Friday, February 25, 2011

Theaetetus and his Beautiful Proof

There can be only five regular polyhedra. Somebody had to prove that once, and the proof is beautiful.

The man who proved it was the little known (except to mathematicians) Theaetetus, instructor/professor at Plato's Academy. Along with Plato, Aristotle and Euclid, he stands out as the greatest of the great from that important time and place.

To read more about him (I'll close with his Wikipedia entry) I strongly recommend reading Chapter 4 of this book:
 Euler's Gem by Dave Richeson

In fact, buy the book and read all the chapters. The proof below comes from Chapter 5 soon thereafter.

The Proof:

Consider a regular polyhedron. Each face is a regular polygon having n sides, and m edges of the polyhedron meet at each vertex (corner).

Because every face must have at least three sides, n is greater than or equal to 3, and because at least three edges meet at each vertex, m is greater than or equal to 3.

Every angle of every face has the same measure, call this angle: theta.

At each vertex there are m faces, each contributing a plane angle with measure theta.

From Euclid's theorem, it follows that m times theta must be  less than 360 degrees.

For which m and which n is this possible?

When n = 3, the faces are equilateral triangles, so theta = 60 degrees. (The measure of an interior angle of a regular n-sided polygon is 180 degrees times (n-2)/n.)

Insisting that m times theta is less than 360 degrees, we have m times 60 degrees is less than 360 degrees, or m is less than 6.

So m = 3, 4 or 5 are the only possibilities.

These values of m yield the tetrahedron, the octahedron, and the icosahedron, respectively.

When n = 4, the faces are square, so theta = 90 degrees.

This implies that m times 90 degrees is less than 360 degrees, or m is less than 4.

So we can only have m=3, and we obtain the cube.

When n = 5, the faces are regular pentagons and theta = 108 degrees. Thus m times 108 degrees is less than 360 degrees, or m is less than 10/3.

So we can have only m = 3, and we obtain the dodecahedron.

When n = 6, the faces are regular hexagons and theta = 120 degrees. But m times 120 degrees being less than 360 degrees implies than m is less than 3, which is impossible.

So there is no regular polygon with hexagonal faces.

We encounter the same problem when n is greater than 6.

Thus there are no other Platonic solids.

Finis.

So-o beautiful.

This is from Book XIII of Euclid's Elements, the final book, which it is believed Euclid wrote directly from Theaetetus' notes. The most important part of Book XIII is considered to be the proof that there are 5 and only 5 regular (Platonic) solids.

"Many historians contend that all of the mathematics in Book X and XIII of the Elements is due to Theaetetus." ... Dave Richeson

Theaetetus, Theaitētos, (ca. 417 BC – 369 BC) of Athens, possibly son of Euphronius, of the Athenian deme Sunium, was a classical Greek mathematician. His principal contributions were on irrational lengths, which was included in Book X of Euclid's Elements, and proving that there are precisely five regular convex polyhedra.

Theaetetus, like Plato, was a student of the Greek mathematician Theodorus of Cyrene. Cyrene was a prosperous Greek colony on the coast of North Africa, in what is now Libya, on the eastern end of the gulf of Sidra. Theodorus had explored the theory of incommensurable quantities, and Theaetetus continued those studies with great enthusiasm; specifically, he classified various forms of irrational numbers according to the way they are expressed as square roots. This theory is presented in great detail in Book X of Euclid's Elements.

Theaetetus was one of the few Greek mathematicians who were actually natives of Athens. Most Greek mathematicians of antiquity came from the numerous Greek cities scattered around the Ionian coast, the Black Sea and the whole Mediterranean basin. Likewise, most Greek scientists came from the scattered Greek cities and not from Athens. Athens, and later Alexandria were centers of attraction because of the philosophical schools of Plato (the Academy) and Aristotle (the Lyceum), and the renowned Museum and Great Library. The Academy of Plato operated in Athens for almost 600 years, and served as educational center even for some of the early fathers of the Christian church.

He evidently resembled Socrates in the snubness of his nose and bulging of his eyes. This and most of what we know of him comes from Plato, who named a dialogue after him, the Theaetetus. He apparently died from wounds and dysentery on his way home after fighting in an Athenian battle at Corinth, now widely presumed to have occurred in 369 BC.

The crater Theaetetus on the Moon is named after him.

Jérôme CHAUVET said...

Hi Steve,

Never heard about him in the past. Has he made important discoveries in other areas of mathematics?

I wonder why some of those intelligent ancient Greeks became so famous and some not... Were the most famous of them more skilled than the less famous ones at making other people talking about them, or are their discoveries alone responsible for their contemporary success?

I have no answer in hand now :(

Best,

Best

Steven Colyer said...

Honestly? I never heard of him either, but he's huge. I was always a Science guy, not a Math guy so much. So, as I dug into Science, I was astounded at how nasty some Scientists could be with each other.

I mean, stop arguing so much and run a damn experiment already and let the chips fall where they may and settle your differences once and for all, m'kay?

Well, I was naive, because I forgot there's this little thing called: "funding", and yup, like the terrible saying goes: "The answer to 99% of questions is: Money." Sigh.

So anyway, before I realized that, it occurred to me that you Scientists really like your Mathematics. I mean you really really REALLY like them, like the Academy of Motion Picture Arts and Sciences likes Sally Field.

For example, it's frigging impossible to understand your papers without understanding Math, so sure enough, I was hooked and they (Mathematicians) reeled me in. Barry Mazur caught me with his prose, and John Baez is reeling me in with his. One great goal of mine, that which keeps me going on dark days, is to understand everything that is discussed at n-Category cafe. And yah, books like Euler's Gem, and the Princeton Companion, reel me in some more.

Plato himself contributed nothing to Math as far as I know, but as Richeson explains in Euler's Gem, he loved the subject and in creating the Academy, gathered around him some of the best. He wrote so much about his love of the five regular polyhedrons for example, that today we refer to them as The Platonic Solids.

Most ancient Greeks were farmers. Come to think of it, most ancient everyone were farmers. Enter the printing press, and the Industrial Revolution, and today, well, I wouldn't know which end of the cow to milk if you paid me. :-)

But I DO know how to spell Theaetetus, even if I'm not sure how to pronounce it. :-)

Oh, and if anyone here wants to pay me, for almost anything, let's talk. I'm currently looking for a job, because I'll need the money, since it looks like my straight A+ daughter is going to go to Cornell next year. :-)
To be a doctor. :-(
Pediatric oncologist. :-)

Jérôme CHAUVET said...

Hi Steve,

If you are actually seeking for a job, then I am so sorry for that. Believe me, I really wish I could help you at the moment. Hope you find a position in the next weeks, or even faster than a week.

Keep hanging on, Steve. You are the best engineer ever !!!

Best,

Steven Colyer said...

Thank you Jerome, much appreciated, but do not worry about me.

I am like a cat. I always land on my feet. At least so far I have. Whichever way the wind blows, c'est la vie.

Sena Erickson said...
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