that if the proof is correct, then noother recognition is needed . . . I’m not interestedinmoney or fame, I don’t want to be on display likean animal in a zoo.” His parents later immigratedto Israel and his sister, a scientist, received her PhDfrom the Weizmann Institute of Science. Perelmanstill tours the world, teaching his brilliance at uni-versities.

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Proving Fermat's Last TheoremTess Solomon '16

Pierre de Fermat (“fur-mah”), a French lawyer, pro-posed Fermats Last Theorem in 1637. It was dis-covered posthumously in the margin of a copy ofArithmetica, an ancient Greek text on mathemat-ics. There Fermat claimed that he had a proof, butthat it was too large to fit in the margin. A prooffor his theoremwould frustratemathematicians for358 years.

The theorem itself is not hard to understand. Itsimply states that no three positive integers a, b, ccan satisfy the equation an + bn = cn for any in-teger value of n greater than two. This equation isbased on the Pythagorean Theorem, which statesthat when a, b are the lengths of the legs of a righttriangle and c is the hypotenuse, then they mustsatisfy the equation a2 + b2 = c2. This particu-lar equation has infinitely many solutions (for in-finitely many right triangles).

Fermat’s LastTheorembecameoneof themostfamous unsolved problems in mathematics. In the300 years after the theorem was proposed, numer-ous incomplete proofs (for example, a proof thatthe theorem held true for n = 3, n = 5, andn = 7) were offered.

In 1984, Gerhard Frey noticed a link betweenthe “Modularity Theorem” and Fermats Last The-orem. Andrew Wiles, an English mathematician,who was 31 at the time, decided to try to prove

theModularityTheorem in order to prove Fermat’sLast Theorem. Although a proof was consideredimpossible by contemporaneous mathematicians,Wilesworkedon it secretly for six years. In 1993, heannounced his findings at a lecture in Cambridge.

Wiles’ proof of Fermat’s LastTheorem receivedworldwide press. As Wiles completed the proof,mathematicians in the lecture hall snappedpicturesof the historic moment, reported the New YorkTimes article from June 24 1993, titled “At Last,Shout of ‘Eureka!’ In Age-Old Math Mystery.” Dr.KennethRibet, a professor at theUniversity ofCal-ifornia at Berkeley, is quoted in the article as saying,“The mathematical landscape has changed. Youdiscover that things that seemedcompletely impos-sible aremore of a reality. This changes theway youapproach problems, what you think is possible.”

An error was found in a peer review of Wiles’paper in September 1993. On September 19, 1994Wiles realized the way to correct his proof. He de-scribed this moment as, “The most important mo-ment of my working life. [It was] so indescribablybeautiful . . . so simple and so elegant.” Wiles’ proofis over 150 pages long. The proof was described asone of the highest achievements in number theory,and as the proof of the century. Wiles received nu-merous awards for his discovery, including beingknighted by the Queen of England.

Mathematical SavantsJacob Berman '16

Over the course of many centuries, a phenomenonhas been observed in which a person with a mentaldisability develops specific prodigious abilities farin excess of what is considered normal. These peo-ple are known as savants, and the triggering eventscan occur from birth, or after a specific accident.The specific ability that the savant excels in canvary, but this article is going to specifically focus onmathematical savants. The causes for mathemati-

cal savant syndrome are largely unknown, althoughthere is some speculation as to why they occur.There are many interesting savants that should beanalyzed, including Jason Padgett and the most fa-mous and unusual of mathematical savants, DanielTammet.

There is no widely accepted cognitive theoryto explain savant syndrome, but there are cer-tainly many different theories. Around half of all

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cases of savant syndrome occur in people withautism. Individuals with autism are often verydetail-oriented, so maybe the way savants developis that they spend so much time focused on onespecific thing. Another hypothesis is that savantshyper-systematize, thereby giving an impression oftalent. There has also been research that provesthat savants access regions of the brain generallyunavailable or untapped by most normal people. Itis also significant to note that the ratio from malesavants to female savants is 6:1, slightly higher thanthe ratio of male autistic people to female autisticpeople, which is 4.3:1.

