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.
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.
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.