Recently I've been trying -- so far unsuccessfully -- to construct a CRF containing J12 (trigonal bipyramid) as cells. The only one I could find is the trivial J12 prism; all other constructions don't seem to work out.

Does anyone know of a 4D CRF that has a 90° dichoral angle between a tetrahedron and some other cell? If so, we might be able to glue two of them together at that other cell to make J12 from the tetrahedra. I can't think of any, though, except the tetrahedral prism, which just gives the J12 prism.

Interestingly enough, if we glue two tetrahedral prisms in a gyrated orientation, we get that segmentotope that's a 4D analogue of the gyrobifastigium (I was trying to find the post with my renders of it but the search function in phpBB is rather crippled ). I believe it was something like triangular_prism||digon or something to that effect.

Another route is to augment the J12 prism, but I think the dichoral angles of the triangular prisms are too big to fit a triangular prism pyramid augment without becoming non-convex.

Any other ideas?