The classification of Platonic solids I | Universal Hyperbolic Geometry 53 | NJ Wildberger

Euclid showed in the last Book XIII of the Elements that there were exactly 5 Platonic solids. Here we go through the argument, but using some modern innovations of notation: in particular instead of talking about angles that sum to 360 degrees around the circle, or perhaps 2 pi radians, we normalize our "turn angle" so that all the way around is exactly 1: the natural unit here. Using turn angles, we can use the Schlafli symbol {p,q} notation to discuss the various possibilities.

First there is the planar situation, where we find the usual three regular tesselations, and then in the spherical case we find exactly five possibilities for {p,q} corresponding to the spherical regular polytopes, i.e. Platonic solids.

However this is really only part of the argument: the mathematical constructions of the Platonic solids still remain--a quite separate question from the physical constructions of these objects. It turns out the the tetrahedron, cube and octahedron have pleasant constructions that generalize very nicely to higher dimensions. However the icosahedron and dodecahedron are quite a different story!

The reality is that the complete argument is much more subtle than is usually conceeded. Euclid's proof, while quite attractive, does not completely work mathematically, convincing though it may be physically.

************************

Here are the Insights into Mathematics Playlists:

Here are the Wild Egg Maths Playlists (some available only to Members!)

************************