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16-cell honeycomb

## Summary

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 ${\displaystyle {\tilde {F}}_{4}}$ = [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

• 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 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]

## 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
${\displaystyle {\tilde {F}}_{4}}$  = [3,3,4,3] {3,3,4,3}
[3,4,3], order 1152
24: 16-cell
${\displaystyle {\tilde {B}}_{4}}$  = [31,1,3,4] = h{4,3,3,4}         =
[3,3,4], order 384
16+8: 16-cell
${\displaystyle {\tilde {D}}_{4}}$  = [31,1,1,1] {3,31,1,1}
= h{4,3,31,1}
=
[31,1,1], order 192
8+8+8: 16-cell
2×½${\displaystyle {\tilde {C}}_{4}}$  = [[(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 ${\displaystyle {\tilde {D}}_{5}}$  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]         ${\displaystyle {\tilde {D}}_{5}}$
<[31,1,3,31,1]>
↔ [31,1,3,3,4]

${\displaystyle {\tilde {D}}_{5}}$ ×21 = ${\displaystyle {\tilde {B}}_{5}}$         ,        ,        ,

,        ,        ,

[[31,1,3,31,1]]         ${\displaystyle {\tilde {D}}_{5}}$ ×22        ,
<2[31,1,3,31,1]>
↔ [4,3,3,3,4]

${\displaystyle {\tilde {D}}_{5}}$ ×41 = ${\displaystyle {\tilde {C}}_{5}}$         ,        ,        ,        ,        ,
[<2[31,1,3,31,1]>]
↔ [[4,3,3,3,4]]

${\displaystyle {\tilde {D}}_{5}}$ ×8 = ${\displaystyle {\tilde {C}}_{5}}$ ×2        ,        ,

Regular and uniform honeycombs in 4-space:

## Notes

1. ^ "The Lattice F4".
2. ^ a b "The Lattice D4".
3. ^ Conway and Sloane, Sphere packings, lattices, and groups, 1.4 n-dimensional packings, p.9
4. ^ Conway and Sloane, Sphere packings, lattices, and groups, 1.5 Sphere packing problem summary of results, p. 12
5. ^ O. R. Musin (2003). "The problem of the twenty-five spheres". Russ. Math. Surv. 58 (4): 794–795. Bibcode:2003RuMaS..58..794M. doi:10.1070/RM2003v058n04ABEH000651.
6. ^ Conway and Sloane, Sphere packings, lattices, and groups, 7.3 The packing D3+, p.119
7. ^ Conway and Sloane, Sphere packings, lattices, and groups, p. 119
8. ^ Conway and Sloane, Sphere packings, lattices, and groups, 7.4 The dual lattice D3*, p.120
9. ^ Conway and Sloane, Sphere packings, lattices, and groups, p. 120
10. ^ 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} = {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)
• 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.
Space Family ${\displaystyle {\tilde {A}}_{n-1}}$  ${\displaystyle {\tilde {C}}_{n-1}}$  ${\displaystyle {\tilde {B}}_{n-1}}$  ${\displaystyle {\tilde {D}}_{n-1}}$  ${\displaystyle {\tilde {G}}_{2}}$  / ${\displaystyle {\tilde {F}}_{4}}$  / ${\displaystyle {\tilde {E}}_{n-1}}$
E2 Uniform tiling {3[3]} δ3 3 3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 4 4
E4 Uniform 4-honeycomb {3[5]} δ5 5 5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 6 6
E6 Uniform 6-honeycomb {3[7]} δ7 7 7 222
E7 Uniform 7-honeycomb {3[8]} δ8 8 8 133331
E8 Uniform 8-honeycomb {3[9]} δ9 9 9 152251521
E9 Uniform 9-honeycomb {3[10]} δ10 10 10
E10 Uniform 10-honeycomb {3[11]} δ11 11 11
En-1 Uniform (n-1)-honeycomb {3[n]} δn n n 1k22k1k21
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