# 16-cell honeycomb

16-cell honeycomb

Perspective projection: the first layer of adjacent 16-cell facets.
Type Regular 4-space honeycomb
Uniform 4-honeycomb
Family Alternated hypercube honeycomb
Schläfli symbol {3,3,4,3}
Coxeter-Dynkin diagram
4-face type {3,3,4}
Cell type {3,3}
Face type {3}
Edge figure cube
Vertex figure
24-cell
Coxeter group ${\tilde{F}}_4$ = [3,3,4,3]
Dual {3,4,3,3}
Properties vertex-transitive, edge-transitive, face-transitive, cell-transitive

In four-dimensional Euclidean geometry, the 16-cell honeycomb is the one of three regular space-filling tessellation (or honeycomb) in Euclidean 4-space. The other two are its dual the 24-cell honeycomb, and the tesseractic honeycomb. This honeycomb is constructed from 16-cell facets, three around every face. It has a 24-cell vertex figure.

This vertex arrangement or lattice is called the B4, D4, or F4 lattice.[1][2]

## Alternate names

• Demitesseractic tetracomb/honeycomb

## Coordinates

As a regular honeycomb, {3,3,4,3}, it has no lower dimensional analogues, but as an alternated form (the demitesseractic honeycomb, h{4,3,3,4}) it is related to the alternated cubic honeycomb.

Vertices can be placed at all integer coordinates (i,j,k,l), such that the sum of the coordinates is even.

## D4 lattice

Its vertex arrangement is called the D4 lattice or F4 lattice.[2] The vertices of this lattice are the centers of the 3-spheres in the densest possible packing of equal spheres in 4-space; its kissing number is 24, which is also the highest possible in 4-space.[3]

=

The D+
4
lattice (also called D2
4
) can be constructed by the union of two 4-demicubic lattices, and is identical to the tesseractic honeycomb:

= =

This packing is only a lattice for even dimensions. The kissing number is 23=8, (2n-1 for n<8, 240 for n=8, and 2n(n-1) for n>8).[4]

The D*
4
lattice (also called D4
4
and C2
4
) can be constructed by the union of all four 5-demicubic honeycombs, 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.

= = .

The kissing number of the D*
4
lattice (and D4 lattice) is 24[5] and its Voronoi tessellation is a 24-cell honeycomb, , containing all rectified 16-cells (24-cell) Voronoi cells, or .[6]

## Symmetry constructions

There are three different symmetry constructions of this tessellation. Each symmetry can be represented by different arrangements of colored 16-cell facets.

Name Coxeter group Schläfli symbol Coxeter diagram Vertex figure
Symmetry
Facets/verf
16-cell honeycomb ${\tilde{F}}_4$ = [3,3,4,3] {3,3,4,3}
[3,4,3], order 1152
24: 16-cell
4-demicube honeycomb ${\tilde{B}}_4$ = [31,1,3,4] = h{4,3,3,4} =
[3,3,4], order 384
16+8: 16-cell
${\tilde{D}}_4$ = [31,1,1,1] {3,31,1,1}
= h{4,3,31,1}
=
[31,1,1], order 192
8+8+8: 16-cell

## Related honeycombs

The [3,4,3,3], , Coxeter group generates 31 permutations of uniform tessellations, 28 are unique in this family and ten are shared in the [4,3,3,4] and [4,3,31,1] families. The alternation (13) is also repeated in other families.

Extended
symmetry
Extended
diagram
Order Honeycombs
[3,3,4,3] ×1

1, 3, 5, 6, 8,
9, 10, 11, 12

[3,4,3,3] ×1

2, 4, 7, 13,
14, 15, 16, 17,
18, 19, 20, 21,
22 23, 24, 25,
26, 27, 28, 29

[(3,3)[3,3,4,3*]]
=[(3,3)[31,1,1,1]]
=[3,4,3,3]

=
=
×4

The [4,3,3,4], , Coxeter group generates 31 permutations of uniform tessellations, 21 with distinct symmetry and 20 with distinct geometry. The expanded tesseractic honeycomb (also known as the stericated tesseractic honeycomb) is geometrically identical to the tesseractic honeycomb. Three of the symmetric honeycombs are shared in the [3,4,3,3] family. Two alternations (13) and (17), and the quarter tesseractic (2) are repeated in other families.

Extended
symmetry
Extended
diagram
Order Honeycombs
[4,3,3,4]: ×1

1, 2, 3, 4,
5, 6, 7, 8,
9, 10, 11, 12,
13

[[4,3,3,4]] ×2 (1), (2), (13), 18
(6), 19, 20
[(3,3)[1+,4,3,3,4,1+]]
= [(3,3)[31,1,1,1]]
= [3,4,3,3]

=
=
×6

14, 15, 16, 17

The [4,3,31,1], , Coxeter group generates 31 permutations of uniform tessellations, 23 with distinct symmetry and 4 with distinct geometry. There are two alternated forms: the alternations (19) and (24) have the same geometry as the 16-cell honeycomb and snub 24-cell honeycomb respectively.

Extended
symmetry
Extended
diagram
Order Honeycombs
[4,3,31,1]: ×1

5, 6, 7, 8

<[4,3,31,1]>:
=[4,3,3,4]

=
×2

9, 10, 11, 12, 13, 14,

(10), 15, 16, (13), 17, 18, 19

[3[1+,4,3,31,1]]
= [3[3,31,1,1]]
= [3,3,4,3]

=
=
×3

1, 2, 3, 4

[(3,3)[1+,4,3,31,1]]
= [(3,3)[31,1,1,1]]
= [3,4,3,3]

=
=
×12

20, 21, 22, 23

This honeycomb is one of ten uniform honeycombs constructed by the ${\tilde{D}}_4$ Coxeter group, all repeated in other families by extended symmetry, seen in the graph symmetry of rings in the Coxeter–Dynkin diagrams. The 10th is constructed as an alternation. As subgroups in Coxeter notation: [3,4,(3,3)*] (index 24), [3,3,4,3*] (index 6), [1+,4,3,3,4,1+] (index 4), [31,1,3,4,1+] (index 2) are all isomorphic to [31,1,1,1]. The ten permutations are listed with its highest extended symmetry relation:

Extended
symmetry
Extended
diagram
Order Honeycombs
[31,1,1,1] ×1 (none)
<[31,1,1,1]>
= [31,1,3,4]

=
×2 (none)
<<[1,131,1]>>
= [4,3,3,4]

=
×4 1, 2
[3[3,31,1,1]]
= [3,4,3,3]

=
×6 3, 4, 5, 6
[<<[1,131,1]>>]
= [[4,3,3,4]]

=
×8 7, 8, 9, 10
[(3,3)[31,1,1,1]]
= [3,3,4,3]

=
×24

Regular and uniform honeycombs in 4-space:

## Notes

1. ^ http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/F4.html
2. ^ a b http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/D4.html
3. ^ O. R. Musin (2003). "The problem of the twenty-five spheres". Russ. Math. Surv. 58: 794–795. doi:10.1070/RM2003v058n04ABEH000651.
4. ^ Conway (1998), p. 119
5. ^ Conway (1998), p. 120
6. ^ Conway (1998), 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} = {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, ISBN 978-0-471-01003-6 [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)
• Richard Klitzing, 4D, Euclidean tesselations x3o3o4o3o - hext - O104
• Conway JH, Sloane NJH (1998). Sphere Packings, Lattices and Groups (3rd ed.). ISBN 0-387-98585-9.