5₂₁ honeycomb

5₂₁ honeycomb
Type	Uniform honeycomb
Family	k₂₁ polytope
Schläfli symbol	{3,3,3,3,3,3^2,1}
Coxeter symbol	5₂₁
Coxeter-Dynkin diagram
8-faces	5₁₁ {3⁷}
7-faces	{3⁶} Note that there are two distinct orbits of this 7-simplex under the honeycomb's full ${\tilde {E}}_{8}$ automorphism group.
6-faces	{3⁵}
5-faces	{3⁴}
4-faces	{3³}
Cells	{3²}
Faces	{3}
Cell figure	1₂₁
Face figure	2₂₁
Edge figure	3₂₁
Vertex figure	4₂₁
Symmetry group	${\tilde {E}}_{8}$ , [3^5,2,1]

In geometry, the 5₂₁ honeycomb is a uniform tessellation of 8-dimensional Euclidean space. The symbol 5₂₁ is from Coxeter, named for the length of the 3 branches of its Coxeter-Dynkin diagram.^[1]

By putting spheres at its vertices one obtains the densest-possible packing of spheres in 8 dimensions. This was proven by Maryna Viazovska in 2016 using the theory of modular forms. Viazovska was awarded the Fields Medal for this work in 2022.

This honeycomb was first studied by Gosset who called it a 9-ic semi-regular figure^[2] (Gosset regarded honeycombs in n dimensions as degenerate n+1 polytopes).

Each vertex of the 5₂₁ honeycomb is surrounded by 2160 8-orthoplexes and 17280 8-simplicies.

The vertex figure of Gosset's honeycomb is the semiregular 4₂₁ polytope. It is the final figure in the k₂₁ family.

This honeycomb is highly regular in the sense that its symmetry group (the affine ${\tilde {E}}_{8}$ Weyl group) acts transitively on the k-faces for k ≤ 6. All of the k-faces for k ≤ 7 are simplices.

Construction

It is created by a Wythoff construction upon a set of 9 hyperplane mirrors in 8-dimensional space.

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

Removing the node on the end of the 2-length branch leaves the 8-orthoplex, 6₁₁.

Removing the node on the end of the 1-length branch leaves the 8-simplex.

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the 4₂₁ polytope.

The edge figure is determined from the vertex figure by removing the ringed node and ringing the neighboring node. This makes the 3₂₁ polytope.

The face figure is determined from the edge figure by removing the ringed node and ringing the neighboring node. This makes the 2₂₁ polytope.

The cell figure is determined from the face figure by removing the ringed node and ringing the neighboring node. This makes the 1₂₁ polytope.

Kissing number

Each vertex of this tessellation is the center of a 7-sphere in the densest packing in 8 dimensions; its kissing number is 240, represented by the vertices of its vertex figure 4₂₁.

E8 lattice

${\tilde {E}}_{8}$ contains ${\tilde {A}}_{8}$ as a subgroup of index 5760.^[3] Both ${\tilde {E}}_{8}$ and ${\tilde {A}}_{8}$ can be seen as affine extensions of $A_{8}$ from different nodes:

${\tilde {E}}_{8}$ contains ${\tilde {D}}_{8}$ as a subgroup of index 270.^[4] Both ${\tilde {E}}_{8}$ and ${\tilde {D}}_{8}$ can be seen as affine extensions of $D_{8}$ from different nodes:

The vertex arrangement of 5₂₁ is called the E8 lattice.^[5]

The E8 lattice can also be constructed as a union of the vertices of two 8-demicube honeycombs (called a D₈² or D₈⁺ lattice), as well as the union of the vertices of three 8-simplex honeycombs (called an A₈³ lattice):^[6]

=

∪

=

∪

Regular complex honeycomb

Using a complex number coordinate system, it can also be constructed as a $\mathbb {C} ^{4}$ regular complex polytope, given the symbol 3{3}3{3}3{3}3{3}3, and Coxeter diagram . Its elements are in relative proportion as 1 vertex, 80 3-edges, 270 ₃{3}₃ faces, 80 ₃{3}₃{3}₃ cells and 1 ₃{3}₃{3}₃{3}₃ Witting polytope cells.^[7]

Related polytopes and honeycombs

The 5₂₁ is seventh in a dimensional series of semiregular polytopes, identified in 1900 by Thorold Gosset. Each member of the sequence has the previous member as its vertex figure. All facets of these polytopes are regular polytopes, namely simplexes and orthoplexes.

k₂₁ figures in n dimensions
Space	Finite						Euclidean	Hyperbolic
E_n	3	4	5	6	7	8	9	10
Coxeter group	E₃=A₂A₁	E₄=A₄	E₅=D₅	E₆	E₇	E₈	E₉ = ${\tilde {E}}_{8}$ = E₈⁺	E₁₀ = ${\bar {T}}_{8}$ = E₈⁺⁺
Coxeter diagram
Symmetry	[3^−1,2,1]	[3^0,2,1]	[3^1,2,1]	[3^2,2,1]	[3^3,2,1]	[3^4,2,1]	[3^5,2,1]	[3^6,2,1]
Order	12	120	1,920	51,840	2,903,040	696,729,600	∞
Graph							-	-
Name	−1₂₁	0₂₁	1₂₁	2₂₁	3₂₁	4₂₁	5₂₁	6₂₁

Notes

^ Coxeter, 1973, Chapter 5: The Kaleidoscope
^ Gosset, Thorold (1900). "On the regular and semi-regular figures in space of n dimensions". Messenger of Mathematics. 29: 43–48.
^ N.W. Johnson: Geometries and Transformations, (2018) 12.5: Euclidean Coxeter groups, p.294
^ Johnson (2011) p.177
^ "The Lattice E8".
^ Kaleidoscopes: Selected Writings of H. S. M. Coxeter, Paper 18, "Extreme forms" (1950)
^ Coxeter Regular Convex Polytopes, 12.5 The Witting polytope

References

Coxeter The Beauty of Geometry: Twelve Essays, Dover Publications, 1999, ISBN 978-0-486-40919-1 (Chapter 3: Wythoff's Construction for Uniform Polytopes)
Coxeter, H. S. M. (1973). Regular Polytopes ((3rd ed.) ed.). New York: Dover Publications. ISBN 0-486-61480-8.
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]
N.W. Johnson: Geometries and Transformations, (2015)

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

[coxeter-1] Coxeter, 1973, Chapter 5: The Kaleidoscope

[gosset-2] Gosset, Thorold (1900). "On the regular and semi-regular figures in space of n dimensions". Messenger of Mathematics. 29: 43–48.

[3] N.W. Johnson: Geometries and Transformations, (2018) 12.5: Euclidean Coxeter groups, p.294

[4] Johnson (2011) p.177

[5] "The Lattice E8".

[6] Kaleidoscopes: Selected Writings of H. S. M. Coxeter, Paper 18, "Extreme forms" (1950)

[7] Coxeter Regular Convex Polytopes, 12.5 The Witting polytope

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