There are different types of savants, the ma-jority are those who are born with savant syn-drome and the minority are those who becomesavants after a specific incident later in life. Oneexample of an “acquired” mathematical savant isJason Padgett. In 2002, something happenedlate at night that would change Padgett’s life for-ever. He was outside a karaoke bar, when hewas hit in the head and mugged. He soon no-ticed his vision start to change. He saw thewater coming from the faucet as perpendicular.He soon started studying math-ematics and physics like he doestoday, and has an amazing apti-tude for drawing fractals. Pad-gett thought that he was alone,until he found out about a morepublicized savant, Daniel Tam-met.

Daniel Tammet, an autisticsavant, has always had an innateway with numbers. Tammet has

written three books, where hedescribes his feeling towards numbers. Each num-ber from 1 to 10,000 has its own feel and look toTammet. He can reportedly “feel” prime numbers,and holds the world record for most digits of piever recited—22,514. Tammet says that he hasn’tmemorizedanything, but rather thedigits of pi sim-ply “come” to him. Tammet also has other talents,which gives him the ability to know eleven differ-ent languages and learn Icelandic in a week. Butthemain difference about Tammet is that he is ableto articulate what exactly he sees or feels differentto others. Most savants have severe mental dis-abilities, but Tammet’s autism isn’t so severe. Re-searchers have been studying Tammet, and are stillsearching for ways to figure out what exactly makesthese savants different.

Being a savant can be viewed both as a verygood thing or a very bad thing. It would be amaz-ing tohave a certain skill, but then again, that comeswith being mentally disabled in something else.Scientists still haven’t developed a strong theory asto how these savants develop. Hopefully DanielTammet will be the key to unlocking this secret.

As Professor Allan Snyder, fromthe Centre for the Mind at theAustralian National Universityin Canberra, explains, “Savantscan’t usually tell us how they dowhat they do,” says Snyder. “Itjust comes to them. Daniel can.He describes what he sees in hishead. That’s why he’s exciting.He could be the Rosetta Stone.”

Biography of Bernhard RiemannSarah Ascherman '16

This past month in math class we learned how toapproximate the area under a curve using RiemannSums. Whichmathematician invented thismethodof approximation? Bernhard Riemann.

Riemannwas amathematicianwhowasborn in

1826 in Breselenz, Germany. He was interested innumber theory and differential geometry. From avery young age, Riemann showed a great interest inmathematics, and his calculation capabilities wereextraordinary for such a young boy. However, Rie-

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mann was very shy and afraid of speaking up andoften had nervous breakdowns. By the time Rie-mann was in high school, he surpassed the mathe-matical knowledge of all his teachers. Surprisingly,when Riemann was 19, he decided that we wantedto study theology and philology. Thankfully for thefuture of mathematics, one of Riemann’s teacherswho had seen his amazing potential in mathemat-ics discouragedRiemann fromwhat he had chosen,and successfully persuadedhim topursue studies in

mathematics. By the age of 29, Riemann was giv-ing mathematical lectures, which later became thebasis of Riemann geometry. Three years later Rie-mann became the head of the mathematics depart-ment of the University of Gottingen. Riemann es-tablished the basis for “modern mathematics,” andalso created the Riemann zeta function, and “Rie-mann hypotheses.” Riemann’s contributions werethus extensive and impressive!

The Math Behind Ludwig van Beethoven's GeniusGabrielle Amar-Ouimet '17

Although ironic, Ludwig Van Beethoven, an 18thcentury composer of some of the most eminentmusic created, spent most of his career going deaf.So, despite his inability to hear, how was he ableto create such intricate and moving compositions?The answers are hidden in the patterns beneath thebeautiful sounds he shaped within his works. Themathematics that Beethoven used in producingmusic can be seen through his Moonlight Sonata,which opens with notes, musical pitches, which aregrouped into triplets:

Though they seem simple, each triplet containsa melodic structure revealing the fascinating rela-tionship between music and math. Through math,Beethoven was able to picture in his mind what hewas composing, without having to physically hearit, and then follow its lines.

Notes are determined by their frequency,which is measured in vibrations per second, orHertz (Hz). The notes on a piano keyboard forma chromatic scale, which divides the octave intosemitones. There are twelve semitones, or halfsteps, to anoctave in the chromatic scale. Thewhitekeys on a keyboard are A, B, C, D, E, F, and G.The black keys are named relative to their adjacentwhite keys. For example, the black key between the

C andDkeys is known as eitherC♯ (C sharp) orD♭(D flat). The frequencies of the notes that make upa chromatic scale formageometric sequencewhereeach term has the form arn where a is frequency,r is some constant ratio, and n is the index of thesequence. The ratio that generates the chromaticscale is r = 21/12.

