The dual of a geodesic polyhedron, e.g. an icosahedron whose faces have been subdivided is in fact a common and excellent choice for geospatial applications, with https://github.com/uber/h3 being a good implementation.

You can also subdivide the faces of a cube into smaller squares with a quadtree, e.g. https://s2geometry.io/

The Disdyakis triacontahedron's faces are probably too long and skinny for most geospatial applications though.

For anyone else not familiar with the exact shape:

https://en.wikipedia.org/wiki/Disdyakis_triacontahedron