Another CRF idea inspired by snub disphenoid

Discussion of known convex regular-faced polytopes, including the Johnson solids in 3D, and higher dimensions; and the discovery of new ones.

Another CRF idea inspired by snub disphenoid

Postby quickfur » Fri Oct 20, 2017 6:59 pm

Today I was looking again at this render of the snub disphenoid:

Image

In my mind I imagined the top part of the snub disphenoid replaced with 3 tetrahedra sharing faces, while the bottom part is replaced by a pair of square pyramids. Now obviously, such a configuration of cells wouldn't close up in 3D, but given that one of the "edges" of the prospective pair of square pyramids is slightly longer than unit edge length, if we were to distort the snub disphenoid so that this edge becomes unit length, then we'd have to break up one of the tetrahedra in order to keep their edges still unit length. IOW, an angle defect is introduced, and a rather shallow one at that. So we could fold it into 4D to make a shallow bowl-shaped configuration of cells.

Or, another way to describe it, we have an edge in 4D, surrounded by 3 tetrahedra and 2 adjacent squippies. Can this configuration be closed up in a CRF way?

To take this idea just one more step further: still imagining that the bottom part of the snub disphenoid is a pair of square pyramids, we see that there is tetragonal symmetry around the shared square face. So, ostensibly, we could have triplets of tetrahedra surrounding the other edges of the square face too. This leads us to a configuration with two squippies sharing a square face, and 12 tetrahedra in 4 sets of 3, each set sharing one of the edges of the square face. Can this be continued, to close up in a CRF way?
quickfur
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