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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.

Ancestors