In the first half of the above measure (measurefifty ofMoonlight Sonata), for example, consists ofthree notes inDmajor, separatedby intervals calledthirds that skip over the next note in the scale. Byplaying the first (D), third (F♯) and fifth (A) notestogether at once, a harmonic pattern called a triad isproduced. The reason for this is that these numbersrepresent the mathematical relationship betweenthe pitch frequencies of different notes, which forma geometric series.

Beginning at the last note of the triad, A (440Hz), the series can be expressed as ar−7, ar−3, ar0,where a is the base frequency (A4was chosen here,so a = 440). By plugging these values as f in thebelow equation, the sound wave for each note canbe graphed, which would have allowed Beethovento see the patterns that he could barely hear. Whenall three waves are graphed, they intersect at theirstarting point of 0ms, and again at 6.8ms:

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Within this span, the D goes through two wavecycles, the F♯ through two and a half and the Athrough three. This pattern is known as conso-nance, the combination of notes that are in har-mony with each other due to the relationship be-tween their frequencies. The secret for creatingpleasing sounding note combinations is just this:frequencies whose wave functionsmatch up at reg-ular intervals.

A more mathematical way of viewing the rela-tionship between these notes is through the ratioof their frequencies. With perfect ratios, the per-fect sounds are created. For example, the ratio ofthe frequencies of note A and note D is just about3/2, the ratio of the frequencies of notes F♯ andD isnearly 5/4, and the ratio of the frequencies of notesA and F♯ is nearly 6/5. This means that every 3rdwave of Amatches upwith every 2ndwave ofD, ev-ery 5th wave of F♯matches up with every 4th wave

of D, and every 6th wave of A matches up with ev-ery 5th wave of F♯. Since every note’s frequencymatches up well with every other note’s frequency(at regular intervals), they all produce an audiblypleasing and satisfying sound.

In addition, Beethoven not only uses conso-nance in his pieces, but also dissonance, the lack ofharmony amongmusical notes. At one point in thepiece, Beethoven combines notes C and B, whosewaves are largely out of sync and distinguishes itwith a consonant triad.

By contrasting this dissonance and conso-nance, Beethoven creates the unquantifiable ele-ments of creativity and emotion to the certainty ofmathematics. Although we can understand the un-derlying patterns that become the structure behindthe music, the reason for why certain sounds strikeor evoke emotions in certain ways is still yet to bediscovered.

References

St. Clair, N. (2014, September 9). Natalya St. Clair: Musicand math: The Genius of Beethoven

Heimiller, Joseph. “WhereMathMeetsMusic.” MusicMas-ter Works, 2002.

http://illuminations.nctm.org/Lessons/SeeingMusic/Chromatic-AS-Hearing.pdf

StoicheometryMa hew Hirschfeld '17

As Ramaz’s sophomores learn each year in theirchemistry curriculum, stoichiometry is the calcula-tion of relative quantities of reactants and productsin chemical reactions. Stoichiometry is foundedonthe Law of Conservation ofMass, which states thatthe total mass of the reactants equals the total massof the products. Scientists have derived from thisthe insight that the relations among quantities ofreactants andproducts typically forma ratio of pos-itive integers. Thismeans that if the amounts of theseparate reactants are known, then the amount ofthe product can be calculated. Conversely, if one

reactant has a known quantity and the quantity ofproduct can be empirically determined, then theamount of the other reactants can also be calcu-lated.

Consider the balanced equation of the reac-tion that occurs as a result of the combustion ofmethane: CH4 + 2O2CO2 + 2H2O. Here, onemolecule of methane reacts with two molecules ofoxygen gas to yield onemolecule of carbon dioxideand two molecules of water. Stoichiometry is usedtomeasure these quantitative relationships and de-termine the amount of products or reactants that

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are produced or needed in a given reaction. Thus,in the example above, stoichiometry may be usedto measure the relationship between the methaneand oxygen as they react to form carbon dioxideand water. Because of the well-known relationshipof moles to atomic weights, the ratios that are ar-rived at can be further used to determine quantitiesby weight in a reaction, which is represented by abalanced equation. Gas stoichiometry deals withreactions involving gases, where the gases are at aknown temperature, pressure, and volume and can

be assumed to be ideal gases. For gases, the vol-ume ratio is ideally the same by the ideal gas law,but the mass ratio of a single reaction has to be cal-culated from the molecular masses of the reactantsand products.

