The 3D regular convex polyhedrons are super famous, have the name: platonic solid, and have been known since antiquity. In particular, there are only 5 of them.
The counts are:
The cool thing is that the 3 that exist in 5+ dimensions are all of one of the three families:
- simplex: triangle, tetrahedron.Non-regular description: take convex hull take D + 1 vertices that are not on a single D-plan.
- hypercube: square, cube. 4D case known as tesseract.Convex hull of all (Cartesian product power) D-tuples.Two are linked iff they differ by a single number. So each vertex has D neighbors.
- cross polytope: square, octahedron.All permutations ofEach edge E is linked to every other edge, except it's opposite -E.