In 7-dimensional
geometry, the 3
31 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 facets, with 56 and 576 of them respectively around each vertex.
Construction
It is created by a
Wythoff construction 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 facet:
:
Removing the node on the end of the 3-length branch leaves the
321 facet:
:
The
vertex figure 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 ...
(1
31).
:
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 (0
31).
:
The cell figure is determined by removing the ringed node of the face figure and ringing the neighboring nodes. This makes
tetrahedral prism ×.
:
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 is 126, represented by the vertices of its
vertex figure 231.
E7 lattice
The 3
31 honeycomb's
vertex arrangement is called the E
7 lattice.
contains
as a subgroup of index 144. Both
and
can be seen as affine extension from
from different nodes:
The E
7 lattice can also be expressed as a union of the vertices of two A
7 lattices, also called A
72:
: = ∪
The E
7* lattice (also called E
72) has double the symmetry, represented by
3,33,3. 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 E
7* lattice is the
132 polytope, and
voronoi tessellation 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 as 3k1 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 ...
.
See also
* 8-polytope
* 133 honeycomb
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
* Coxeter ''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