In terms of etymology, the term stoichiometrywas first usedby JeremiasBenjaminRichter in 1792when the first volume of Richter’s Stoichiometry orthe Art ofMeasuring the Chemical Elementswas pub-lished. The term is derived from the Greek wordsstoicheion, “element”—andmetron, “measure”.

Flavors of RecursionMichael Rosenberg '15

Any programmer worth his salt knows what a re-cursive function is, but there are different types ofrecursive functions that may not be so popular orobvious. For the uninitiated, a recursive functionis defined byMerriamWebster as “a computer pro-gramming technique involving the use of a proce-dure, subroutine, function, or algorithm that callsitself one or more times until a specified conditionis met at which time the rest of each repetition isprocessed from the last one called to the first.” Thatwas a mouthful. In short, a recursive function is afunction that calls itself until it has reached its goal.As an example, here is an implementation of a fac-torial function in the simplest recursive form:

f(x) =

1, if x = 01, if x = 1x× f(x− 1)

To every recursive function there are twostates, it can either be in its winding state or its un-winding state. The winding state is defined as thestate when the function is still calling itself withinitself, increasing the depth. The unwinding state iswhen the function is returning from the recursivecalls. Here is the expansion of 5!:

Winding

f(5)5 × f(4)

5 × (4 × f(3))5 × (4 × (3 × f(2)))

5 × (4 × (3 × (2 × f(1))))

Unwinding

5 × (4 × (3 × (2 × 1)))5 × (4 × (3 × 2))

5 × (4 × 6)5 × 24

120

As you can see, all of the actual arithmetic ofmultiplication only happens in the unwinding partof the recursive process in this case. The details ofhow this algorithm is run on a computer ismore re-lated to computer science, but suffice it to say thatthis is a highly impractical algorithm for efficientcomputation. This inefficiency relates to the typeof recursion the algorithm uses.

The first improvement that can bemade to thisrecursive function is making it tail recursive. Tailrecursion simply means that the last operation per-formed before calling itself and increasing its depthmust be the function call itself, read: not multiply-ing a number by the result of that function call likeour previous function did. Changing the algorithm

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like this is strictly mathematically equivalent, how-ever it has a large impact on how efficiently it is runon a computer. A simple way to rewrite the func-tion to be tail recursive is to add something calledan accumulator to the parameters so that it can . . .accumulate the values as it is passed to each subse-quently deeper function call. Here it is:

f(x, a) =

a, if x = 0a, if x = 1f(x− 1, x× a)

Where x is the number to compute the factorial of and a is the accumulator(should be 1 to start)

The reason why such a form improves effi-ciency is that most computer execution environ-ments are able tomuchmore heavily optimize codeusing tail recursion as opposed tonon-tail recursivecode. The expansion of this function is almost themirror of the first one:

f(5, 1)f(4, 5)f(3, 20)

f(2, 60)f(1, 120)120

120120

120120

The reason I say it is, in a way, the mirror imageof the first function is because here, all the actualmultiplication is doneduring thewinding stage andit simply returns one after another as soon as it hitsthe bottom of the winding phase.

The last form of recursion I will discuss isreally an extension of tail recursion but with it’sown properties. This form is called continuation-passing style, or CPS, and it differs from the pre-vious two in that it takes in a function as a param-eter to call with the result of the factorial. So if I

write: g(x) = x + 1 and then call this new fac-torial function like f(5, g), it should call g with thevalue 120 and return 121. Instead of defining afunction that can be called by name, we will useanother method which allows us to call functionsanonymously. Using lambda syntax, we can definefunctions that are to be called by other functions.The identity function iswritten in this lambda formas λx . x where everything with a lambda next to itis a parameter and everything after the last dot iswhat the function does. For instance, if I were todefine a function f to only apply the given functiong to a given number n, it would be defined simplyas f(n, g) = g(n). This way, I can pass to it any ar-bitrary function to operate on n. So f(4, λx . x× 3)would evaluate to 12. This is how a continuation-passing style factorial function would be defined:

f(x, g) =

g(1), if x = 0g(1), if x = 1f(x− 1, λn . g(x× n))

