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 ...
, the 3
21 polytope is a
uniform 7-polytope
In seven-dimensional geometry, a 7-polytope is a polytope contained by 6-polytope facets. Each 5-polytope ridge being shared by exactly two 6-polytope facets.
A uniform 7-polytope is one whose symmetry group is transitive on vertices and whose ...
, constructed within the symmetry of the
E7 group. It was discovered by
Thorold Gosset
John Herbert de Paz Thorold Gosset (16 October 1869 – December 1962) was an English lawyer and an amateur mathematician. In mathematics, he is noted for discovering and classifying the semiregular polytopes in dimensions four and higher, and ...
, published in his 1900 paper. He called it an
7-ic semi-regular figure.
[Gosset, 1900]
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 3
21, describing its bifurcating
Coxeter-Dynkin diagram, with a single ring on the end of one of the 3-node sequences.
The rectified 3
21 is constructed by points at the mid-edges of the 3
21. The birectified 3
21 is constructed by points at the triangle face centers of the 3
21. The trirectified 3
21 is constructed by points at the tetrahedral centers of the 3
21, and is the same as the rectified 1
32.
These polytopes are part of a family of 127 (2
7-1) convex
uniform polytopes in 7-dimensions, made of
uniform 6-polytope
In six-dimensional geometry, a uniform 6-polytope is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform 5-polytopes.
The complete set of convex uniform 6-polytopes has not been determined, bu ...
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 (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...
s, defined by all permutations of rings in this
Coxeter-Dynkin diagram: .
321 polytope
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 ...
, the 3
21 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 ...
. It has 56 vertices, and 702 facets: 126
311 and 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.
For visualization this 7-dimensional polytope is often displayed in a special skewed orthographic projection direction that fits its 56 vertices within an 18-gonal regular polygon (called a
Petrie polygon
In geometry, a Petrie polygon for a regular polytope of dimensions is a skew polygon in which every consecutive sides (but no ) belongs to one of the facets. The Petrie polygon of a regular polygon is the regular polygon itself; that of a reg ...
). Its 756 edges are drawn between 3 rings of 18 vertices, and 2 vertices in the center. Specific higher elements (faces, cells, etc.) can also be extracted and drawn on this projection.
The 1-
skeleton
A skeleton is the structural frame that supports the body of an animal. There are several types of skeletons, including the exoskeleton, which is the stable outer shell of an organism, the endoskeleton, which forms the support structure inside ...
of the 3
21 polytope is the
Gosset graph.
This polytope, along with the
7-simplex
In 7-dimensional geometry, a 7- simplex is a self-dual regular 7-polytope. It has 8 vertices, 28 edges, 56 triangle faces, 70 tetrahedral cells, 56 5-cell 5-faces, 28 5-simplex 6-faces, and 8 6-simplex 7-faces. Its dihedral angle is cos−1(1/ ...
, 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, represented by
331 and Coxeter-Dynkin diagram: .
Alternate names
*It is also called the Hess polytope for
Edmund Hess
Edmund Hess (17 February 1843 – 24 December 1903) was a German mathematician who discovered several regular polytopes.
See also
* Schläfli–Hess polychoron
* Hess polytope
References
* ''Regular Polytopes
In mathematics, a re ...
who first discovered it.
*It was enumerated by
Thorold Gosset
John Herbert de Paz Thorold Gosset (16 October 1869 – December 1962) was an English lawyer and an amateur mathematician. In mathematics, he is noted for discovering and classifying the semiregular polytopes in dimensions four and higher, and ...
in his 1900 paper. He called it an ''7-ic semi-regular figure''.
*
E. L. Elte
Emanuel Lodewijk Elte (16 March 1881 in Amsterdam – 9 April 1943 in Sobibor extermination camp, Sobibór)[ Em ...]
named it V
56 (for its 56 vertices) in his 1912 listing of semiregular polytopes.
*
H.S.M. Coxeter called it 3
21 due to its bifurcating
Coxeter-Dynkin diagram, having 3 branches of length 3, 2, and 1, and having a single ring on the final node of the 3 branch.
* ''Hecatonicosihexa-pentacosiheptacontihexa-exon'' (Acronym Naq) - 126-576 facetted polyexon (Jonathan Bowers)
Coordinates
The 56 vertices can be most simply represented in 8-dimensional space, obtained by the 28 permutations of the coordinates and their opposite:
: ± (-3, -3, 1, 1, 1, 1, 1, 1)
Construction
Its construction is based on the
E7 group.
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 ...
named it as 3
21 by its bifurcating
Coxeter-Dynkin diagram, with a single ring on the end of the 3-node sequence.
