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 B4, D4, or F4 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 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 D+
4 lattice (also called D2
4) can be constructed by the union of two D4 lattices, and is identical to the C4 lattice:
[6]
The kissing number for D+
4 is 23 = 8, (2n – 1 for n < 8, 240 for n = 8, and 2n(n – 1) for n > 8).
[7]
The related D*
4 lattice (also called D4
4 and C2
4) 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]
The
kissing number of the D*
4 lattice (and D4 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 |
= [31,1,3,4] | = h{4,3,3,4} | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() [3,3,4], order 384 |
16+8: 16-cell |
= [31,1,1,1 | {3,31,1,1} = h{4,3,31,1} |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() [31,1,1], order 192 |
8+8+8: 16-cell |
2×½ = [[(4,3,3,4,2+)]] | ht0,4{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 |
[31,1,3,31,1] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
<[31,1,3,31,1]> ↔ [31,1,3,3,4] |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ↔ ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
×21 = | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]()
|
[[31,1,3,31,1]] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
×22 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
<2[31,1,3,31,1]> ↔ [4,3,3,3,4] |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ↔ ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
×41 = | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
[<2[31,1,3,31,1]>] ↔ [[4,3,3,3,4]] |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ↔ ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
×8 = ×2 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Regular and uniform honeycombs in 4-space:
Space | Family | / / | ||||
---|---|---|---|---|---|---|
E2 | Uniform tiling | {3[3]} | δ3 | hδ3 | qδ3 | Hexagonal |
E3 | Uniform convex honeycomb | {3[4]} | δ4 | hδ4 | qδ4 | |
E4 | Uniform 4-honeycomb | {3[5]} | δ5 | hδ5 | qδ5 | 24-cell honeycomb |
E5 | Uniform 5-honeycomb | {3[6]} | δ6 | hδ6 | qδ6 | |
E6 | Uniform 6-honeycomb | {3[7]} | δ7 | hδ7 | qδ7 | 222 |
E7 | Uniform 7-honeycomb | {3[8]} | δ8 | hδ8 | qδ8 | 133 • 331 |
E8 | Uniform 8-honeycomb | {3[9]} | δ9 | hδ9 | qδ9 | 152 • 251 • 521 |
E9 | Uniform 9-honeycomb | {3[10]} | δ10 | hδ10 | qδ10 | |
E10 | Uniform 10-honeycomb | {3[11]} | δ11 | hδ11 | qδ11 | |
En-1 | Uniform (n-1)- honeycomb | {3[n]} | δn | hδn | qδn | 1k2 • 2k1 • k21 |