In 7-dimensional

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 3₃₁ honeycomb is a uniform honeycomb, also given by

Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more ...

and is composed of 3₂₁ and

7-simplex In 7-dimensional geometry, a 7-simplex is a self-dual regular 7-polytope. It has 8 vertices, 28 edges, 56 triangle faces, 70 tetrahedral cells, 56 5-cell 5-faces, 28 5-simplex 6-faces, and 8 6-simplex 7-faces. Its dihedral angle is cos−1(1/7) ...

facets A facet is a flat surface of a geometric shape, e.g., of a cut gemstone. Facet may also refer to: Arts, entertainment, and media * ''Facets'' (album), an album by Jim Croce * ''Facets'', a 1980 album by jazz pianist Monty Alexander and his tri ...

, with 56 and 576 of them respectively around each vertex.

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 8

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 7-dimensional space. The facet information can be extracted from its Coxeter-Dynkin diagram. : Removing the node on the short branch leaves the

6-simplex In geometry, a 6-simplex is a self-dual regular 6-polytope. It has 7 vertices, 21 edges, 35 triangle faces, 35 tetrahedral cells, 21 5-cell 4-faces, and 7 5-simplex 5-faces. Its dihedral angle is cos−1(1/6), or approximately 80.41°. Alter ...

facet: : Removing the node on the end of the 3-length branch leaves the 3₂₁ facet: : 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 2₃₁ 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 by removing the ringed node and ringing the neighboring node. This makes

6-demicube In geometry, a 6-demicube or demihexteract is a uniform 6-polytope, constructed from a ''6-cube'' ( hexeract) with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes. E. L. Elte i ...

(1₃₁). : The

face 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 by removing the ringed node and ringing the neighboring node. This makes

rectified 5-simplex In five-dimensional geometry, a rectified 5-simplex is a convex uniform 5-polytope, being a rectification of the regular 5-simplex. There are three unique degrees of rectifications, including the zeroth, the 5-simplex itself. Vertices of the ''re ...

(0₃₁). : The cell figure is determined by removing the ringed node of the face figure and ringing the neighboring nodes. This makes

tetrahedral prism In geometry, a tetrahedral prism is a convex uniform 4-polytope. This 4-polytope has 6 polyhedral cells: 2 tetrahedra connected by 4 triangular prisms. It has 14 faces: 8 triangular and 6 square. It has 16 edges and 8 vertices. It is one of 18 u ...

×. :

Kissing number

Each vertex of this tessellation is the center of a 6-sphere in the densest known packing in 7 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 of ...

is 126, represented by the vertices of its

2₃₁.

E7 lattice

The 3₃₁ honeycomb's

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 ...

is called the E₇ lattice.

_7

contains

_7

as a subgroup of index 144. Both

_7

and

_7

can be seen as affine extension from

A_7

from different nodes: The E₇ lattice can also be expressed as a union of the vertices of two A₇ lattices, also called A₇²: : = ∪ The E₇^* lattice (also called E₇²) has double the symmetry, represented by 3,3^3,3. The

Voronoi cell In mathematics, a Voronoi diagram is a partition of a plane into regions close to each of a given set of objects. In the simplest case, these objects are just finitely many points in the plane (called seeds, sites, or generators). For each seed t ...

of the E₇^* lattice is the 1₃₂ polytope, and

voronoi tessellation Voronoi or Voronoy is a Slavic masculine surname; its feminine counterpart is Voronaya. It may refer to *Georgy Voronoy (1868–1908), Russian and Ukrainian mathematician **Voronoi diagram **Weighted Voronoi diagram ** Voronoi deformation density ** ...

the 1₃₃ honeycomb.The Voronoi Cells of the E6* and E7* Lattices
, Edward Pervin 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 honeycombs

It is in a dimensional series of uniform polytopes and honeycombs, expressed by

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 ...

as 3_k1 series. A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral

hosohedron In spherical geometry, an -gonal hosohedron is a tessellation of lunes on a spherical surface, such that each lune shares the same two polar opposite vertices. A regular -gonal hosohedron has Schläfli symbol with each spherical lune havin ...

References

H. S. M. 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 t ...

, ''Regular Polytopes'', 3rd Edition, Dover New York, 1973 *

''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,
GoogleBook
** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', ath. Zeit. 200 (1988) 3–45*

R. T. Worley R. or r. may refer to: * ''Reign'', the period of time during which an Emperor, king, queen, etc., is ruler. * ''Rex (title), Rex'', abbreviated as R., the Latin word meaning King * ''Regina'', abbreviated as R., the Latin word meaning Queen regna ...

, ''The Voronoi Region of E7*''. SIAM J. Discrete Math., 1.1 (1988), 134-141. * p124-125, 8.2 The 7-dimensinoal lattices: E7 and E7* * {{Honeycombs 8-polytopes

Construction

Kissing number

E7 lattice

Related honeycombs

See also

References