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 ...
, .
Removing the node on the end of the 2-length branch leaves the
6-orthoplex
In geometry, a 6-orthoplex, or 6-cross polytope, is a regular 6-polytope with 12 vertices, 60 edges, 160 triangle faces, 240 tetrahedron cells, 192 5-cell ''4-faces'', and 64 ''5-faces''.
It has two constructed forms, the first being regular wi ...
in its alternated form: 3
11, .
Every simplex facet touches a 6-orthoplex facet, while alternate facets of the orthoplex touch either a simplex or another orthoplex.
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 (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...
is determined by removing the ringed node and ringing the neighboring node. This makes
221 polytope, .
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 refl ...
orders.
Images
Related polytopes
The 3
21 is fifth in a dimensional series of
semiregular polytope
In geometry, by Thorold Gosset's definition a semiregular polytope is usually taken to be a polytope that is vertex-transitive and has all its facets being regular polytopes. E.L. Elte compiled a longer list in 1912 as ''The Semiregular Polyt ...
s. Each progressive
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 ...
is constructed
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 (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...
of the previous polytope.
Thorold Gosset
John Herbert de Paz Thorold Gosset (16 October 1869 – December 1962) was an English lawyer and an amateur mathematician. In mathematics, he is noted for discovering and classifying the semiregular polytopes in dimensions four and higher, and ...
identified this series in 1900 as containing all
regular polytope
In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or -faces (for all , where is the dimension of the polytope) — cells, ...
facets, containing all
simplex
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
es and
orthoplex
In geometry, a cross-polytope, hyperoctahedron, orthoplex, or cocube is a regular, convex polytope that exists in ''n''- dimensional Euclidean space. A 2-dimensional cross-polytope is a square, a 3-dimensional cross-polytope is a regular octahed ...
es.
It is 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 3
k1 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 hav ...
.)
Rectified 321 polytope
Alternate names
* Rectified hecatonicosihexa-pentacosiheptacontihexa-exon as a rectified 126-576 facetted polyexon (acronym ranq) (Jonathan Bowers)
Construction
Its construction is based on the
E7 group.
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 ...
named it as 3
21 by its bifurcating
Coxeter-Dynkin diagram, with a single node on the end of the 3-node sequence.
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 ...
, .
Removing the node on the end of the 2-length branch leaves the
rectified 6-orthoplex in its alternated form: t
13
11, .
Removing the node on the end of the 3-length branch leaves the
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 (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...
is determined by removing the ringed node and ringing the neighboring node. This makes
5-demicube prism, .
Images
Birectified 321 polytope
Alternate names
* Birectified hecatonicosihexa-pentacosiheptacontihexa-exon as a birectified 126-576 facetted polyexon (acronym branq) (Jonathan Bowers)
[Klitzing, (o3o3o3o *c3x3o3o - branq)]
Construction
Its construction is based on the
E7 group.
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 ...
named it as 3
21 by its bifurcating
Coxeter-Dynkin diagram, with a single node on the end of the 3-node sequence.
The facet information can be extracted from its
Coxeter-Dynkin diagram, .
Removing the node on the short branch leaves the
birectified 6-simplex, .
Removing the node on the end of the 2-length branch leaves the
birectified 6-orthoplex
In six-dimensional geometry, a rectified 6-orthoplex is a convex uniform 6-polytope, being a rectification of the regular 6-orthoplex.
There are unique 6 degrees of rectifications, the zeroth being the 6-orthoplex, and the 6th and last being the ...
in its alternated form: t
2(3
11), .
Removing the node on the end of the 3-length branch leaves the
rectified 221 polytope in its alternated form: t
1(2
21), .
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 (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...
is determined by removing the ringed node and ringing the neighboring node. This makes
rectified 5-cell
In four-dimensional geometry, the rectified 5-cell is a uniform 4-polytope composed of 5 regular tetrahedral and 5 regular octahedral cells. Each edge has one tetrahedron and two octahedra. Each vertex has two tetrahedra and three octahedra. In t ...
-triangle duoprism, .
Images
See also
*
List of E7 polytopes
Notes
References
*
T. Gosset: ''On the Regular and Semi-Regular Figures in Space of n Dimensions'', Messenger of Mathematics, Macmillan, 1900
*
* 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) 3-45See p342 (figure 3.7c) by Peter mcMullen: (18-gonal node-edge graph of 3
21)
* o3o3o3o *c3o3o3x - naq, o3o3o3o *c3o3x3o - ranq, o3o3o3o *c3x3o3o - branq
External links
''Gosset’s Polytopes''in vZome
{{DEFAULTSORT:3 21 Polytope
7-polytopes