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Concept
24-cell honeycomb
Summary
In four-dimensional Euclidean geometry, the 24-cell honeycomb, or icositetrachoric honeycomb is a regular space-filling tessellation (or honeycomb) of 4-dimensional Euclidean space by regular 24-cells. It can be represented by Schläfli symbol {3,4,3,3}.
The dual tessellation by regular 16-cell honeycomb has Schläfli symbol {3,3,4,3}. Together with the tesseractic honeycomb (or 4-cubic honeycomb) these are the only regular tessellations of Euclidean 4-space.
The 24-cell honeycomb can be constructed as the Voronoi tessellation of the D4 or F4 root lattice. Each 24-cell is then centered at a D4 lattice point, i.e. one of
These points can also be described as Hurwitz quaternions with even square norm.
The vertices of the honeycomb lie at the deep holes of the D4 lattice. These are the Hurwitz quaternions with odd square norm.
It can be constructed as a birectified tesseractic honeycomb, by taking a tesseractic honeycomb and placing vertices at the centers of all the square faces. The 24-cell facets exist between these vertices as rectified 16-cells. If the coordinates of the tesseractic honeycomb are integers (i,j,k,l), the birectified tesseractic honeycomb vertices can be placed at all permutations of half-unit shifts in two of the four dimensions, thus: (i+1⁄2,j+1⁄2,k,l), (i+1⁄2,j,k+1⁄2,l), (i+1⁄2,j,k,l+1⁄2), (i,j+1⁄2,k+1⁄2,l), (i,j+1⁄2,k,l+1⁄2), (i,j,k+1⁄2,l+1⁄2).
Each 24-cell in the 24-cell honeycomb has 24 neighboring 24-cells. With each neighbor it shares exactly one octahedral cell.
It has 24 more neighbors such that with each of these it shares a single vertex.
It has no neighbors with which it shares only an edge or only a face.
The vertex figure of the 24-cell honeycomb is a tesseract (4-dimensional cube). So there are 16 edges, 32 triangles, 24 octahedra, and 8 24-cells meeting at every vertex. The edge figure is a tetrahedron, so there are 4 triangles, 6 octahedra, and 4 24-cells surrounding every edge. Finally, the face figure is a triangle, so there are 3 octahedra and 3 24-cells meeting at every face.
Official source
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