I have to shamefully admit I never owned a D20 until recently. While at the game shop I noticed they sold sets of dice as D4, D6, D8, D12, and D20. Curious. What is so special about these shapes?

What is interesting is these shapes are Platonic solids. Platonic solids look “nice” because every face is the same shape and every angle between every shape is the same. And what is unbelievable there are only 5 of them! That bears repeating. There is a finite number of such shapes, and there are only FIVE OF THEM! Of all the possible shapes in 3D there are only 5 such shapes? Euclid proved this in Elements.

So my first thought was if that was true then what about that trick of subdividing an icosahedron to tessellate a sphere? Why doesn’t that create a platonic solid with 80 sides? Its because the first subdivision creates a shape where some vertices terminate 5 edges and others terminate 6 edges.

This is very sad as it mean we can never have an uniform Geodesic Grid larger than 20 faces. This is because subdivision of a single triangle becomes a four triangles in the shape of a triforce. This means two adjacent subdivided triangles now share a vertex with 6 edges. There does not exist a regular shape with each vertex having 6 edges.

So D20 are popular because that’s as large as nice shapes get.