16-cell honeycomb
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.
16-cell honeycomb | |
---|---|
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 | |
4-face type | {3,3,4} |
Cell type | {3,3} |
Face type | {3} |
Edge figure | cube |
Vertex figure | 24-cell |
Coxeter group | = [3,3,4,3] |
Dual | {3,4,3,3} |
Properties | vertex-transitive, edge-transitive, face-transitive, cell-transitive, 4-face-transitive |
Its dual is the 24-cell honeycomb. Its vertex figure is a 24-cell. The vertex arrangement is called the B_{4}, D_{4}, or F_{4} 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.
D_{4} lattice
The vertex arrangement of the 16-cell honeycomb is called the D_{4} lattice or F_{4} 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 R^{4}, as proved by Oleg Musin in 2003.[4][5]
The D^{+}
_{4} lattice (also called D^{2}
_{4}) can be constructed by the union of two D_{4} lattices, and is identical to the tesseractic honeycomb:[6]
∪ = =
This packing is only a lattice for even dimensions. The kissing number is 2^{3} = 8, (2^{n – 1} for n < 8, 240 for n = 8, and 2n(n – 1) for n > 8).[7]
The D^{*}
_{4} lattice (also called D^{4}
_{4} and C^{2}
_{4}) can be constructed by the union of all four D_{4} lattices, but it is identical to the D_{4} lattice: It is also the 4-dimensional body centered cubic, the union of two 4-cube honeycombs in dual positions.[8]
∪ ∪ ∪ = = ∪ .
The kissing number of the D^{*}
_{4} lattice (and D_{4} lattice) is 24[9] and its Voronoi tessellation is a 24-cell honeycomb,
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 |
---|---|---|---|---|
= [3,3,4,3] | {3,3,4,3} | [3,4,3], order 1152 | 24: 16-cell | |
= [3^{1,1},3,4] | = h{4,3,3,4} | [3,3,4], order 384 | 16+8: 16-cell | |
= [3^{1,1,1,1}] | {3,3^{1,1,1}} = h{4,3,3^{1,1}} | [3^{1,1,1}], order 192 | 8+8+8: 16-cell | |
2×½ = [[(4,3,3,4,2<sup>+</sup>)]] | ht_{0,4}{4,3,3,4} | 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.
This honeycomb is one of 20 uniform honeycombs constructed by the Coxeter group, all but 3 repeated in other families by extended symmetry, seen in the graph symmetry of rings in the Coxeter–Dynkin diagrams. The 20 permutations are listed with its highest extended symmetry relation:
D5 honeycombs | |||
---|---|---|---|
Extended symmetry |
Extended diagram |
Extended group |
Honeycombs |
[3^{1,1},3,3^{1,1}] | |||
<[3^{1,1},3,3^{1,1}]> ↔ [3^{1,1},3,3,4] |
↔ |
×2_{1} = | |
[[3^{1,1},3,3^{1,1}]] | ×2_{2} | ||
<2[3^{1,1},3,3^{1,1}]> ↔ [4,3,3,3,4] |
↔ |
×4_{1} = | |
[<2[3^{1,1},3,3^{1,1}]>] ↔ [[4,3,3,3,4]] |
↔ |
×8 = ×2 |
See also
Regular and uniform honeycombs in 4-space:
Notes
- http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/F4.html
- http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/D4.html
- 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
- O. R. Musin (2003). "The problem of the twenty-five spheres". Russ. Math. Surv. 58: 794–795. Bibcode:2003RuMaS..58..794M. doi:10.1070/RM2003v058n04ABEH000651.
- Conway and Sloane, Sphere packings, lattices, and groups, 7.3 The packing D_{3}^{+}, 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 D_{3}^{*}, p.120
- Conway and Sloane, Sphere packings, lattices, and groups, p. 120
- Conway and Sloane, Sphere packings, lattices, and groups, p. 466
References
- Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8
- pp. 154–156: Partial truncation or alternation, represented by h prefix: h{4,4} = {4,4}; h{4,3,4} = {3^{1,1},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, ISBN 978-0-471-01003-6
- (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)
- Klitzing, Richard. "4D Euclidean tesselations". x3o3o4o3o - hext - O104
- Conway JH, Sloane NJH (1998). Sphere Packings, Lattices and Groups (3rd ed.). ISBN 0-387-98585-9.
Fundamental convex regular and uniform honeycombs in dimensions 2-9 | ||||||
---|---|---|---|---|---|---|
Space | Family | / / | ||||
E^{2} | Uniform tiling | {3^{[3]}} | δ_{3} | hδ_{3} | qδ_{3} | Hexagonal |
E^{3} | Uniform convex honeycomb | {3^{[4]}} | δ_{4} | hδ_{4} | qδ_{4} | |
E^{4} | Uniform 4-honeycomb | {3^{[5]}} | δ_{5} | hδ_{5} | qδ_{5} | 24-cell honeycomb |
E^{5} | Uniform 5-honeycomb | {3^{[6]}} | δ_{6} | hδ_{6} | qδ_{6} | |
E^{6} | Uniform 6-honeycomb | {3^{[7]}} | δ_{7} | hδ_{7} | qδ_{7} | 2_{22} |
E^{7} | Uniform 7-honeycomb | {3^{[8]}} | δ_{8} | hδ_{8} | qδ_{8} | 1_{33} • 3_{31} |
E^{8} | Uniform 8-honeycomb | {3^{[9]}} | δ_{9} | hδ_{9} | qδ_{9} | 1_{52} • 2_{51} • 5_{21} |
E^{9} | Uniform 9-honeycomb | {3^{[10]}} | δ_{10} | hδ_{10} | qδ_{10} | |
E^{n-1} | Uniform (n-1)-honeycomb | {3^{[n]}} | δ_{n} | hδ_{n} | qδ_{n} | 1_{k2} • 2_{k1} • k_{21} |