| A ~ 2 {\displaystyle {\tilde {A}}_{2}} | A ~ 3 {\displaystyle {\tilde {A}}_{3}} |
|---|---|
| Triangular tiling | Tetrahedral-octahedral honeycomb |
| With red and yellow equilateral triangles | With cyan and yellow tetrahedra, and red rectified tetrahedra (octahedra) |
In geometry, the simplicial honeycomb (or n-simplex honeycomb) is a dimensional infinite series of honeycombs, based on the A ~ n {\displaystyle {\tilde {A}}_{n}} affine Coxeter group symmetry. It is represented by a Coxeter-Dynkin diagram as a cyclic graph of n + 1 nodes with one node ringed. It is composed of n-simplex facets, along with all rectified n-simplices. It can be thought of as an n-dimensional hypercubic honeycomb that has been subdivided along all hyperplanes x + y + ⋯ ∈ Z {\displaystyle x+y+\cdots \in \mathbb {Z} } , then stretched along its main diagonal until the simplices on the ends of the hypercubes become regular. The vertex figure of an n-simplex honeycomb is an expanded n-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
| n | A ~ 2 + {\displaystyle {\tilde {A}}_{2+}} | Tessellation | Vertex figure | Facets per vertex figure | Vertices per vertex figure | Edge figure |
|---|---|---|---|---|---|---|
| 1 | A ~ 1 {\displaystyle {\tilde {A}}_{1}} | Apeirogon | Line segment | 2 | 2 | Point |
| 2 | A ~ 2 {\displaystyle {\tilde {A}}_{2}} | Triangular tiling2-simplex honeycomb | Hexagon(Truncated triangle) | 3+3 triangles | 6 | Line segment |
| 3 | A ~ 3 {\displaystyle {\tilde {A}}_{3}} | Tetrahedral-octahedral honeycomb3-simplex honeycomb | Cuboctahedron(Cantellated tetrahedron) | 4+4 tetrahedron6 rectified tetrahedra | 12 | Rectangle |
| 4 | A ~ 4 {\displaystyle {\tilde {A}}_{4}} | 4-simplex honeycomb | Runcinated 5-cell | 5+5 5-cells10+10 rectified 5-cells | 20 | Triangular antiprism |
| 5 | A ~ 5 {\displaystyle {\tilde {A}}_{5}} | 5-simplex honeycomb | Stericated 5-simplex | 6+6 5-simplex15+15 rectified 5-simplex20 birectified 5-simplex | 30 | Tetrahedral antiprism |
| 6 | A ~ 6 {\displaystyle {\tilde {A}}_{6}} | 6-simplex honeycomb | Pentellated 6-simplex | 7+7 6-simplex21+21 rectified 6-simplex35+35 birectified 6-simplex | 42 | 4-simplex antiprism |
| 7 | A ~ 7 {\displaystyle {\tilde {A}}_{7}} | 7-simplex honeycomb | Hexicated 7-simplex | 8+8 7-simplex28+28 rectified 7-simplex56+56 birectified 7-simplex70 trirectified 7-simplex | 56 | 5-simplex antiprism |
| 8 | A ~ 8 {\displaystyle {\tilde {A}}_{8}} | 8-simplex honeycomb | Heptellated 8-simplex | 9+9 8-simplex36+36 rectified 8-simplex84+84 birectified 8-simplex126+126 trirectified 8-simplex | 72 | 6-simplex antiprism |
| 9 | A ~ 9 {\displaystyle {\tilde {A}}_{9}} | 9-simplex honeycomb | Octellated 9-simplex | 10+10 9-simplex45+45 rectified 9-simplex120+120 birectified 9-simplex210+210 trirectified 9-simplex252 quadrirectified 9-simplex | 90 | 7-simplex antiprism |
| 10 | A ~ 10 {\displaystyle {\tilde {A}}_{10}} | 10-simplex honeycomb | Ennecated 10-simplex | 11+11 10-simplex55+55 rectified 10-simplex165+165 birectified 10-simplex330+330 trirectified 10-simplex462+462 quadrirectified 10-simplex | 110 | 8-simplex antiprism |
| 11 | A ~ 11 {\displaystyle {\tilde {A}}_{11}} | 11-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:
| A ~ 2 {\displaystyle {\tilde {A}}_{2}} | A ~ 4 {\displaystyle {\tilde {A}}_{4}} | A ~ 6 {\displaystyle {\tilde {A}}_{6}} | A ~ 8 {\displaystyle {\tilde {A}}_{8}} | A ~ 10 {\displaystyle {\tilde {A}}_{10}} | ... | |||||
|---|---|---|---|---|---|---|---|---|---|---|
| A ~ 3 {\displaystyle {\tilde {A}}_{3}} | A ~ 3 {\displaystyle {\tilde {A}}_{3}} | A ~ 5 {\displaystyle {\tilde {A}}_{5}} | A ~ 7 {\displaystyle {\tilde {A}}_{7}} | A ~ 9 {\displaystyle {\tilde {A}}_{9}} | ... | |||||
| C ~ 1 {\displaystyle {\tilde {C}}_{1}} | C ~ 2 {\displaystyle {\tilde {C}}_{2}} | C ~ 3 {\displaystyle {\tilde {C}}_{3}} | C ~ 4 {\displaystyle {\tilde {C}}_{4}} | C ~ 5 {\displaystyle {\tilde {C}}_{5}} | ... |
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
- 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, ISBN 0-486-61480-8
- 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, ISBN 978-0-471-01003-6 [1]
- (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]
| ||||||
|---|---|---|---|---|---|---|
| Space | Family | A ~ n − 1 {\displaystyle {\tilde {A}}_{n-1}} | C ~ n − 1 {\displaystyle {\tilde {C}}_{n-1}} | B ~ n − 1 {\displaystyle {\tilde {B}}_{n-1}} | D ~ n − 1 {\displaystyle {\tilde {D}}_{n-1}} | G ~ 2 {\displaystyle {\tilde {G}}_{2}} / F ~ 4 {\displaystyle {\tilde {F}}_{4}} / E ~ n − 1 {\displaystyle {\tilde {E}}_{n-1}} |
| E2 | Uniform tiling | 0[3] | δ3 | hδ3 | qδ3 | Hexagonal |
| E3 | Uniform convex honeycomb | 0[4] | δ4 | hδ4 | qδ4 | |
| E4 | Uniform 4-honeycomb | 0[5] | δ5 | hδ5 | qδ5 | 24-cell honeycomb |
| E5 | Uniform 5-honeycomb | 0[6] | δ6 | hδ6 | qδ6 | |
| E6 | Uniform 6-honeycomb | 0[7] | δ7 | hδ7 | qδ7 | 222 |
| E7 | Uniform 7-honeycomb | 0[8] | δ8 | hδ8 | qδ8 | 133 • 331 |
| E8 | Uniform 8-honeycomb | 0[9] | δ9 | hδ9 | qδ9 | 152 • 251 • 521 |
| E9 | Uniform 9-honeycomb | 0[10] | δ10 | hδ10 | qδ10 | |
| E10 | Uniform 10-honeycomb | 0[11] | δ11 | hδ11 | qδ11 | |
| En-1 | Uniform (n-1)-honeycomb | 0[n] | δn | hδn | qδn | 1k2 • 2k1 • k21 |