1₃₃ honeycomb

1₃₃ honeycomb
(no image)
Type	Uniform tessellation
Schläfli symbol	{3,3^3,3}
Coxeter symbol	1₃₃
Coxeter-Dynkin diagram	or
7-face type	1₃₂
6-face types	1₂₂ 1₃₁
5-face types	1₂₁ {3⁴}
4-face type	1₁₁ {3³}
Cell type	1₀₁
Face type	{3}
Cell figure	Square
Face figure	Triangular duoprism
Edge figure	Tetrahedral duoprism
Vertex figure	Trirectified 7-simplex
Coxeter group	${\tilde {E}}_{7}$ , [[3,3^3,3]]
Properties	vertex-transitive, facet-transitive

In 7-dimensional geometry, 1₃₃ is a uniform honeycomb, also given by Schläfli symbol {3,3^3,3}, and is composed of 1₃₂ facets.

Construction

It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 7-dimensional space.

The facet information can be extracted from its Coxeter-Dynkin diagram.

Removing a node on the end of one of the 3-length branch leaves the 1₃₂, its only facet type.

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the trirectified 7-simplex, 0₃₃.

The edge figure is determined by removing the ringed nodes of the vertex figure and ringing the neighboring node. This makes the tetrahedral duoprism, {3,3}×{3,3}.

Kissing number

Each vertex of this polytope corresponds to the center of a 6-sphere in a moderately dense sphere packing, in which each sphere is tangent to 70 others; the best known for 7 dimensions (the kissing number) is 126.

Geometric folding

The ${\tilde {E}}_{7}$ group is related to the ${\tilde {F}}_{4}$ by a geometric folding, so this honeycomb can be projected into the 4-dimensional demitesseractic honeycomb.

${\tilde {E}}_{7}$	${\tilde {F}}_{4}$

{3,3^3,3}	{3,3,4,3}

E₇^* lattice

${\tilde {E}}_{7}$ contains ${\tilde {A}}_{7}$ as a subgroup of index 144.^[1] Both ${\tilde {E}}_{7}$ and ${\tilde {A}}_{7}$ can be seen as affine extension from $A_{7}$ from different nodes:

The E₇^* lattice (also called E₇²)^[2] has double the symmetry, represented by [[3,3^3,3]]. The Voronoi cell of the E₇^* lattice is the 1₃₂ polytope, and voronoi tessellation the 1₃₃ honeycomb.^[3] The E₇^* lattice is constructed by 2 copies of the E₇ lattice vertices, one from each long branch of the Coxeter diagram, and can be constructed as the union of four A₇^* lattices, also called A₇⁴:

∪

=

∪

= dual of

.

Related polytopes and honeycombs

The 1₃₃ is fourth in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 1_3k series. The final is a noncompact hyperbolic honeycomb, 1₃₄.

1_3k dimensional figures
Space	Finite				Euclidean	Hyperbolic
n	4	5	6	7	8	9
Coxeter group	A₃A₁	A₅	D₆	E₇	${\tilde {E}}_{7}$ =E₇⁺	${\bar {T}}_{8}$ =E₇⁺⁺
Coxeter diagram
Symmetry	[3^−1,3,1]	[3^0,3,1]	[3^1,3,1]	[3^2,3,1]	[[3^3,3,1]]	[3^4,3,1]
Order	48	720	23,040	2,903,040	∞
Graph					-	-
Name	1_3,-1	1₃₀	1₃₁	1₃₂	1₃₃	1₃₄

Rectified 1₃₃ honeycomb

Rectified 1₃₃ honeycomb
(no image)
Type	Uniform tessellation
Schläfli symbol	{3^3,3,1}
Coxeter symbol	0₃₃₁
Coxeter-Dynkin diagram	or
7-face type	Trirectified 7-simplex Rectified 1_32
6-face types	Birectified 6-simplex Birectified 6-cube Rectified 1_22
5-face types	Rectified 5-simplex Birectified 5-simplex Birectified 5-orthoplex
4-face type	5-cell Rectified 5-cell 24-cell
Cell type	{3,3} {3,4}
Face type	{3}
Vertex figure	{}×{3,3}×{3,3}
Coxeter group	${\tilde {E}}_{7}$ , [[3,3^3,3]]
Properties	vertex-transitive, facet-transitive

The rectified 1₃₃ or 0₃₃₁, Coxeter diagram has facets and , and vertex figure .

Notes

^ N.W. Johnson: Geometries and Transformations, (2018) 12.4: Euclidean Coxeter groups, p.294
^ "The Lattice E7".
^ The Voronoi Cells of the E6* and E7* Lattices Archived 2016-01-30 at the Wayback Machine, Edward Pervin

References

H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
Coxeter The Beauty of Geometry: Twelve Essays, Dover Publications, 1999, ISBN 978-0-486-40919-1 (Chapter 3: Wythoff's Construction for Uniform Polytopes)
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]
Klitzing, Richard. "7D Heptacombs o3o3o3o3o3o3o *d3x - linoh".
Klitzing, Richard. "7D Heptacombs o3o3o3x3o3o3o *d3o - rolinoh".

v t e 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
E^n-1	Uniform (n-1)-honeycomb	0_[n]	δ_n	hδ_n	qδ_n	1_k2 • 2_k1 • k₂₁

[1] N.W. Johnson: Geometries and Transformations, (2018) 12.4: Euclidean Coxeter groups, p.294

[2] "The Lattice E7".

[3] The Voronoi Cells of the E6* and E7* Lattices Archived 2016-01-30 at the Wayback Machine, Edward Pervin

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