Ciro Santilli 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 per dimension are:
Table 1. Number of regular polytopes per dimension.
| Dimension | Count |
|---|---|
| 2 | Infinite |
| 3 | 5 |
| 4 | 6 |
| >4 | 3 |
The cool thing is that the 3 that exist in 5+ dimensions are all of one of the three families:Then, the 2 3D missing ones have 4D analogues and the sixth one in 4D does not have a 3D analogue: the 24-cell. Yes, this is the kind of irregular stuff Ciro Santilli lives for.
The name does not imply regular by default. For regular ones, you should say "regular polytope".
Non-regular description: take convex hull take D + 1 vertices that are not on a single D-plan.
Convex hull of all (Cartesian product power) D-tuples, e.g. in 3D:
( 1, 1, 1)
( 1, 1, -1)
( 1, -1, 1)
( 1, -1, -1)
(-1, 1, 1)
(-1, 1, -1)
(-1, -1, 1)
(-1, -1, -1)From this we see that there are vertices.
Two vertices are linked iff they differ by a single number. So each vertex has D neighbors.
The non-regular version of the hypercube.
Examples: square, octahedron.
Each edge E is linked to every other edge, except it's opposite -E.