16-cell honeycomb

16-cell honeycomb
File:Demitesseractic tetra hc.png 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	File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png = File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.png = File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.png File:CDel label2.pngFile:CDel branch hh.pngFile:CDel 4a4b.pngFile:CDel nodes.pngFile:CDel split2.pngFile:CDel node.png
4-face type	{3,3,4} File:Schlegel wireframe 16-cell.png
Cell type	{3,3} File:Tetrahedron.png
Face type	{3}
Edge figure	cube
Vertex figure	File:24-cell t0 F4.svg 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

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.

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.

D₄ lattice

The vertex arrangement of the 16-cell honeycomb is called the D₄ lattice or F₄ lattice. The vertices of this lattice are the centers of the 3-spheres in the densest known packing of equal spheres in 4-space; its kissing number is 24, which is also the same as the kissing number in R⁴, as proved by Oleg Musin in 2003. 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:

File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.png ∪ File:CDel nodes 01rd.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.png = File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.png = File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png

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

File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.png ∪ File:CDel nodes 01rd.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.png ∪ File:CDel nodes.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes 10lu.png ∪ File:CDel nodes.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes 01ld.png = File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.png = File:CDel nodes 10r.pngFile:CDel 4a4b.pngFile:CDel nodes.pngFile:CDel split2.pngFile:CDel node.png ∪ File:CDel nodes 01r.pngFile:CDel 4a4b.pngFile:CDel nodes.pngFile:CDel split2.pngFile:CDel node.png.

The kissing number of the D^*
₄ lattice (and D₄ lattice) is 24 and its Voronoi tessellation is a 24-cell honeycomb, File:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes.pngFile:CDel 4a4b.pngFile:CDel nodes.png, containing all rectified 16-cells (24-cell) Voronoi cells, File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png or File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png.

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
${\tilde{F}}_4$ = [3,3,4,3]	{3,3,4,3}	File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png [3,4,3], order 1152	24: 16-cell
${\tilde{B}}_4$ = [31,1,3,4]	= h{4,3,3,4}	File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png = File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png	File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png [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}	File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.png = File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.png	File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes.png [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}	File:CDel label2.pngFile:CDel branch hh.pngFile:CDel 4a4b.pngFile:CDel nodes.pngFile:CDel split2.pngFile:CDel node.png		8+4+4: 4-demicube 8: 16-cell

Related honeycombs

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 ${\tilde{D}}_5$ 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:

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, 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} = {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, 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.

Fundamental convex regular and uniform honeycombs in 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