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 ...
, 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 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 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 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: .
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 of ...
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 t ...
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 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, 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 line ...
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 refle ...
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 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
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 ...
, 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 line ...
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 refle ...
orders.
[Coxeter, Regular Polytopes, 11.8 Gossett figures in six, seven, and eight dimensions, p. 202-203]
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* o3o3o3x *c3o3o3o - lin, o3o3x3o *c3o3o3o - rolin
{{Polytopes
7-polytopes