I know this looks very complicated, but it’smuch easier to take in when you see the expansionof the function. We’re just going to use the identityfunction as the initial parameter so we can get justthe result of the factorial:

f(5, λx . x)f(4, λx . g(5 × x))

f(3, λx . g′(4 × x))f(2, λx . g′′(3 × x))

f(1, λx . g′′′(2 × x))g′′′′(1)

Where each subsequent prime added to the g refers to the lambda functiongiven to the function one-up and g refers to the first lambda functionwhichis the identity function.

Once the function finally hits 1, it calls itslambda function with 1 as the parameter and letsit unroll, here it is:

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g′′′′(1)g′′′(2 × 1)

g′′(3 × (2 × 1))g′(4 × (3 × (2 × 1)))

g(5 × (4 × (3 × (2 × 1))))g(120) = 120

This is not a very good example of the power

of CPS, but it’s mostly to show the form of it.CPS is essentially a tail recursive function wherethe accumulator has been generalized to be nestedfunctions instead of nested operations onnumbers.CPS is a powerful form that affords a programmerthe ability to explicitly define the path of his execu-tion and still be optimized because it is still a tailrecursive function.

Distance between the Earth and the SunHenry Koffler '18

The Greeks, who invented and defined geometry,trigonometry, geography and architecture, also es-tablished astronomy. The more the Greeks ob-served, the more questions arose. One issue of de-bate was the distance between the earth and thesun. The point in earth’s orbit that is closest to thesun is called the perihelion. The distance betweenthe earth’s perihelion and the sun is about 91 mil-lion miles. The point in earth’s orbit that is farthestfrom the sun is called the aphelion. The earth’saphelion is about 94.5 million miles away from thesun. The first person with a recorded attempt atcalculating the distance was Archimedes in the 3rdcentury bce. Besides building a working heat ray,this is one of his many accomplishments. Unfor-tunately, Archimedes’ estimate was wrong . . . bya lot. The next big astronomer who attempted tocrack the code was Cristiaan Huygens in 1659. Be-

ing a smart man, he correctly guessed (by chance)the size of Venus. Using this newfound knowl-edge, plus the phases of Venus in relation to Earthand the Sun, he found the angles in a Venus-Earth-Sun right triangle. Using the Pythagorean theo-rem, he was able to find the distance from Earthto the sun. However, due to the fact that Huy-gen’ method was partially guesswork and not com-pletely scientific, he usually does not get full creditfor his discoveries even though he got very closeto right answer. Although their answer was furtheraway from the actual measurement than Huygen’s,Cassini and Richer get the first scientific credit fortheir trigonometrical solution in 1672. Nowadays,scientists use space probes and radar as a more ac-curate methods of calculation for the distance be-tween the earth and the sun. The scientistsmeasurethe time it takes to bounce photons off the sun.

EulerBenjamin Kaplan '16

Leonard Euler was a world-renowned mathemati-cian who lived in the 18th century in Switzerland.He is credited with inventing lots of the notationthat we use now. Euler was born in 1707 in Basel,Switzerland. One of his early influences was afriend of the family, Johann Bernoulli, another fa-mousmathematician. Eulerwent on to enroll in theUniversity of Basel at the age of 13. His dissertationis considered to beon the same level as that ofNew-

ton and Descartes. Beside mathematics, Euler alsostudied theology, Greek, and Hebrew. Thanks tohis relationshipwithBernoulli, Euler received a po-sition in the mathematics department of the Impe-rial Russian Academy of Science. He later on con-tinued his education and research at the Universityof Berlin

At the Imperial Academy and the University ofBerlin, Eulermade someof hismost famousdiscov-

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eries and contributed greatly to the world of math-ematics. The first and most basic contribution isthe notation of a function as f(x). Other notationthat he created includes the capital sigma (

∑) for

summation, and the letter i for the unit imaginarynumber. The contribution that he is most notablefor is e as the base of the natural logarithm (ln). eis a constant that is equal to approximately 2.718,and is defined as the limit as limn→∞(1 + 1/n)n.e is later used in Euler’s identity which states thateiπ + 1 = 0. This is a result of De Moivre’s theo-rem.

