@johncarlosbaez@mathstodon.xyz Here are some photos of a 3D print
@saulsch@mathstodon.xyz and I made, of a toroidal half of the 120-cell. On the torus boundary, you can see the 10x10 grid of vertices that correspond to
@jasonhise@mathstodon.xyz 's 10x10 grid of tetrahedra. I also found a screenshot with solid dodecahedra poking through the torus boundary.
At the core of the solid torus, there is a ring of ten dodecahedra. And there is a dual ring of dodecahedra in the other toroidal half. The two rings face each other, equidistant from the 10x10 grid of vertices.
That core ring of dodecahedra is surrounded by five other rings of dodecahedra, that together make up the toroidal half of the 120-cell. These twist around in a discrete version of the Hopf fibration. (The figure here is from our paper at
https://www.ams.org/journals/notices/201511/rnoti-p1309.pdf) Each of the five other rings has a corresponding dual ring in the other half, and a corresponding equidistant 10x10 grid of vertices. Since the pairs of opposite rings of dodecahedra partition the 120-cell, and there is a fixed-point free action moving one pair of rings to another, I think that should be enough to imply that the corresponding 10x10 grids are disjoint.
So, I don't think we ever thought about these sheets of tetrahedra before, but they're not so far away from these patterns of rings in the 120-cell.
