24-Cell Sections and Net

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The sections of the 24-cell, a four-dimensional regular polytope built out of 24 octahedral cells, are 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 a fully intact octahedron, Section II a truncated octahedron with each face being a square, and Section III a cuboctahedron consisting entirely of equilateral triangles. However, all three sections are scaled equally as parts of the 24-cell. The triangles represent faces of the octahedral cells, while the hexagons and squares symbolize equatorial sections of the octahedra, with the hexagons being cut by a plane exactly halfway between two faces, and the square by a plane midway between two vertices. A partial "net" for the 24-cell is visually represented in the images by an assemblage of octahedra, constructed from 24 copies of Section I. The central uncolored cell forms the "south pole," with white cells surrounding it, followed by dark brown equatorial cells, then blue cells, and finally light brown cells forming the "north pole." Consistent coloring is used throughout the models, where the uncolored cell at the center or "south pole" serves as a focal point. The next layer consists of white cells, followed by dark brown cells (the equatorial cells), then blue cells, and ultimately light brown cells (the "north pole"). In ascending order, the five polyhedra are: the fully intact octahedron (Section I), the white truncated octahedron (Section II), the cuboctahedron consisting of triangles and squares (Section III), another truncated octahedron with alternating blue and dark brown cells (Section II), and finally a light brown fully intact octahedron (Section I). A choice had to be made when deciding on face color where the white and blue cells meet on either side of the dark brown equatorial cells. This resulted in the triangles on the cuboctahedron being colored with either blue or white, depending on the model. The models were created after studying Coxeter's <i>Regular Polytopes</i> during a graduate class.