TODO understand and explain definition.
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:
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".
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)The non-regular version of the hypercube.
Examples: square, octahedron.
Each edge E is linked to every other edge, except it's opposite -E.
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A regular polytope is a multi-dimensional geometric figure that is highly symmetrical, with identical shapes and arrangements in its structure. In general, a regular polytope can be defined as a convex polytope that is both uniform (its faces are the same type of regular polygon) and vertex-transitive (the structure looks the same from any vertex).