by Mercurial, the Spectre » Sun Dec 24, 2017 8:23 pm
When you snub a generic polychoron/honeycomb of the form {p,q,r}, that is, alternating is omnitruncation, you will have 6 different edge lengths to work out on. You have only four degrees of freedom between variations, so the other two are neglected. That is, they cannot have equal edges in general.
For the s3s3s4s (snub tesseract), you have to consider the topology of the x3x3x4x (omnitruncated tesseract).
The x3x3x4x has:
x3x3x - truncated octahedron (Td symmetry)
x3x2x - hexagonal prism (D3h symmetry)
x2x4x - octagonal prism (D4h symmetry)
x3x4x - truncated cuboctahedron (Oh symmetry)
while the s3s3s4s has:
s3s3s - icosahedron (T symmetry)
s3s2s - octahedron (D3 symmetry)
s2s4s - square antiprism (D4 symmetry)
s3s4s - snub cube (O symmetry)
To rigorously prove that s3s3s4s cannot be made uniform, look at the truncated cuboctahedron of which the uniform snub cube is based (look at Wikipedia). The vertices of the related x3x4x are simply the union of the snub cube's enantiomorphs. The x4x (octagon) in the unique x3x4x, from which its alternation is the uniform snub cube, is not a regular octagon. Rather, it is merely an octagon with equal angles and 2 different kinds of edges that alternate with each other with a symmetry equal to a square.
Now look at the uniform square antiprism. It is derived from x2x4x, except that x4x is always a regular octagon (since a uniform antiprism's two bases must be opposite with respect to each other in a dual fashion). In the omnitruncated tesseract, you have the fact that every x2x4x is connected to x3x4x through an octagon. Since individually, the octagons needed in making them uniform are different but dependent on one in the omnitruncate itself, it cannot be made uniform because it only takes on one value, not two.
The only uniform cases therefore belong to s2s2s2s (hex), s3s3s *b3s (sadi) and s5s2s(5/3)s (gudap), because the base polytope is regular (meaning equal faces) or because there exists a higher symmetry in which the number of edge lengths is the same or lesser than the degrees of freedom, from which a solution can be easily determined.
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