In 6-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 2
21 polytope is a
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 ...
, constructed within the symmetry of the
E6 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
6-ic semi-regular figure. It is also called the
Schläfli polytope.
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
21, describing its bifurcating
Coxeter-Dynkin diagram, with a single ring on the end of one of the 2-node sequences. He also studied its connection with the 27 lines on the
cubic surface
In mathematics, a cubic surface is a surface in 3-dimensional space defined by one polynomial equation of degree 3. Cubic surfaces are fundamental examples in algebraic geometry. The theory is simplified by working in projective space rather th ...
, which are naturally in correspondence with the vertices of 2
21.
The rectified 2
21 is constructed by points at the mid-edges of the 2
21. The birectified 2
21 is constructed by points at the triangle face centers of the 2
21, and is the same as the rectified 1
22.
These polytopes are a part of family of 39 convex
uniform polytopes in 6-dimensions, made of
uniform 5-polytope
In geometry, a uniform 5-polytope is a five-dimensional uniform polytope. By definition, a uniform 5-polytope is vertex-transitive and constructed from uniform 4-polytope Facet (geometry), facets.
The complete set of convex uniform 5-polytopes ...
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: .
2_21 polytope
The 2
21 has 27 vertices, and 99 facets: 27
5-orthoplex
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 with ...
es and 72
5-simplices. 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 (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...
is a
5-demicube.
For visualization this 6-dimensional polytope is often displayed in a special skewed orthographic projection direction that fits its 27 vertices within a 12-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 216 edges are drawn between 2 rings of 12 vertices, and 3 vertices projected into the center. Higher elements (faces, cells, etc.) can also be extracted and drawn on this projection.
The
Schläfli graph
In the mathematical field of graph theory, the Schläfli graph, named after Ludwig Schläfli, is a 16- regular undirected graph with 27 vertices and 216 edges. It is a strongly regular graph with parameters srg(27, 16, 10, 8).
...
is the 1-skeleton of this polytope.
Alternate names
*
E. L. Elte
Emanuel Lodewijk Elte (16 March 1881 in Amsterdam – 9 April 1943 in Sobibor extermination camp, Sobibór)[ Em ...]
named it V
27 (for its 27 vertices) in his 1912 listing of semiregular polytopes.
* Icosihepta-heptacontidi-peton - 27-72 facetted polypeton (acronym jak) (Jonathan Bowers)
Coordinates
The 27 vertices can be expressed in 8-space as an edge-figure of the
421 polytope:
(-2, 0, 0, 0,-2, 0, 0, 0),
( 0,-2, 0, 0,-2, 0, 0, 0),
( 0, 0,-2, 0,-2, 0, 0, 0),
( 0, 0, 0,-2,-2, 0, 0, 0),
( 0, 0, 0, 0,-2, 0, 0,-2),
( 0, 0, 0, 0, 0,-2,-2, 0)
( 2, 0, 0, 0,-2, 0, 0, 0),
( 0, 2, 0, 0,-2, 0, 0, 0),
( 0, 0, 2, 0,-2, 0, 0, 0),
( 0, 0, 0, 2,-2, 0, 0, 0),
( 0, 0, 0, 0,-2, 0, 0, 2)
(-1,-1,-1,-1,-1,-1,-1,-1),
(-1,-1,-1, 1,-1,-1,-1, 1),
(-1,-1, 1,-1,-1,-1,-1, 1),
(-1,-1, 1, 1,-1,-1,-1,-1),
(-1, 1,-1,-1,-1,-1,-1, 1),
(-1, 1,-1, 1,-1,-1,-1,-1),
(-1, 1, 1,-1,-1,-1,-1,-1),
( 1,-1,-1,-1,-1,-1,-1, 1),
( 1,-1, 1,-1,-1,-1,-1,-1),
( 1,-1,-1, 1,-1,-1,-1,-1),
( 1, 1,-1,-1,-1,-1,-1,-1),
(-1, 1, 1, 1,-1,-1,-1, 1),
( 1,-1, 1, 1,-1,-1,-1, 1),
( 1, 1,-1, 1,-1,-1,-1, 1),
( 1, 1, 1,-1,-1,-1,-1, 1),
( 1, 1, 1, 1,-1,-1,-1,-1)
Construction
Its construction is based on the
E6 group.
The facet information can be extracted from its
Coxeter-Dynkin diagram, .
Removing the node on the short branch leaves the
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-s ...
, .
Removing the node on the end of the 2-length branch leaves the
5-orthoplex
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 with ...
in its alternated form: (2
11), .
Every simplex facet touches a 5-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
5-demicube (1
21 polytope), . The edge-figure is the vertex figure of the vertex figure, a
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 ...
, (0
21 polytope), .