Other fields that he studied in great detail in-clude number theory, graph, theory, and appliedmathematics.

Other notable achieve-ments of Euler in-clude winning the ParisAcademy Prize Competi-tion, solving the problemof the Seven Bridges ofKönigsberg, helping cre-ate the latitude table, andcontributing to the use ofmathematics in music the-ory. As a whole, Eulercontributed significantly to mathematics, and thatwhich he taught and discovered is still used todayby modern mathematicians for all different appli-cations.

Mathematical Card TricksMa hew Hirschfeld '17

One of the most childish, yet pride-instilling dis-plays of one’s cleverness is the performance of acard trick, outwitting and baffling all those in at-tendance. Additionally, card tricks are an amusingway tomake friends, becomemore familiarwith ac-quaintances, or to simply spend time with the peo-ple you love, especially during the holidays. Hereare twomathematical card tricks that are sure to im-press:

The21-Card Trick

1. Hand your friend a stack of 21 playing cards.

2. Instruct him to pick one out, without show-ingor telling youwhich cardhe chose, and toplace the card back into the stack at random.

3. Deal the cards out face-up in three columns,working row-by-row (1st column-2ndcolumn-3rd column, 1-2-3, 1-2-3, etc). Youshould have three columns of seven cards infront of you.

4. Have your friend tell youwhichpile containshis card (without telling youwhich card it is,of course).

5. Gather the three columns into one stack ofcards again. This time, be sure to put the pilethat holds your friend’s card in themiddle ofthe three piles. For example, if the first pilecontained his card, then you could pick upthe third pile first, then the first pile (con-taining the card), and then the secondpile—or the second pile, then the first, and thenthe third. It is very important that the pilecontaining his card goes into the middle.

6. Deal out the cards again, in the same 3-column, 7-row fashion as last time.

7. Have your friend tell youwhichpile containshis card.

8. Again, gather the three columns into onestack of cards, making sure to put the pilethat holds your friend’s card in themiddle ofthe three piles.

9. For the final time, deal out the cards in thesame way you have done thus far.

10. Have your friend tell youwhichpile containshis card.

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11. Gather the three columns again, paying spe-cial attention that you put the column withhis card in the middle of the three piles. Youshould have dealt the cards out a total of 3times.

12. Your friend’s card should be the 11th cardfrom the top. Do not flip the deck over, orelse the eleventh card will not be his card.

The “Spelling Out” Trick

1. Hold the outspread deck in front of you andask the viewer to select 9 cards.

2. Take the 9 cards and deal them out face-down into 3 piles of three.

3. Ask the viewer to choose a pile.

4. Flip the pile over, showing the viewer thecard on the bottom, and announce the nameof the card aloud. The name of the card isits number, then the word “of,” and then theplural form of its suit.

5. Place thepile face-downagain andgather thethree piles into one stack of cards. No other

rules matter when gathering the cards, ex-ceptmake sure to put the pile selected by theviewer on top of the other 2 piles.

6. Announce the number of the card and spellit out, placing a card face-down for each let-ter.

7. Collect the cards dealt andplace themon thebottom of the stack.

8. Say the world, “of,” and spell it out, placing acard face-down for each letter.

9. Collect the cards dealt andplace themon thebottom of the stack.

10. Announce the plural form of the suit of thecard and spell it out, placing a card face-down for each letter.

11. Collect the cards dealt andplace themon thebottom of the stack.

12. The viewers card, which is the card you justspelled out, will be the 5th card from the top.

13. Deal out 4 cards face-down. Deal the 5thface-up and ask the viewer, “Is this yourcard?”

The Mathematics of a SukkahJasmine Levine '18

The Bible, in Leviticus 23: 42-43, states, “For aseven day period shall you live in booths. Every res-ident among Israelites shall live in booths in orderthat your generations should know that I had thechildren of Israel live in booths when I took themout of the land Egypt.” From these verses the Rab-bis decreed that on the 15th day of themonthof theHebrew month of Tishrei, Jews “dwell” in booths,known as sukkot, which are temporary structures.In countries that have mild temperatures at thattimeof year,many ferventlyobservant Jews actuallysleep in their sukkot to fulfill the obligation to actu-ally “dwell” in the booths. Those less scrupulous

in their observance, or those who live in climateswhere it is too cold to actually sleep in the booths,eat their meals and have social gatherings in thesebooths.

