Omnitruncated 5-simplex honeycomb

Omnitruncated 5-simplex honeycomb
(No image)
Type	Uniform honeycomb
Family	Omnitruncated simplectic honeycomb
Schläfli symbol	t₀₁₂₃₄₅{3^[6]}
Coxeter–Dynkin diagram	Template:CDD
5-face types	t₀₁₂₃₄{3,3,3,3}
4-face types	t₀₁₂₃{3,3,3} {}×t₀₁₂{3,3} {6}×{6}
Cell types	t₀₁₂{3,3} {4,3} {}x{6}
Face types	{4} {6}
Vertex figure	Irr. 5-simplex
Symmetry	${\tilde{A}}_{5}$ ×12, [6[3^[6]]]
Properties	vertex-transitive

In five-dimensional Euclidean geometry, the omnitruncated 5-simplex honeycomb or omnitruncated hexateric honeycomb is a space-filling tessellation (or honeycomb). It is composed entirely of omnitruncated 5-simplex facets.

The facets of all omnitruncated simplectic honeycombs are called permutahedra and can be positioned in n+1 space with integral coordinates, permutations of the whole numbers (0,1,..,n).

A₅^* lattice

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.

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

${\tilde{A}}_{5}$	Template:CDD
${\tilde{C}}_{3}$	Template:CDD

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] Template:Webarchive
- (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	Template:Dark mode invert	Template:Dark mode invert	Template:Dark mode invert	Template:Dark mode invert	Template:Dark mode invert / Template:Dark mode invert / Template:Dark mode invert
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]	δ_n	hδ_n	qδ_n	1_k2 • 2_k1 • k₂₁