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 3₂₁ 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, ...

, 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 3₂₁, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of one of the 3-node sequences. The rectified 3₂₁ is constructed by points at the mid-edges of the 3₂₁. The birectified 3₂₁ is constructed by points at the triangle face centers of the 3₂₁. The trirectified 3₂₁ is constructed by points at the tetrahedral centers of the 3₂₁, and is the same as the rectified 1₃₂. These polytopes are part of a family of 127 (2⁷-1) 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 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: .

3₂₁ polytope

In 7-dimensional

, the 3₂₁ 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. The uniform polytopes in two dimensions are the regular polygons (the definition is different in 2 dimen ...

. 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 ...

). 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 3₂₁ polytope is the

Gosset graph The Gosset graph, named after Thorold Gosset, is a specific regular graph (1-skeleton of the 7-dimensional 321 polytope) with 56 vertices and valency 27. Construction The Gosset graph can be explicitly constructed as follows: the 56 vertice ...

. 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 regu ...

who first discovered it. *It was enumerated by

in his 1900 paper. He called it an ''7-ic semi-regular figure''. * E. L. Elte named it V₅₆ (for its 56 vertices) in his 1912 listing of semiregular polytopes. * H.S.M. Coxeter called it 3₂₁ 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 3₂₁ 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

, . 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: 3₁₁, . Every simplex facet touches a 6-orthoplex facet, while alternate facets of the orthoplex touch either a simplex or another orthoplex. The

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 refle ...

orders.

Images

Related polytopes

The 3₂₁ 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

is constructed

of the previous polytope.

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

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 ha ...

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.

named it as 3₂₁ 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

, . Removing the node on the end of the 2-length branch leaves the

rectified 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 th ...

in its alternated form: t₁3₁₁, . Removing the node on the end of the 3-length branch leaves the 2₂₁, . The

is determined by removing the ringed node and ringing the neighboring node. This makes

5-demicube In five-dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a ''5-hypercube'' (penteract) with alternated vertices removed. It was discovered by Thorold Gosset. Since it was the only semiregular 5- ...

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.

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: t₂(3₁₁), . Removing the node on the end of the 3-length branch leaves the rectified 2₂₁ polytope in its alternated form: t₁(2₂₁), . The

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

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 3₂₁) * o3o3o3o *c3o3o3x - naq, o3o3o3o *c3o3x3o - ranq, o3o3o3o *c3x3o3o - branq

External links

''Gosset’s Polytopes''
in vZome {{DEFAULTSORT:3 21 Polytope 7-polytopes

321 polytope

Alternate names

Coordinates

Construction

Images

Related polytopes

Rectified 321 polytope

Alternate names

Construction

Images

Birectified 321 polytope

Alternate names

Construction

Images

See also

Notes

References

External links

3₂₁ polytope

Rectified 3₂₁ polytope

Birectified 3₂₁ polytope