Triangular tilingTetrahedral-octahedral honeycombWith red and yellow equilateral trianglesWith cyan and yellow tetrahedra, and red rectified tetrahedra (octahedra)
In geometry, the simplicial honeycomb (or -simplex honeycomb) is a dimensional infinite series of honeycombs, based on the affine Coxeter group symmetry. It is represented by a Coxeter-Dynkin diagram as a cyclic graph of nodes with one node ringed. It is composed of -simplex facets, along with all rectified -simplices. It can be thought of as an -dimensional hypercubic honeycomb that has been subdivided along all hyperplanes , then stretched along its main diagonal until the simplices on the ends of the hypercubes become regular. The vertex figure of an -simplex honeycomb is an expanded -simplex.
In 2 dimensions, the honeycomb represents the triangular tiling, with Coxeter graph filling the plane with alternately colored triangles. In 3 dimensions it represents the tetrahedral-octahedral honeycomb, with Coxeter graph filling space with alternately tetrahedral and octahedral cells. In 4 dimensions it is called the 5-cell honeycomb, with Coxeter graph , with 5-cell and rectified 5-cell facets. In 5 dimensions it is called the 5-simplex honeycomb, with Coxeter graph , filling space by 5-simplex, rectified 5-simplex, and birectified 5-simplex facets. In 6 dimensions it is called the 6-simplex honeycomb, with Coxeter graph , filling space by 6-simplex, rectified 6-simplex, and birectified 6-simplex facets.
By dimension nTessellationVertex figureFacets per vertex figureVertices per vertex figureEdge figure1ApeirogonLine segment22Point2Triangular tiling2-simplex honeycombHexagon(Truncated triangle)3+3 triangles6Line segment3Tetrahedral-octahedral honeycomb3-simplex honeycombCuboctahedron(Cantellated tetrahedron)4+4 tetrahedra6 rectified tetrahedra12Rectangle44-simplex honeycombRuncinated 5-cell5+5 5-cells10+10 rectified 5-cells20Triangular antiprism55-simplex honeycombStericated 5-simplex6+6 5-simplex15+15 rectified 5-simplex20 birectified 5-simplex30Tetrahedral antiprism66-simplex honeycombPentellated 6-simplex7+7 6-simplex21+21 rectified 6-simplex35+35 birectified 6-simplex424-simplex antiprism77-simplex honeycombHexicated 7-simplex8+8 7-simplex28+28 rectified 7-simplex56+56 birectified 7-simplex70 trirectified 7-simplex565-simplex antiprism88-simplex honeycombHeptellated 8-simplex9+9 8-simplex36+36 rectified 8-simplex84+84 birectified 8-simplex126+126 trirectified 8-simplex726-simplex antiprism99-simplex honeycombOctellated 9-simplex10+10 9-simplex45+45 rectified 9-simplex120+120 birectified 9-simplex210+210 trirectified 9-simplex252 quadrirectified 9-simplex907-simplex antiprism1010-simplex honeycombEnnecated 10-simplex11+11 10-simplex55+55 rectified 10-simplex165+165 birectified 10-simplex330+330 trirectified 10-simplex462+462 quadrirectified 10-simplex1108-simplex antiprism1111-simplex honeycomb............
Projection by folding
The (2n−1)-simplex honeycombs and 2n-simplex honeycombs can be projected into the n-dimensional hypercubic honeycomb by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same vertex arrangement:
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Kissing number
These honeycombs, seen as tangent n-spheres located at the center of each honeycomb vertex have a fixed number of contacting spheres and correspond to the number of vertices in the vertex figure. This represents the highest kissing number for 2 and 3 dimensions, but falls short on higher dimensions. In 2-dimensions, the triangular tiling defines a circle packing of 6 tangent spheres arranged in a regular hexagon, and for 3 dimensions there are 12 tangent spheres arranged in a cuboctahedral configuration. For 4 to 8 dimensions, the kissing numbers are 20, 30, 42, 56, and 72 spheres, while the greatest solutions are 24, 40, 72, 126, and 240 spheres respectively.
See also Hypercubic honeycomb Alternated hypercubic honeycomb Quarter hypercubic honeycomb Truncated simplicial honeycomb Omnitruncated simplicial honeycomb
References George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs) Branko Grünbaum, Uniform tilings of 3-space. Geombinatorics 4(1994), 49 - 56. Norman Johnson Uniform Polytopes, Manuscript (1991) Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10] (1.9 Uniform space-fillings) (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]