24-Cell Sections and Net

The sections of the 24-cell (a four-dimensional regular polytope built out of 24 octahedral cells) as cut by three-dimensional hyperplanes parallel to one cell and intersecting vertices and midpoints of edges, from an initial cell at the "south pole" (Section I) to the "equator" (Section III).

Section I is an octahedron, Section II a truncated octahedron, and Section III a cuboctahedron, but the three sections are to scale as sections of the 24-cell. The triangles are faces of the octahedral cells, while the hexagons and squares are equatorial sections of the octahedra, the hexagons being cut by a plane midway between two faces, and the square by a plane midway between two vertices.

The assemblage of octahedra shown in the images represents the a partial "net" for the 24-cell, built from 24 copies of Section I, with the octahedral cell forming the "south pole" at the center, and the light brown octahedron at the base as the "north pole."

The models are all colored consistently, with the cell at the center or "south pole" being uncolored, those around it white, the next layer dark brown (the "equatorial" cells), then blue, and finally light brown (the "north pole"). The five polyhedra are the five sections, ascending through the white octahedron (Section I), the white and dark brown truncated octahedron (Section II), the blue and dark brown cuboctahedron (Section III), the blue and dark brown truncated octahedron (Section II), and the light brown octahedron (Section I). Where the white and blue cells meet on either side of the dark brown equatorial cells, a choice had to be made for face color. So the triangles on the cuboctahedron could have been either blue or white.

Created after reading Coxeter's Regular Polytopes with a graduate class.