In 7-dimensional
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, 2
31 is a
uniform polytope
In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform Facet (mathematics), facets. Here, "vertex-transitive" means that it has symmetries taking every vertex to every other vertex; the sam ...
, constructed from the
E7 group.
Its
Coxeter symbol
Harold Scott MacDonald "Donald" Coxeter (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician. He is regarded as one of the greatest geometers of the 20th century.
Coxeter was born in England and educated ...
is 2
31, describing its bifurcating
Coxeter-Dynkin diagram, with a single ring on the end of the 2-node branch.
The rectified 2
31 is constructed by points at the mid-edges of the 2
31.
These polytopes are part of a family of 127 (or 2
7−1) convex
uniform polytopes in seven dimensions, made of
uniform polytope
In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform Facet (mathematics), facets. Here, "vertex-transitive" means that it has symmetries taking every vertex to every other vertex; the sam ...
facets and
vertex figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a general -polytope is sliced off.
Definitions
Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected ed ...
s, defined by all combinations of rings in this
Coxeter-Dynkin diagram: .
231 polytope
The 2
31 is composed of 126
vertices, 2016
edges
Enhanced Data rates for GSM Evolution (EDGE), also known as 2.75G and under various other names, is a 2G digital mobile phone technology for packet switched data transmission. It is a subset of General Packet Radio Service (GPRS) on the GS ...
, 10080
faces
The face is the front of the head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affect the ...
(triangles), 20160
cells
Cell most often refers to:
* Cell (biology), the functional basic unit of life
* Cellphone, a phone connected to a cellular network
* Clandestine cell, a penetration-resistant form of a secret or outlawed organization
* Electrochemical cell, a d ...
(
tetrahedra
In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...
), 16128 4-faces (
4-simplexes), 4788 5-faces (756
pentacross
In five-dimensional geometry, a 5-orthoplex, or 5-cross polytope, is a five-dimensional polytope with 10 vertices, 40 edges, 80 triangle faces, 80 tetrahedron cells, 32 5-cell 4-faces.
It has two constructed forms, the first being regular with ...
es, and 4032
5-simplex
In five-dimensional geometry, a 5-simplex is a self-dual regular 5-polytope. It has six vertices, 15 edges, 20 triangle faces, 15 tetrahedral cells, and 6 5-cell facets. It has a dihedral angle of cos−1(), or approximately 78.46°.
The ...
es), 632 6-faces (576
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°.
A ...
es and 56
221). Its
vertex figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a general -polytope is sliced off.
Definitions
Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected ed ...
is a
6-demicube
In geometry, a 6-demicube, demihexeract or hemihexeract 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 demihypercub ...
.
Its 126 vertices represent the root vectors of the
simple Lie group
In mathematics, a simple Lie group is a connected non-abelian Lie group ''G'' which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian symm ...
E7.
This polytope is the
vertex figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a general -polytope is sliced off.
Definitions
Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected ed ...
for a
uniform tessellation of 7-dimensional space,
331.
Alternate names
*
named it V
126 (for its 126 vertices) in his 1912 listing of semiregular polytopes.
* It was called 2
31 by
Coxeter
Harold Scott MacDonald "Donald" Coxeter (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician. He is regarded as one of the greatest geometers of the 20th century.
Coxeter was born in England and educated ...
for its bifurcating
Coxeter-Dynkin diagram, with a single ring on the end of the 2-node sequence.
* Pentacontahexa-pentacosiheptacontahexa-exon (Acronym: laq) - 56-576 facetted polyexon (Jonathan Bowers)
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 7
hyperplane
In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension. Like a plane in space, a hyperplane is a flat hypersurface, a subspace whose dimension is ...
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°.
A ...
. There are 576 of these facets. These facets are centered on the locations of the vertices of the
321 polytope, .
Removing the node on the end of the 3-length branch leaves the
221. There are 56 of these facets. These facets are centered on the locations of the vertices of the
132 polytope, .
The
vertex figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a general -polytope is sliced off.
Definitions
Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected ed ...
is determined by removing the ringed node and ringing the neighboring node. This makes the
6-demicube
In geometry, a 6-demicube, demihexeract or hemihexeract 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 demihypercub ...
, 1
31, .
Seen in a
configuration matrix, the element counts can be derived by mirror removal and ratios of
Coxeter group
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean ref ...
orders.
Images
Related polytopes and honeycombs
Rectified 231 polytope
The rectified 2
31 is a
rectification
Rectification has the following technical meanings:
Mathematics
* Rectification (geometry), truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points
* Rectifiable curve, in mathematics
* Recti ...
of the 2
31 polytope, creating new vertices on the center of edge of the 2
31.
Alternate names
* Rectified pentacontahexa-pentacosiheptacontahexa-exon - as a rectified 56-576 facetted polyexon (Acronym: ) (Jonathan Bowers)
[Klitzing, (o3x3o3o *c3o3o3o - rolaq)]
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 7
hyperplane
In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension. Like a plane in space, a hyperplane is a flat hypersurface, a subspace whose dimension is ...
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
rectified 6-simplex
In six-dimensional geometry, a rectified 6-simplex is a convex uniform 6-polytope, being a rectification of the regular 6-simplex.
There are three unique degrees of rectifications, including the zeroth, the 6-simplex itself. Vertices of the ''r ...
, .
Removing the node on the end of the 2-length branch leaves the,
6-demicube
In geometry, a 6-demicube, demihexeract or hemihexeract 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 demihypercub ...
,
.
Removing the node on the end of the 3-length branch leaves the
rectified 221, .
The
vertex figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a general -polytope is sliced off.
Definitions
Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected ed ...
is determined by removing the ringed node and ringing the neighboring node.
:
Images
See also
*
List of E7 polytopes
In 7-dimensional geometry, there are 127 uniform 7-polytope, uniform polytopes with E7 symmetry. The three simplest forms are the 3_21 polytope, 321, 2_31 polytope, 231, and 1_32 polytope, 132 polytopes, composed of 56, 126, and 576 vertex (geomet ...
Notes
References
*
*
H. S. M. Coxeter, ''Regular Polytopes'', 3rd Edition, Dover New York, 1973
* 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
wiley.com
** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'',
ath. Zeit. 200 (1988) 3-45* x3o3o3o *c3o3o3o - laq, o3x3o3o *c3o3o3o - rolaq
{{Polytopes
7-polytopes