Seen in a
configuration matrix, the element counts can be derived from the
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
Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow. The number of vertices by color are given in parentheses.
Geometric folding
The 2
21 is related to the
24-cell by a geometric
folding
Fold, folding or foldable may refer to:
Arts, entertainment, and media
* ''Fold'' (album), the debut release by Australian rock band Epicure
* Fold (poker), in the game of poker, to discard one's hand and forfeit interest in the current pot
*Abov ...
of the E6/F4
Coxeter-Dynkin diagrams. This can be seen in the
Coxeter plane
In mathematics, the Coxeter number ''h'' is the order of a Coxeter element of an irreducible Coxeter group. It is named after H.S.M. Coxeter.
Definitions
Note that this article assumes a finite Coxeter group. For infinite Coxeter groups, there ar ...
projections. The 24 vertices of the 24-cell are projected in the same two rings as seen in the 2
21.
This polytope can tessellate Euclidean 6-space, forming the
222 honeycomb with this Coxeter-Dynkin diagram: .
Related complex polyhedra
The
regular complex polygon
In geometry, a regular complex polygon is a generalization of a regular polygon in real space to an analogous structure in a complex Hilbert space, where each real dimension is accompanied by an imaginary one. A regular polygon exists in 2 real ...
333, , in
has a real representation as the ''2
21'' polytope, , in 4-dimensional space. It is called a
Hessian polyhedron
In geometry, the Hessian polyhedron is a regular complex polyhedron 333, , in \mathbb^3. It has 27 vertices, 72 3 edges, and 27 33 faces. It is self-dual.
Coxeter named it after Ludwig Otto Hesse for sharing the ''Hessian configuration'' \left ...
after
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 ...
. It has 27 vertices, 72 3-edges, and 27 33 faces. Its
complex reflection group In mathematics, a complex reflection group is a finite group acting on a finite-dimensional complex vector space that is generated by complex reflections: non-trivial elements that fix a complex hyperplane pointwise.
Complex reflection groups arise ...
is
3 sub>3
sub>3, order 648.
Related polytopes
The 2
21 is fourth 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.
The 2
21 polytope is fourth in dimensional series 2
k2.
The 2
21 polytope is second in dimensional series 2
2k.
Rectified 2_21 polytope
The rectified 2
21 has 216 vertices, and 126 facets: 72
rectified 5-simplices, and 27
rectified 5-orthoplex
In five-dimensional geometry, a rectified 5-orthoplex is a convex uniform 5-polytope, being a rectification of the regular 5-orthoplex.
There are 5 degrees of rectifications for any 5-polytope, the zeroth here being the 5-orthoplex itself, and ...
es and 27
5-demicubes . 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 (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...
is a
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 ...
prism.
Alternate names
* Rectified icosihepta-heptacontidi-peton as a rectified 27-72 facetted polypeton (acronym rojak) (Jonathan Bowers)
[Klitzing, (o3x3o3o3o *c3o - rojak)]
Construction
Its construction is based on the
E6 group and information can be extracted from the ringed
Coxeter-Dynkin diagram representing this polytope: .
Removing the ring on the short branch leaves the
rectified 5-simplex
In five-dimensional geometry, a rectified 5-simplex is a convex uniform 5-polytope, being a rectification of the regular 5-simplex.
There are three unique degrees of rectifications, including the zeroth, the 5-simplex itself. Vertices of the '' ...
, .
Removing the ring on the end of the other 2-length branch leaves the
rectified 5-orthoplex
In five-dimensional geometry, a rectified 5-orthoplex is a convex uniform 5-polytope, being a rectification of the regular 5-orthoplex.
There are 5 degrees of rectifications for any 5-polytope, the zeroth here being the 5-orthoplex itself, and ...
in its alternated form: t
1(2
11), .
Removing the ring on the end of the same 2-length branch leaves the
5-demicube: (1
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 ring and ringing the neighboring ring. 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 ...
prism, t
1x, .
Images
Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow.
Truncated 2_21 polytope
The truncated 2
21 has 432 vertices, 5040 edges, 4320 faces, 1350 cells, and 126 4-faces. 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 (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...
is a
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 ...
pyramid.
Images
Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow, green, cyan, blue, purple.
See also
*
List of E6 polytopes
Notes
References
*
T. Gosset: ''On the Regular and Semi-Regular Figures in Space of n Dimensions'', Messenger of Mathematics, Macmillan, 1900
*
* 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 17)
Harold Scott MacDonald Coxeter, Coxeter, ''The Evolution of Coxeter-Dynkin diagrams'',
ieuw Archief voor Wiskunde 9 (1991) 233-248See figure 1: (p. 232) (Node-edge graph of polytope)
* x3o3o3o3o *c3o - jak, o3x3o3o3o *c3o - rojak
{{Polytopes
6-polytopes