In
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 5
21 honeycomb is a
uniform tessellation of 8-dimensional Euclidean space. The symbol 5
21 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 5
21 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
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 ...
, 6
11.
:
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
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 ...
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
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 ...
421.
E8 lattice
contains
as a subgroup of index 5760. Both
and
can be seen as affine extensions of
from different nodes:
contains
as a subgroup of index 270. Both
and
can be seen as affine extensions of
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 5
21 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 D
82 or D
8+ lattice), as well as the union of the vertices of three
8-simplex honeycombs (called an A
83 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
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 5
21 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
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 ...
. 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.
See also
*
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 ...
*
152 honeycomb
*
251 honeycomb
Notes
References
*
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 ...
''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