Mercurial, the Spectre wrote:[...]

But if you want facets to be semiregular or regular, here is the following.

In 3D you get the family of uniform polytopes because a uniform polygon is a regular polygon or a star polygon.

In 4D you get the uniform polychora because a semiregular polyhedron has regular polygons and are the same as a uniform polyhedron.

In 5D you are limited to the 6 regular polychora, rectified 5-cell, rectified 600-cell, and snub 24-cell because the cells must be regular polyhedra. For 5D polytopes this means the 3 regular polytera alongside the 5-demicube, rectates of the 5-simplex (rectified 5-simplex, birectified 5-simplex), and rectates of the 5-orthoplex (rectified 5-orthoplex, birectified 5-cube/orthoplex)

In 6D, 7D, and 8D, you have the Gosset polytopes (up to 4_21) that contain semiregular facets alongside the 3 regular polytopes and the demihypercube.

Eventually you are restricted to the 3 regular polytopes and the demihypercube as the number of dimensions approaches infinity.

Mercurial, in 3D you were allowing non-convex figures. But at 5D you were swapping to convex ones only. I think those ought to be added there too.

In fact, uberscetch's definition looks like a direct extension of Gosset's definition of semiregularity:

semiregilarity asks for

1) vertex-uniform = vertex-transitive = isogonal

2) all facets (i.e. (D-1)-faces) are regular polytopes

whereas uberscetch's definition asks for

1) vertex-transitive

2) all facets are semiregular polytopes

Thus we could re-write his definition into

1) all 0-faces are symmetry-equivalent

2) all (D-2)-faces are regular polytopes

That is, for D=4 you'd get that just the 2-faces are to be regular, which are the polygons. Therefore uberscetch here is quite near to the idea of CRFs. But in contrast, the setting of CRFs additionally restricts to convex shapes only, whereas OTOH it is more liberate in not asking for vertex-transitiveness at all.

Mercurial is right about the higher-dimensional restrictiveness of his definition. Within 3D and 4D the set of Wythoffians obviously is contained. But also figures beyond. But Wythoffians generally would allow for prisms as 3-faces. But for D >= 5 uberscetch's definition would disallow those prismatic 3-faces! That is from D=5 on not even all Wythoffians are contained within uberscetch's definition!

--- rk