Today I was thinking about various 3D crown jewels, and how they might be derived from "simpler" shapes.

J84 (snub disphenoid) - there are various different ways to analyze the snub disphenoid, but recently I noticed that it can be derived from a pentagonal bipyramid: imagine a paper model of a pentagonal bipyramid, where you select a pair of adjacent edges on the pentagonal cross-section. Use scissors to cut it up along those edges, then push out the now-split vertex between them, until you can fit in a new edge. The result is a snub disphenoid. I'm not sure if the icosahedral bipyramid in 4D has enough degrees of freedom to allow an analogous construction, but it might be worth trying?

J85 (sphenocorona) - if we start with a pentagonal bipyramid, and this time cut four consecutive edges along the pentagonal cross-section, then we can stretch the now cut vertices apart and fill in the gaps with a triangle-square-square-triangle patch to make a sphenocorona. This does require a great deal of freedom of deformation, though; I'm not sure if it's possible in 4D?

J86 (sphenomegacorona) - it seems to be an analogous derivation to J84, applied to an icosahedron instead of a pentagonal bipyramid.

J87 (hebesphenomegacorona) - seems to be an analogous derivation to J85, applied to an icosahedron instead of a pentagonal bipyramid, and the inserted patch has 3 squares instead of just two.

J88 (disphenocingulum) - appears to be a distorted hexagonal antiprism capped by two "spheno" wedges. Not sure what the 4D analogue would be; perhaps some kind of duoprism suitably deformed in order to fit odd-shaped augments with non-flat joining surfaces? Sorta like D4.8.x in the sense of having non-flat joining surface, except simpler.

The question of interest, of course, is whether the 4D analogues of the pentagonal bipyramid / icosahedron have enough degrees of freedom to allow similar kinds of cuttings and deformations? Thanks to the Blind couple's results, we know that any crown jewels in this direction cannot consist only of tetrahedra or other regular cells; in all likelihood the "pushing apart" stage of the derivation will introduce triangular prisms and maybe other kinds of prisms. But still, these derivations are a lot simpler than the currently-known BT polychora, so perhaps there's a chance we can find something here?

Of particular interest is, suppose we cut off a piece of the 600-cell's surface to get a bunch of linked tetrahedra. What is the maximum number of tetrahedra in the patch such that it will have at least 1 degree of freedom? When does a patch containing tetrahedra, 5 around each edge, become forced to take on the exact curvature of the 600-cell? Such "flexible patches" of tetrahedra may be useful in constructing 4D crown jewels, since their extra degree(s) of freedom may permit enough deformation to insert unusual cell shapes to close them up in a CRF way.

Furthermore, suppose we have a CRF that contains two conjoined icosahedral cells, with dichoral symmetry between them. Suppose we now perform the ike -> J86 derivation on these cells, in correponding positions. Will it be possible to insert CRF cells to close up the new shape? If so, this will produce a non-trivial CRF with two J86 cells. The idea behind this derivation is that if the higher interconnectivity of 4D polychora makes it hard for direct generalization of the ike->J86 deformation, maybe we can use the 3D ike->J86 deformation, but applies to ikes within a 4D polytope, then patch up the result with new cells to close it up.

Thoughts?