Journey Through Genius - Chapter 1 - Reading Check

1. How do we know the Babylonians were aware of some version of the Pythagorean Theorem?

The Plimption-322 , p. 5

2. Roughly how old is the Plimpton-322 tablet?

1800 BC

3. Consider Thales, Pythagoras and Hippocrates. Where and when did they live?

Thales (600 BC - Miletus - Asia Minor) Pythagoras (572 BC - Samos / Asia Minor / Eastern Greece) Hippocrates (440 BC - Chio - Asia Minor)

4. Order the following sets of numbers: Real, constructible, transcendental, algebraic.

constructible < algebraic < trascendental < real

5. How many lunes are quadrable?

5

6. What great ancient civilization used a base 60 number system?

Bablyonians

7. What is our first great theorem and to whom do we attribute the proof?

Hippocrates of Chios - His lune, on a square inscribed inside a circle is quadrable.

8. To what ancient civilization do we celebrate for the way they used the simple and elementary as a foundation for the complex and intricate?

Greek

9. Why do you think the Greeks regarded quadrature so highly?

It represented the triumph of reason and reflected the inherent simplicity in nature.

10. Give a rough argument that the circle cannot be squared.

The number is not algebraic, therefore it cannot be constructible. If the circle could be squared, then we could construct a square with area pi, and thus an edge of length sqrt(pi), implying that pi is constructible.

1.2.3.

a.b.c.d.e.f.g.h.i.

i.j.k.l.m.

i.n.o.p.

i.ii.iii.iv.v.vi.vii.viii.

Visited Egypt and perhaps BabyloniaPythagoras - 580 BC - Samos

Wandered for years, eventually settling in Southern Italy. School of Pythagoras -

Aims: political, philosophical, political300 young aristocrats, living in secret society (fraternity)All worldly goods were held commonVegan - our souls transmigrate to other animalsQuadrivium - arithmetic, music, geometry and astrologyStudents were divided - listeners and mathematici. After 3 years of listening in mute obedience form behind curtain, a pupil could come into the inner circle.Women, by law forbidden to attend public meetings were admitted to lectures. (28 women in the mathematici category)At 60, he married one his pupils. Some say she was his daughter, others say they never married. Who knows?He did write down his teachings, the members weren’t allowed to spread what they learned.

Story: a pupil was drowned in a shipwreck as punishment by the gods for boasting about the dodecahedron. tetracyts - triangular 10 - fire, water, air, earth, and the five pointed star were beloved symbolsRefused to eat beans, drink wine, pick up anything that had fallen, or stir a fire with an iron.Damon & PythiasDeath: the Pythagoreans were (locally) powerful, in a violent revolt their house was burned

Some say his followers helped him escape, but in his ensuing flight, he reached a field of sacred beans and chose to die at the hands of his enemies.“Knowledge is the greatest purification” (mathematics), “Everything is number” (positive integer)Music: notes sounded by taut string depended on length, and the octave, the fourth, the fifth, simple ratios give pleasing soundsnumbers had meaning if not symbols/numerals

1 = reason because reason could produce only one consistent body of truths2 = man3 = woman4 = justice, product of equals5 = marriage, union of 2 & 36 = creationevens (2+) were separable & prolific so feminine & earth, somewhat less highly regardedodds (bunch of dudes) were masculine and divine

Triangle & Polygons & Sums & Differences … Curvilinear?

1.

2.

3.

4.

5.a.b.c.

6.a.b.c.d.e.f.g.

a merchant, was swindled and lost it all.

Hippocrtates’ lune (born: 430 BC)

was one of the first paid teachers

Aristotle thought him foolish

In the 5th century, theorems piled up, requiring organization. He pioneered this, but his work was lost.

In Athens, 3 challenging problems had risenSquare the circleDuplicate the cubeTrisect any angle

His lune is quadrable proof requires these simple ideas

Ratio of areas of circles is ratio of squares of diameters * (proof?)AB = sqrt2*AO = sqrt2*/2 * ACAB^2 / AC^2 = 1/2Small semicircle = 1/2 large semicircle = AFBO quadrantLune = triangle

Thales TheoremPythagorean Identity