by **quickfur** » Sat May 07, 2016 2:50 am

Today I got another crown jewel idea.

First, observe some patterns in 3D:

1) Start with an octahedron, pick a vertex, and a pair of opposite incident edges. Imagine cutting the octahedron with scissors along these two edges, then pulling apart the polyhedron, keeping all faces attached except at the cut edges. The polyhedron would deform as the vertex, now divided into two vertices, are pulled apart. The hole is in the shape of a pair of isosceles triangles, which, when the vertices are a unit length apart, become equilateral triangles that can be filled up with new faces. The result is a pentagonal bipyramid.

2) Start with an octahedron again, and this time cut it along a 3 edges lying on a plane, keeping the last edge at the back attached. It now becomes two square pyramids joined at an edge, and as the pyramids are pulled apart the result gap can be filled up with a triangular prism. The result is a biaugmented triangular prism.

3) Start with an icosahedron, and pick an edge. Cut it at this edge, and pull it apart into a square-shaped gap. The two triangles at the far ends of the edge deform into squares. The result is a hebesphenocorona.

Here are some comments on these constructions: in case (1), the chosen vertex expands into an edge, whereas the two incident edges expand into triangles. In case (2), the middle edge expands into a square, while the two side edges expand into triangles. In case (3), the chosen edge expands into a square, whereas two existing triangles deform into other squares.

So here's the idea for an analogous 4D construction: start with a 16-cell, and choose a triangular face. Cut the polychoron along this face, as well as along 3 adjacent triangles. Pull this triangle apart, producing a hole in the shape of a triangular prism. The 3 adjacent triangles should distort into square pyramids, and the original vertices should stretch into new edges. These new edges are where 3 new tetrahedral cells would appear, sitting between the new square pyramids (there were triangular faces originally where these new tetrahedra lie). Assuming the distortion can be done in a CRF way, the resulting polychoron should have 1 triangular prism, 3 square pyramids, 3 + 16 = 19 tetrahedra.

Does this construction work? Can the result be CRF? As far as I can tell, this works topologically, and the result ought to be convex, since 19 tetrahedra plus a few odd cells seem far short of the 600-cell, which is also convex, so the curvature of the result ought to be well within convex bounds. The only thing I'm unsure about is if all the edges can be made unit length: we may assume a rigid triangular prism, so the existence of the square faces is assured, so the question of whether it's possible to close up the polytope with tetrahedra in a CRF way, given the initial configuration of a triangular prism + 3 square pyramids, is reduced to whether all edges can be made unit length.