Английская Википедия:16-cell honeycomb
| 16-cell honeycomb | |
|---|---|
| Файл:Demitesseractic tetra hc.png Perspective projection: the first layer of adjacent 16-cell facets. | |
| Type | Regular 4-honeycomb Uniform 4-honeycomb |
| Family | Alternated hypercube honeycomb |
| Schläfli symbol | {3,3,4,3} |
| Coxeter diagrams | Шаблон:CDD Шаблон:CDD = Шаблон:CDD Шаблон:CDD = Шаблон:CDD Шаблон:CDD |
| 4-face type | {3,3,4} Файл:Schlegel wireframe 16-cell.png |
| Cell type | {3,3} Файл:Tetrahedron.png |
| Face type | {3} |
| Edge figure | cube |
| Vertex figure | Файл:24-cell t0 F4.svg 24-cell |
| Coxeter group | <math>{\tilde{F}}_4</math> = [3,3,4,3] |
| Dual | {3,4,3,3} |
| Properties | vertex-transitive, edge-transitive, face-transitive, cell-transitive, 4-face-transitive |
In four-dimensional Euclidean geometry, the 16-cell honeycomb is one of the three regular space-filling tessellations (or honeycombs), represented by Schläfli symbol {3,3,4,3}, and constructed by a 4-dimensional packing of 16-cell facets, three around every face.
Its dual is the 24-cell honeycomb. Its vertex figure is a 24-cell. The vertex arrangement is called the B4, D4, or F4 lattice.[1][2]
Alternate names
- Hexadecachoric tetracomb/honeycomb
- Demitesseractic tetracomb/honeycomb
Coordinates
Vertices can be placed at all integer coordinates (i,j,k,l), such that the sum of the coordinates is even.
D4 lattice
The vertex arrangement of the 16-cell honeycomb is called the D4 lattice or F4 lattice.[2] The vertices of this lattice are the centers of the 3-spheres in the densest known packing of equal spheres in 4-space;[3] its kissing number is 24, which is also the same as the kissing number in R4, as proved by Oleg Musin in 2003.[4][5]
The related DШаблон:Sup sub lattice (also called DШаблон:Sup sub) can be constructed by the union of two D4 lattices, and is identical to the C4 lattice:[6]
The kissing number for DШаблон:Sup sub is 23 = 8, (2n – 1 for n < 8, 240 for n = 8, and 2n(n – 1) for n > 8).[7]
The related DШаблон:Sup sub lattice (also called DШаблон:Sup sub and CШаблон:Sup sub) can be constructed by the union of all four D4 lattices, but it is identical to the D4 lattice: It is also the 4-dimensional body centered cubic, the union of two 4-cube honeycombs in dual positions.[8]
- Шаблон:CDD ∪ Шаблон:CDD ∪ Шаблон:CDD ∪ Шаблон:CDD = Шаблон:CDD = Шаблон:CDD ∪ Шаблон:CDD.
The kissing number of the DШаблон:Sup sub lattice (and D4 lattice) is 24[9] and its Voronoi tessellation is a 24-cell honeycomb, Шаблон:CDD, containing all rectified 16-cells (24-cell) Voronoi cells, Шаблон:CDD or Шаблон:CDD.[10]
Symmetry constructions
There are three different symmetry constructions of this tessellation. Each symmetry can be represented by different arrangements of colored 16-cell facets.
| Coxeter group | Schläfli symbol | Coxeter diagram | Vertex figure Symmetry |
Facets/verf |
|---|---|---|---|---|
| <math>{\tilde{F}}_4</math> = [3,3,4,3] | {3,3,4,3} | Шаблон:CDD | Шаблон:CDD [3,4,3], order 1152 |
24: 16-cell |
| <math>{\tilde{B}}_4</math> = [31,1,3,4] | = h{4,3,3,4} | Шаблон:CDD = Шаблон:CDD | Шаблон:CDD [3,3,4], order 384 |
16+8: 16-cell |
| <math>{\tilde{D}}_4</math> = [31,1,1,1] | {3,31,1,1} = h{4,3,31,1} |
Шаблон:CDD = Шаблон:CDD | Шаблон:CDD [31,1,1], order 192 |
8+8+8: 16-cell |
| 2×½<math>{\tilde{C}}_4</math> = [[(4,3,3,4,2+)]] | ht0,4{4,3,3,4} | Шаблон:CDD | 8+4+4: 4-demicube 8: 16-cell |
Related honeycombs
It is related to the regular hyperbolic 5-space 5-orthoplex honeycomb, {3,3,3,4,3}, with 5-orthoplex facets, the regular 4-polytope 24-cell, {3,4,3} with octahedral (3-orthoplex) cell, and cube {4,3}, with (2-orthoplex) square faces.
It has a 2-dimensional analogue, {3,6}, and as an alternated form (the demitesseractic honeycomb, h{4,3,3,4}) it is related to the alternated cubic honeycomb.
See also
Regular and uniform honeycombs in 4-space:
- Tesseractic honeycomb
- 24-cell honeycomb
- Truncated 24-cell honeycomb
- Snub 24-cell honeycomb
- 5-cell honeycomb
- Truncated 5-cell honeycomb
- Omnitruncated 5-cell honeycomb
Notes
References
- Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, Шаблон:ISBN
- pp. 154–156: Partial truncation or alternation, represented by h prefix: h{4,4} = {4,4}; h{4,3,4} = {31,1,4}, h{4,3,3,4} = {3,3,4,3}, ...
- 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, Шаблон:ISBN [1]
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
- George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)
- Шаблон:KlitzingPolytopes x3o3o4o3o - hext - O104
- Шаблон:Cite book
- ↑ Шаблон:Cite web
- ↑ 2,0 2,1 Шаблон:Cite web
- ↑ Conway and Sloane, Sphere packings, lattices, and groups, 1.4 n-dimensional packings, p.9
- ↑ Conway and Sloane, Sphere packings, lattices, and groups, 1.5 Sphere packing problem summary of results, p. 12
- ↑ Шаблон:Cite journal
- ↑ Conway and Sloane, Sphere packings, lattices, and groups, 7.3 The packing D3+, p.119
- ↑ Conway and Sloane, Sphere packings, lattices, and groups, p. 119
- ↑ Conway and Sloane, Sphere packings, lattices, and groups, 7.4 The dual lattice D3*, p.120
- ↑ Conway and Sloane, Sphere packings, lattices, and groups, p. 120
- ↑ Conway and Sloane, Sphere packings, lattices, and groups, p. 466