When theRabbismandatedwhatwasmeantbythe term to “dwell” in sukkot they also designatedthe exact specifications as to the size of a sukkah,which leads us to the subject of the mathematics ofthe sukkah. How big or how small must a sukkahbe? This is a relatively simple question to answer. Asukkahmust be no more than 20 high, no less than10 low and at least 7 by 7. The problem is that thesenumbers are not given in the units of measure that

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we use in our dailymeasurement operations. In de-creeing the measurement of a sukkah, the Rabbisestablished the size in terms of amot and tephachim,Biblical andRabbinicmeasurements used to definethe size of the tabernacle built by the Jewish peoplewhile living in the wilderness during their 40 yearsof wandering.

So exactly how much is an amah (amot beingthe Hebrew plural of amah)? Well that dependsuponwhichopinion you accept as there is disagree-ment as to the exact size of an amah. An amah isthe length of an adult, male forearm. So how longis that? That is where the disagreement comes inwith the range being from 18 to 22 inches. Hence,an amah can be from exactly a foot and a half toalmost two feet. Similarly a tephach (singular fortephachim) is the size of an adultmale fist. The con-sensus is that a tephach is about 3.5 inches.

Taking this into consideration we have to con-vert amot and tephachim into inches and feet in or-der to determine the size of a sukkah. If we say thata sukkah cannot be taller than 20 amot, then in feet

the sukkah must be no taller than 30 feet high us-ing 18 inches as the size of an amah. If we use 22inches as the size of an amah then the sukkah canbe as tall as 36.66 feet. The sukkah’s height can beno lower than 10 tephachim. If we take a tephach tobe 3.5 inches, then the minimum height a sukkahmust be is 35 inches. Similarly theminimum lengthandwidth of a sukkah is 7 tephachim by 7 tephachimor 24.5 inches by 24.5 inches. The dimensions of asukkah, as determined by theRabbis, are, therefore,35 inches to 30 and 36.66 feet tall by 24.5 incheslong by 24.5 inches wide.

There are noupper limits on the size of a sukkahin terms of length or width. In fact, the Rabbisteach that the Messiah will construct a sukkah thatis large enough to accommodate the needs of mil-lions. That is an interesting structure to contem-plate. What ismore intriguing is contemplating theminimum requirements for a sukkah and trying tounderstand how someone is supposed to “dwell” ina structure that is 35”× 24.5”× 24.5”.

The Real Meaning of NumbersTyler Mandelbaum '17

In the Hebrew languageevery letter in the al-phabet corresponds to anumber. This means thatevery word in a sentencebecomes a series of num-bers. Many times in Ju-daism, we either look at

numbers one at a time or added together. Themeanings of the words that share numbers arethought to be immensely related or identical. It isalso commonly thought that the direction of theshapes and lines in a letter has deep spiritual mean-ings. The letters in the alphabet are divided intothree categories: ones, tens, and hundreds. The let-ters go up in value until 400. An apparent ques-tion is raised from this; what about all of the othernumbers above 400? One answer is that Hebrew

is a spiritual language and with just the 22 lettersof the alphabet, everything can be described fromthe start of creation until infinity. An example ofhow people can look at these words in relation tonumbers is with the words in Hebrew for nature,God(א-להים), and cup .(כוס) Eachof thesewordsadds up in value to the number 86. Kabbalahwouldstate that all of these words have very similarmean-ings, which is obviously very strange consideringhow different all of these words seem to be. We canlook at this like a logic statement.

Nature and God are known to be one and thesame. A cup, in kabbalah, stands for a kli, whichmeans a desire to receive. Therefore, nature andour kli are the same (because of their equal values).When we match our desires (kli) with those of na-ture, we will also match them with God (becausenature and God are synonymous). So, the equa-

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tionwould look like this this: IfA = B, andB = C,then A = C.

In the Torah, God is referred to as “ayn sof ”—having no end. This is essentially calling Hasheminfinite. This name simply suggests that God existsand will forever exist without implying anything

about His character.

