16-cell honeycomb

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	${\tilde {F}}_{4}$= [3,3,4,3]
Dual	{3,4,3,3}
Properties	vertex-transitive,edge-transitive,face-transitive,cell-transitive,4-face-transitive

Infour-dimensionalEuclidean geometry,the16-cell honeycombis one of the three regular space-fillingtessellations(orhoneycombs), represented bySchläfli symbol{3,3,4,3}, and constructed by a 4-dimensional packing of16-cellfacets,three around every face.

Its dual is the24-cell honeycomb.Its vertex figure is a24-cell.Thevertex arrangementis called the B₄,D₄,orF₄lattice.

Alternate names

• Hexadecachoric tetracomb/honeycomb
• Demitesseractic tetracomb/honeycomb

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

D₄lattice

Thevertex arrangementof the 16-cell honeycomb is called theD₄latticeor F₄lattice.The vertices of this lattice are the centers of the3-spheresin the densest knownpackingof equal spheres in 4-space;itskissing numberis 24, which is also the same as the kissing number inR⁴,as proved by Oleg Musin in 2003.

The related D⁺
₄lattice (also called D²
₄) can be constructed by the union of two D₄lattices, and is identical to the C₄lattice:

∪==

The kissing number for D⁺
₄is 2³= 8, (2^{n– 1}forn< 8, 240 forn= 8, and 2n(n– 1) forn> 8).

The related D^*
₄lattice (also called D⁴
₄and C²
₄) can be constructed by the union of all four D₄lattices, but it is identical to theD₄lattice:It is also the 4-dimensionalbody centered cubic,the union of two4-cube honeycombsin dual positions.

∪∪∪==∪.

Thekissing numberof the D^*
₄lattice (and D₄lattice) is 24and itsVoronoi tessellationis a24-cell honeycomb,,containing all rectified 16-cells (24-cell)Voronoi cells,or.

Symmetry constructions

There are three different symmetry constructions of this tessellation. Each symmetry can be represented by different arrangements of colored16-cellfacets.

Coxeter group	Schläfli symbol	Coxeter diagram	Vertex figure Symmetry	Facets/verf
${\tilde {F}}_{4}$=[3,3,4,3]	{3,3,4,3}		[3,4,3], order 1152	24:16-cell
${\tilde {B}}_{4}$=[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
2×½${\tilde {C}}_{4}$= [[(4,3,3,4,2+)]]	ht0,4{4,3,3,4}			8+4+4:4-demicube 8:16-cell

Related honeycombs

It is related to the regular hyperbolic 5-space5-orthoplex honeycomb,{3,3,3,4,3}, with5-orthoplexfacets, the regular 4-polytope24-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 analternatedform (thedemitesseractic honeycomb,h{4,3,3,4}) it is related to thealternated cubic honeycomb.

This honeycomb is one of20 uniform honeycombsconstructed by the${\tilde {D}}_{5}$Coxeter group,all but 3 repeated in other families by extended symmetry, seen in the graph symmetry of rings in theCoxeter–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]		${\tilde {D}}_{5}$
<[31,1,3,31,1]> ↔ [31,1,3,3,4]	↔	${\tilde {D}}_{5}$×21=${\tilde {B}}_{5}$	,,, ,,,
[[31,1,3,31,1]]		${\tilde {D}}_{5}$×22	,
<2[31,1,3,31,1]> ↔ [4,3,3,3,4]	↔	${\tilde {D}}_{5}$×41=${\tilde {C}}_{5}$	,,,,,
[<2[31,1,3,31,1]>] ↔ [[4,3,3,3,4]]	↔	${\tilde {D}}_{5}$×8 =${\tilde {C}}_{5}$×2	,,

See also

Regular and uniform honeycombs in 4-space:

• Tesseractic honeycomb
• 24-cell honeycomb
• Truncated 24-cell honeycomb
• Snub 24-cell honeycomb
• 5-cell honeycomb
• Truncated 5-cell honeycomb
• Omnitruncated 5-cell honeycomb

References

• Coxeter, H.S.M.Regular Polytopes,(3rd edition, 1973), Dover edition,ISBN0-486-61480-8
  • pp. 154–156: Partial truncation or alternation, represented byhprefix: 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,ISBN978-0-471-01003-6[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.).ISBN0-387-98585-9.

Fundamental convexregularanduniform honeycombsin dimensions 2–9
Space	Family	${\tilde {A}}_{n-1}$	${\tilde {C}}_{n-1}$	${\tilde {B}}_{n-1}$	${\tilde {D}}_{n-1}$	${\tilde {G}}_{2}$/${\tilde {F}}_{4}$/${\tilde {E}}_{n-1}$
E2	Uniform tiling	0[3]	δ3	hδ3	qδ3	Hexagonal
E3	Uniform convex honeycomb	0[4]	δ4	hδ4	qδ4
E4	Uniform 4-honeycomb	0[5]	δ5	hδ5	qδ5	24-cell honeycomb
E5	Uniform 5-honeycomb	0[6]	δ6	hδ6	qδ6
E6	Uniform 6-honeycomb	0[7]	δ7	hδ7	qδ7	222
E7	Uniform 7-honeycomb	0[8]	δ8	hδ8	qδ8	133•331
E8	Uniform 8-honeycomb	0[9]	δ9	hδ9	qδ9	152•251•521
E9	Uniform 9-honeycomb	0[10]	δ10	hδ10	qδ10
E10	Uniform 10-honeycomb	0[11]	δ11	hδ11	qδ11
En-1	Uniform (n-1)-honeycomb	0[n]	δn	hδn	qδn	1k2•2k1•k21