It’s reasonable to ask why we can’t have more faces.

20 is definitely as high as we can go though.

The Platonic solids are the only regular tilings of the sphere. If we are only interested in triangulations, we can ignore the cube and dodecahedron.

For it to be regular, each vertex needs to be surrounded by the same number of faces.

tetrahedron has *3* equilateral triangles around each vertex,

octahedron has *4*

icosahedron has *5*

Trying to continue this pattern - if you put **6** equilateral triangles around a vertex, that makes 360°, so there is no angle defect, and therefore no Gaussian curvature - it is what we call intrinsically flat, and there is no way to cover a sphere with such vertices.

Another rule that any mesh with the topology of a sphere has to follow is:

(number of faces)-(number of edges)+(number of vertices)=2

If each vertex had 6 triangles around it, there would be no way to meet this condition.

It is actually possible to make a surface with more than 20 faces, all equilateral triangles and the topology of the sphere, but by giving up the properties of smoothness, vertices lying on the sphere surface, and regularity:

This has a mix of valence 5,6 and 7 vertices. (I generated it by remeshing a sphere, then equalizing the edge lengths, keeping the vertices as close to the original sphere as possible)

It could be interesting to try and find some more symmetrical and nicer looking ‘crinkly’ triangulated spheres like this.