The (regular, 3D) Platonic Solids All faces, all edges, all corners, are the same. They are composed of regular 2D polygons: There were infinitely many 2D n-gons! How many of these regular 3D solids are there? OK – Let’s step back and look at some beautiful geometry that we _can_ see: the Platonic Solids! They have been known to mankind for 2 millennia. “Regular” means that all faces, edges, and corners are indistinguishable; they are all exactly the same. The surfaces of these objects are composed of regular 2D polygons. There are infinitely many of those 2D n-gons. But how many are there of these regular 3D solids? >>> There are indeed just the 5 shown here. But could you prove this to your friends -- or to your grandmother? -- Soon you will be able to do this!

Making a Corner for a Platonic Solid Put at least 3 polygons around a shared vertex to form a real physical 3D corner! Putting 3 squares around a vertex leaves a large (90º) gap; Forcefully closing this gap makes the structure pop out into 3D space, forming the corner of a cube. We can also do this with 3 pentagons:  dodecahedron. In order to construct the surface of some Platonic solid, we have to assemble some 2D polygons so that they form real physical 3D corners; we thus need at least 3 faces coming together at each corner. When we put 3 squares around a shared vertex, then there is still a large gap left. When we forcefully close this gap, the structure pops out into the 3rd dimension, and a real cube corner is formed. We can also use 3 pentagons, and the result will be the dodecahedron.

Why Only 5 Platonic Solids? Lets try to build all possible ones: from triangles: 3, 4, or 5 around a corner: from squares: only 3 around a corner: from pentagons: only 3 around a corner: from hexagons:  “floor tiling”, does not bend! higher n-gons:  do not fit around a vertex without undulations (forming saddles);  Now the edges are no longer all alike! Now we can see why there are only 5 Platonic solids: When we use triangles for the surface facets, we have a choice, we can use 3 or 4 or 5 around a point to form a regular corner. This leads to the Tetrahedron, the Octahedron, and the Icosahedron which uses 20 triangles. … But if we use squares there is only one option: 3 squares around a corner – forming a cube. And we have already seen that with 3 pentagons we can make a dodecahedron. All larger n-gons are too “round” and are not able to make true 3D corners.