References

http://www.kabbalah.info/eng/content/view/frame/60270?/eng/content/view/full/60270&main

Internal Rate of ReturnDJ Presser '16

It is very important in investing to figure out howlucrative an investment could be or how lucrativean investment that you made was. One way thatpeople evaluate this is with the gross rate of return.The rate of return simply evaluates what percentyou make on your original investment yearly. Forinstance, if I buy one share of Google tomorrow for$500, and I thought that I could sell it for $750 inexactly one year from now, my rate return of returnwould be 50%. The way you derive the gross rateof return is by taking the price that you sold thestock(or any object) for and subtracting the origi-nal price of the object and divide the difference bytheoriginal price

($750−$500

$500

). While rate of return

is an okay way to evaluate how an investment per-formed over the lifetime of the investment, it failsto account for the length of the investment and thevalue of time as money.

There is a different financial accounting for-mula that is able to account for the length of the in-vestment. The way that one would evaluate for thiswould be by using the internal rate of return (IRR)formula. For example, if I am a factory owner andI need to buy some new equipment, I need to fig-ure out which machine it makes the most sense tobuy. According to my calculations, I figure out thatMachine #1 costs $500,000, it will produce about$95,000 worth of profit every year, and it will lastfor approximately ten years. In my research, I alsofind a different machine that costs $1,000,000, butit will produce $150,000 of profit for fifteen years.In this case, it is extremely important to accountfor the length of time, because it could drasticallychange my investment decision.

In the case ofMachine #1, I would bemaking a

gross rate of return of 95% per year. If I were to buyMachine #2, I would be making a rate of return of75% per year. However, if I were to account for thefact that Machine #2 has five extra years of produc-tion, my rate of return would seemmuch closer. Toevaluate IRR you must use the formula:

0 = C0+C1

1 + IRR+

C2

(1 + IRR)2+. . .+

Cn

(1 + IRR)n

If you were to plug the specifics ofMachine #1 intothe equation,

0 = −500, 000+95, 0001 + IRR

+ . . .+95, 000

(1 + IRR)10

you would find that the IRR for Machine #1 is13.77%. If you were to plug the specifics of Ma-chine #2 into the equation,

0 = −1, 000, 000+150, 0001 + IRR

+. . .+150, 000

(1 + IRR)15

you would find that the IRR for Machine #2 is12.82%.

When looking for an investment or evaluatingyour investment opportunities, youwant to choosethe one with the highest IRR. The higher the IRR,themore you aremakingper year, while accountingfor the value ofmoney over time. In the example ofthemachines, whether evaluated using gross returnor IRR would lead you to making the same invest-ment, by using IRR, you are able to see how closethe difference between the decisions truly is.

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Puzzle

The goal of this KenKen puzzle is to fill a grid withdigits 1 through 5 so that no digit appears more thanonce in any row or any column. The grids are dividedinto groups of cells called “cages,” and the numbers inthe cells of each cagemust produce a certain specified“target number” when combined using the specifiedmathematical operation.

Trivia

• [0, 1) ⊂ R is not homeomorphic to the unit circleS1 ⊂ R2 butR ∪ {∞} is.

• Depending on the space in which a sequence is de-fined, it may approach more than one limit point.

• The number of prime numbers less than n is approxi-mated by the function π(x) = x

ln(x) .

•∑∞

n=11n2 = π2

6

• A continuous bounded function defined on abounded interval can be approximated to an arbitraryprecision by a polynomial of sufficiently high degree(this is the Weierstraß Theorem).

• Let S = [0, 1]∩ (R−Q). Then∫S 1 dx =

∫ 10 1 dx =

1.

Image Sources

Cover Mandelbrot Photohttps://www.youtube.com/watch?v=j1pjw4qxjM4

Eratosthenes (p.1)https://www.childrensmuseum.org/legacy-games/cosmicquest/assets/erosthanis.jpg

Perelman (p.3)http://upload.wikimedia.org/wikipedia/commons/4/43/Perelman%2C_Grigori_%281966%29.jpg

Kim Peek (p.5)http://historicmysteries.com/wp-content/

uploads/2010/03/savant.jpg

Euler (p.8)http://www.mathematik.ch/mathematiker/Euler.jpg

Gematria (p.13)http://www.biblewheel.com/images/ethkolfigure8.gif

KenKen (p.15)Generated at http://linuxdingsda.de/~wintix/kenken/index.php

All uncredited images are public domain or of unknown source

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