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 |
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 B_{4}, D_{4}, or F_{4} lattice.^{[1]}^{[2]}
Vertices can be placed at all integer coordinates (i,j,k,l), such that the sum of the coordinates is even.
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 related D^{+}
_{4} lattice (also called D^{2}
_{4}) can be constructed by the union of two D_{4} lattices, and is identical to the C_{4} lattice:^{[6]}
The kissing number for D^{+}
_{4} is 2^{3} = 8, (2^{n – 1} for n < 8, 240 for n = 8, and 2n(n – 1) for n > 8).^{[7]}
The related 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, , containing all rectified 16-cells (24-cell) Voronoi cells, or .^{[10]}
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^{+})]] | ht_{0,4}{4,3,3,4} | 8+4+4: 4-demicube 8: 16-cell |
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 | , , |
Regular and uniform honeycombs in 4-space:
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^{10} | Uniform 10-honeycomb | {3^{[11]}} | δ_{11} | hδ_{11} | qδ_{11} | |
E^{n-1} | Uniform (n-1)-honeycomb | {3^{[n]}} | δ_{n} | hδ_{n} | qδ_{n} | 1_{k2} • 2_{k1} • k_{21} |