- The Pythagorean theorem and its proof
- Triangulation
- The "Art Museum Theorem" and its proof
- The Golden Rectangle and the re-appearence of the Golden Ratio (!)
- The Golden Spiral
- Polygons and regular polygons
- Polyhedra and regular polyhedra
Of course another name for a "regular polyhedron" is a "Platonic solid." As a general reference on the Platonic solids, you might look at this fun website:
http://www.learner.org/interactives/geometry/platonic.html
Some of the exercises there make for good practice.
A point I have emphasized in class and on this blog is the difference between the use of math, and the knowledge (or theory) of math. It is, for instance, the difference between knowing that a 3-4-5 triangle is a right triangle, and knowing why right triangles satisfy a2+b2=c2.
What's the difference? Why is that important? Consider the following. The ancient Babylonians knew about many Pythagorean triples: they had 3-4-5, 5-12-13, and many others. But after the Pythagoreans developed the theory of right triangles, it became possible to produce infinitely many pythagorean triples! Plato himself developed such a method, as did Euclid (see this for a "fun" description of Euclid's method).
This represents a massive advance on the ancient science. What made the advance possible went beyond the heuristic techniques practiced by prior societies: with the concept of proof came theoretical knowledge, and with theory, the techniques of mathematics exploded in fecund usefulness.
Still, the roots of the Pythagorean theorem trace back back to profoundly ancient times: times as ancient to Pythagoras as the Roman legions are to us. The first set of Pythagorean triples are seen in Mesopotamian cuneiform tablets from the 20th century BC! Yes, people were studying right triangles 4000 years ago. They appear in formal Chinese writing somewhat later: around 200 BC, just after the Warring States period, but some centuries before their contact with the Greco-Roman world. But knowledge of geometry was likely developed much earlier. Sadly, much was lost to history when the Qing dynasty, in a bid to limit intellectual diversity after the destructive fracturing and perpetual warfare of the prior epoch, oversaw the burning of nearly all books in China around 210BC---a great deal of science and literature was lost permanently.
It is important to note that none of these earlier sources provided any actual proof of the Pythagorean theorem, at least in total. The first recorded proof appears in the writing of Euclid in 300 BC (although the Zho Bi Suan Jing gives a formal proof that 3-4-5 triangles are right).
With math so ancient, there is one question: why!!!???
I am sure many undergraduates in math classes have wondered this. Was math developed for torture? As a religious expression of perfection? As a high intellectual exercise? As a tool of the elite to oppress the working classes?
The Babylonians used Pythagorean triples, and an evidently well-developed computational system for understanding them, for the purpose of parcelling off land. In order to allot land equitably and to make efficient use of space, it was important to be able to mark off right angles. Without modern surveying equipment, this is a difficult task! Actually creating a right angle with real accuracy is hard to do! Sure, a person can eye-ball it fairly well, but when dealing with potentially large parcels of land, small errors in the angle could create very large deviations of actual area. That could cost people a great deal in lost produce, and could cost the state in lost taxes.
The Pythagorean triple is a fantastic way to precisely mark off right angles, and occurs simply by measuring lengths, not angles. One takes three cords of, say, lengths 30 meters, 40 meters, and 50 meters---it was well within the capability of ancient societies to manufacture such cord lengths with a great deal of precision. Then your people extend the cords to maximum length without stretching, until they just meet to produce a big triangle. The angle between the 30m length and the 50m length will be a perfect right angle. Land apportionment problem solved!
Now, is this why right triangles and Pythagorean triples were developed? Or did the bean-counters and actuaries of Babylon simply take advantage of math that was developed for other reasons? A certain answer is impossible, as there is no historical record that far back but only archaeological remains---various tablets and stones with precious scant detail. But what is clear is this: societies of the time had use for this math, reasons to develop it, and to nurture its progress.
Figures such as triangles were studied in pre-history times: one reason, as I have discussed, is their usefulness in dividing up land resources. But what about solid shapes? When and where did they come into play? What was their usefulness? We'll discuss more about this next time.
