geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, the 5₂₁ honeycomb is a uniform tessellation of 8-dimensional Euclidean space. The symbol 5₂₁ is from

Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington to ...

, named for the length of the 3 branches of its Coxeter-Dynkin diagram.Coxeter, 1973, Chapter 5: The Kaleidoscope By putting spheres at its vertices one obtains the densest-possible packing of spheres in 8 dimensions. This was proven by

Maryna Viazovska Maryna Sergiivna Viazovska ( uk, Марина Сергіївна Вязовська, ; born 2 December 1984) is a Ukrainian mathematician known for her work in sphere packing. She is full professor and Chair of Number Theory at the Institute of M ...

in 2016 using the theory of

modular forms In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition. The theory of ...

. Viazovska was awarded the

Fields Medal The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of the International Mathematical Union (IMU), a meeting that takes place every four years. The name of the award ho ...

for this work in 2022. This honeycomb was first studied by Gosset who called it a ''9-ic semi-regular figure'' (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 In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...

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

_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 In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling. It is often referred to as Wythoff's kaleidoscopic construction. Construction process ...

upon a set of 9

hyperplane In geometry, a hyperplane is a subspace whose dimension is one less than that of its ''ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyper ...

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 In geometry, an 8-orthoplex or 8-cross polytope is a regular 8-polytope with 16 vertices, 112 edges, 448 triangle faces, 1120 tetrahedron cells, 1792 5-cells ''4-faces'', 1792 ''5-faces'', 1024 ''6-faces'', and 256 ''7-faces''. It has two constr ...

, 6₁₁. : Removing the node on the end of the 1-length branch leaves the

8-simplex In geometry, an 8- simplex is a self-dual regular 8-polytope. It has 9 vertices, 36 edges, 84 triangle faces, 126 tetrahedral cells, 126 5-cell 4-faces, 84 5-simplex 5-faces, 36 6-simplex 6-faces, and 9 7-simplex 7-faces. Its dihedral angle is ...

. : The

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

edge figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Definitions Take some corner or vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw lines ...

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 In geometry, the kissing number of a mathematical space is defined as the greatest number of non-overlapping unit spheres that can be arranged in that space such that they each touch a common unit sphere. For a given sphere packing (arrangement o ...

is 240, represented by the vertices of its

4₂₁.

E8 lattice

_8

contains

_8

as a subgroup of index 5760. Both

_8

and

_8

can be seen as affine extensions of

A_8

from different nodes:

_8

contains

_8

as a subgroup of index 270. Both

_8

and

_8

can be seen as affine extensions of

D_8

from different nodes: The

vertex arrangement In geometry, a vertex arrangement is a set of points in space described by their relative positions. They can be described by their use in polytopes. For example, a ''square vertex arrangement'' is understood to mean four points in a plane, equ ...

of 5₂₁ is called the

E8 lattice In mathematics, the E lattice is a special lattice in R. It can be characterized as the unique positive-definite, even, unimodular lattice of rank 8. The name derives from the fact that it is the root lattice of the E root system. The normIn t ...

. 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): : = ∪ = ∪ ∪

Regular complex honeycomb

Using a

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...

coordinate system, it can also be constructed as a

\mathbb^4

regular complex polytope, given the symbol 33333, and

Coxeter diagram Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington t ...

. Its elements are in relative proportion as 1 vertex, 80 3-edges, 270 ₃₃ faces, 80 ₃₃₃ cells and 1 ₃₃₃₃

Witting polytope In 4-dimensional complex geometry, the Witting polytope is a regular complex polytope, named as: 3333, and Coxeter diagram . It has 240 vertices, 2160 3 edges, 2160 Möbius–Kantor polygon, 33 faces, and 240 Hessian polyhedron, 333 cells. It is s ...

cells.Coxeter Regular Convex Polytopes, 12.5 The Witting polytope

Related polytopes and honeycombs

The 5₂₁ is seventh in a dimensional series of

semiregular polytope In geometry, by Thorold Gosset's definition a semiregular polytope is usually taken to be a polytope that is vertex-transitive and has all its facets being regular polytopes. E.L. Elte compiled a longer list in 1912 as ''The Semiregular Polyt ...

s, identified in 1900 by

Thorold Gosset John Herbert de Paz Thorold Gosset (16 October 1869 – December 1962) was an English lawyer and an amateur mathematician. In mathematics, he is noted for discovering and classifying the semiregular polytopes in dimensions four and higher, and ...

. Each member of the sequence has the previous member as its

. All facets of these polytopes are

regular polytope In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or -faces (for all , where is the dimension of the polytope) — cells, ...

s, namely

simplex In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...

es and

orthoplex In geometry, a cross-polytope, hyperoctahedron, orthoplex, or cocube is a regular, convex polytope that exists in ''n''- dimensional Euclidean space. A 2-dimensional cross-polytope is a square, a 3-dimensional cross-polytope is a regular octahed ...

es.

Notes

References

''The Beauty of Geometry: Twelve Essays'', Dover Publications, 1999, (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,

** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', ath. Zeit. 200 (1988) 3-45* N.W. Johnson: ''Geometries and Transformations'', (2015) {{Honeycombs 9-polytopes