24-cell Honeycomb - Cross-sections

Cross-sections

One way to visualize 4-dimensional figures is to consider various 3-dimensional cross-sections. Applying this technique to the 24-cell honeycomb gives rise to various 3-dimensional honeycombs with varying degrees of regularity.

A vertex-first cross-section is one orthogonal to a line joining opposite vertices of one of the 24-cells. For instance, one could take any of the coordinate hyperplanes in the coordinate system given above (i.e. the planes determined by x_i = 0). The cross-section of {3,4,3,3} by one of these hyperplanes gives a rhombic dodecahedral honeycomb. Each of the rhombic dodecahedra corresponds to a maximal cross-section of one of the 24-cells intersecting the hyperplane (the center of each such 24-cell lies in the hyperplane). Accordingly, the rhombic dodecahedral honeycomb is the Voronoi tessellation of the D₃ root lattice (a face-centered cubic lattice). Shifting this hyperplane halfway to one of the vertices (e.g. x_i = ½) gives rise to a regular cubic honeycomb. In this case the center of each 24-cell lies off the hyperplane. Shifting again, so the hyperplane intersects the vertex, gives another rhombic dodecahedral honeycomb but with new 24-cells (the former ones having shrunk to points). In general, for any integer n, the cross-section through x_i = n is a rhombic dodecahedral honeycomb, and the cross-section through x_i = n + ½ is a cubic honeycomb. As the hyperplane moves through 4-space, the cross-section morphs between the two periodically.

A cell-first cross-section is one parallel to one of the octahedral cells of a 24-cell. Consider, for instance, the hyperplane orthogonal to (1,1,0,0). The cross-section of {3,4,3,3} by this hyperplane is a rectified cubic honeycomb. Each cuboctahedron in this honeycomb is a maximal cross-section of a 24-cell whose center lies in the plane. Meanwhile, each octahedron is a boundary cell of a 24-cell whose center lies off the plane. Shifting this hyperplane till it lies halfway between the center of a 24-cell and the boundary, one obtains a bitruncated cubic honeycomb. The cuboctahedra have shrunk, and the octahedra have grown until they are both truncated octahedra. Shifting again, so the hyperplane intersects the boundary of the central 24-cell gives a rectified cubic honeycomb again, the cuboctahedra and octahedra having swapped positions. As the hyperplane sweeps through 4-space, the cross-section morphs between these two honeycombs periodically.