5-simplex honeycomb

5-simplex honeycomb
(No image)
Type	Uniform 5-honeycomb
Family	Simplectic honeycomb
Schläfli symbol	{3[6]} = 0[6]
Coxeter diagram	Template:CDD
5-face types	{34} , t1{34} t2{34}
4-face types	{33} , t1{33}
Cell types	{3,3} , t1{3,3}
Face types	{3}
Vertex figure	t0,4{34}
Coxeter groups	${\tilde{A}}_5$×2, <[3[6]]>
Properties	vertex-transitive

In five-dimensional Euclidean geometry, the 5-simplex honeycomb or hexateric honeycomb is a space-filling tessellation (or honeycomb or pentacomb). Each vertex is shared by 12 5-simplexes, 30 rectified 5-simplexes, and 20 birectified 5-simplexes. These facet types occur in proportions of 2:2:1 respectively in the whole honeycomb.

A5 lattice

This vertex arrangement is called the A₅ lattice or 5-simplex lattice. The 30 vertices of the stericated 5-simplex vertex figure represent the 30 roots of the ${\tilde{A}}_5$ Coxeter group. It is the 5-dimensional case of a simplectic honeycomb.

The ATemplate:Sup sub lattice is the union of two A₅ lattices:

Template:CDD ∪ Template:CDD

The ATemplate:Sup sub is the union of three A₅ lattices:

Template:CDD ∪ Template:CDD ∪ Template:CDD.

The ATemplate:Sup sub lattice (also called ATemplate:Sup sub) is the union of six A₅ lattices, and is the dual vertex arrangement to the omnitruncated 5-simplex honeycomb, and therefore the Voronoi cell of this lattice is an omnitruncated 5-simplex.

Template:CDD ∪ Template:CDD ∪ Template:CDD ∪ Template:CDD ∪ Template:CDD ∪ Template:CDD = dual of Template:CDD

Projection by folding

The 5-simplex honeycomb can be projected into the 3-dimensional cubic honeycomb by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same vertex arrangement:

${\tilde{A}}_5$	Template:CDD
${\tilde{C}}_3$	Template:CDD

See also

Regular and uniform honeycombs in 5-space:

• 5-cubic honeycomb
• 5-demicube honeycomb
• Truncated 5-simplex honeycomb
• Omnitruncated 5-simplex honeycomb

References

• Norman Johnson Uniform Polytopes, Manuscript (1991)
• 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, Template:ISBN [1]
  • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2,10] (1.9 Uniform space-fillings)
  • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]

Template:Navbar-collapsible
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