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 ...
, 2
31 is a
uniform polytope
In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons (the definition is different in 2 dimensions to exclude vert ...
, constructed from the
E7 group.
Its
Coxeter symbol
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 ...
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 7-dimensions, made of
uniform polytope
In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons (the definition is different in 2 dimensions to exclude vert ...
facets and
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 of a polyhedron. Mark a point somewhere along each connected edge. Draw line ...
s, defined by all permutations of rings in this
Coxeter-Dynkin diagram: .
2_31 polytope
The 2
31 is composed of 126
vertices, 2016
edges, 10080
faces
The face is the front of an animal's 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 affe ...
(Triangles), 20160
cells (
tetrahedra
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
), 16128 4-faces (
3-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 wit ...
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 5 ...
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°.
Alte ...
es and 56
221). 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 of a polyhedron. Mark a point somewhere along each connected edge. Draw line ...
is a
6-demicube.
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 polyhedron or polytope is sliced off.
Definitions
Take some corner or vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw line ...
for a
uniform tessellation of 7-dimensional space,
331.
Alternate names
*
E. L. Elte 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 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 ...
for its bifurcating
Coxeter-Dynkin diagram, with a single ring on the end of the 2-node sequence.
* ''Pentacontihexa-pentacosiheptacontihexa-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 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°.
Alte ...
. 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 polyhedron or polytope is sliced off.
Definitions
Take some corner or vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw line ...
is determined by removing the ringed node and ringing the neighboring node. This makes the
6-demicube, 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 refle ...
orders.
Images
Related polytopes and honeycombs
Rectified 2_31 polytope
The rectified 2
31 is a
rectification of the 2
31 polytope, creating new vertices on the center of edge of the 2
31.
Alternate names
* Rectified pentacontihexa-pentacosiheptacontihexa-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 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 ''rect ...
, .
Removing the node on the end of the 2-length branch leaves the,
6-demicube,
.
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 polyhedron or polytope is sliced off.
Definitions
Take some corner or vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw line ...
is determined by removing the ringed node and ringing the neighboring node.
:
Images
See also
*
List of E7 polytopes
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,
** (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