Brad, Jim, good, glad you have a perfectionist working for you. If my
geometry was going into a talk at MoMA, of course I’d demand to be the speaker. Not likely this year or ever, haha...
Did you think of taking a plane intersection with a tiling of rhombic dodecahedra (RD) along the same symmetry plane?
Hadn’t thought about the specifics of the symmetrical case, though I did think about the average case. Start from triangular tiling through three vertices of each RD.
Call the set of all vertices L1. As the plane moves up or down, choose up or down triangles and shrink them linearly to a set of centroids L2u or L2d. The sets L2u + L1 and L2d + L1 are the vertices of two translation-Eq. regular hexagonal tilings.
I can’t really envision this; I’d need to see a picture. (Or pass a RD through the Wall of Fire.) The two intersections are similar, but with noticeable differences.
Your preferred cubic intersection geometry has three different polygons, while the RD intersection has only two. Also, the cubic intersection continually changes, while the RD intersection freezes the hexagonal tiling for a finite interval.
I also liked George’s idea about intersecting fractals. Has anyone tried taking plane sections of the Icosahedral Danzer tetrahedra tiling? (I’m guessing no.)
Not that I know of. Jim