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 ...
, 1
32 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 ver ...
, 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 ...
is 1
32, describing its bifurcating
Coxeter-Dynkin diagram, with a single ring on the end of one of the 1-node sequences.
The rectified 1
32 is constructed by points at the mid-edges of the 1
32.
These polytopes are part of a family of 127 (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 ver ...
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 lines ...
s, defined by all permutations of rings in this
Coxeter-Dynkin diagram: .
1_32 polytope
This polytope can
tessellate
A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety o ...
7-dimensional space, with symbol 1
33, and Coxeter-Dynkin diagram, . It is 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 ...
of the dual
E7* lattice.
Alternate names
*
named it V
576 (for its 576 vertices) in his 1912 listing of semiregular polytopes.
*
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 ...
called it 1
32 for its bifurcating
Coxeter-Dynkin diagram, with a single ring on the end of the 1-node branch.
* ''Pentacontihexa-hecatonicosihexa-exon'' (Acronym lin) - 56-126 facetted polyexon (Jonathan Bowers)
Images
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 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 hype ...
mirrors in 7-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
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,
Removing the node on the end of the 3-length branch leaves the
122,
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 lines ...
is determined by removing the ringed node and ringing the neighboring node. This makes the
birectified 6-simplex, 0
32,
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.
Related polytopes and honeycombs
The 1
32 is third 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 1
3k series. The next figure is the Euclidean honeycomb
133 and the final is a noncompact hyperbolic honeycomb, 1
34.
Rectified 1_32 polytope
The rectified 1
32 (also called 0
321) is a
rectification of the 1
32 polytope, creating new vertices on the center of edge of the 1
32. Its vertex figure is a duoprism prism, the product of a regular tetrahedra and triangle, doubled into a prism: ××.
Alternate names
* Rectified pentacontihexa-hecatonicosihexa-exon for rectified 56-126 facetted polyexon (acronym rolin) (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 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 hype ...
mirrors in 7-dimensional space. These mirrors are represented by its
Coxeter-Dynkin diagram, , and the ring represents the position of the active mirror(s).
Removing the node on the end of the 3-length branch leaves the
rectified 122 polytope,
Removing the node on the end of the 2-length branch leaves the
demihexeract, 1
31,
Removing the node on the end of the 1-length branch leaves the
birectified 6-simplex,
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 lines ...
is determined by removing the ringed node and ringing the neighboring node. This makes the tetrahedron-triangle duoprism prism, ××,
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.
[Coxeter, Regular Polytopes, 11.8 Gossett figures in six, seven, and eight dimensions, p. 202-203]
Images
See also
*
List of E7 polytopes
In 7-dimensional geometry, there are 127 uniform polytopes with E7 symmetry. The three simplest forms are the 321, 231, and 132 polytopes, composed of 56, 126, and 576 vertices respectively.
They can be visualized as symmetric orthographic p ...
Notes
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
* 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* o3o3o3x *c3o3o3o - lin, o3o3x3o *c3o3o3o - rolin
{{Polytopes
7-polytopes