2 _{21} | Rectified 2 _{21} | |

(1 _{22}) | Birectified 2 _{21}(Rectified 1 _{22}) | |

orthogonal projections in E_{6} Coxeter plane |
---|

In 6-dimensional geometry, the **2 _{21}** polytope is a uniform 6-polytope, constructed within the symmetry of the E

- 2 21 polytope
- Alternate names
- Coordinates
- Construction
- Images
- Geometric folding
- Related complex polyhedra
- Related polytopes
- Rectified 2 21 polytope
- Alternate names 2
- Construction 2
- Images 2
- Truncated 2 21 polytope
- Images 3
- See also
- Notes
- References

Its Coxeter symbol 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

The **rectified 2 _{21}** is constructed by points at the mid-edges of the

These polytopes are a part of family of 39 convex uniform polytopes in 6-dimensions, made of uniform 5-polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: .

2_{21} polytope | |
---|---|

Type | Uniform 6-polytope |

Family | k_{21} polytope |

Schläfli symbol | {3,3,3^{2,1}} |

Coxeter symbol | 2_{21} |

Coxeter-Dynkin diagram | or |

5-faces | 99 total: 27 2 _{11} 72 {3 ^{4}} |

4-faces | 648: 432 {3 ^{3}} 216 {3 ^{3}} |

Cells | 1080 {3,3} |

Faces | 720 {3} |

Edges | 216 |

Vertices | 27 |

Vertex figure | 1 (5-demicube)_{21} |

Petrie polygon | Dodecagon |

Coxeter group | E_{6}, [3^{2,2,1}], order 51840 |

Properties | convex |

The **2 _{21}** has 27 vertices, and 99 facets: 27 5-orthoplexes and 72 5-simplices. Its vertex figure 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). 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 is the 1-skeleton of this polytope.

- E. L. Elte named it V
_{27}(for its 27 vertices) in his 1912 listing of semiregular polytopes.^{ [3] } **Icosihepta-heptacontidi-peton**- 27-72 facetted polypeton (acronym jak) (Jonathan Bowers)^{ [4] }

The 27 vertices can be expressed in 8-space as an edge-figure of the 4_{21} 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)

Its construction is based on the E_{6} group.

The facet information can be extracted from its Coxeter-Dynkin diagram, .

Removing the node on the short branch leaves the 5-simplex, .

Removing the node on the end of the 2-length branch leaves the 5-orthoplex 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 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, (0_{21} polytope), .

Seen in a configuration matrix, the element counts can be derived from the Coxeter group orders.^{ [5] }

E_{6} | k-face | f_{k} | f_{0} | f_{1} | f_{2} | f_{3} | f_{4} | f_{5} | k-figure | notes | |||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

D_{5} | ( ) | f_{0} | 27 | 16 | 80 | 160 | 80 | 40 | 16 | 10 | h{4,3,3,3} | E_{6}/D_{5} = 51840/1920 = 27 | |

A_{4}A_{1} | { } | f_{1} | 2 | 216 | 10 | 30 | 20 | 10 | 5 | 5 | r{3,3,3} | E_{6}/A_{4}A_{1} = 51840/120/2 = 216 | |

A_{2}A_{2}A_{1} | {3} | f_{2} | 3 | 3 | 720 | 6 | 6 | 3 | 2 | 3 | {3}x{ } | E_{6}/A_{2}A_{2}A_{1} = 51840/6/6/2 = 720 | |

A_{3}A_{1} | {3,3} | f_{3} | 4 | 6 | 4 | 1080 | 2 | 1 | 1 | 2 | { }v( ) | E_{6}/A_{3}A_{1} = 51840/24/2 = 1080 | |

A_{4} | {3,3,3} | f_{4} | 5 | 10 | 10 | 5 | 432 | * | 1 | 1 | { } | E_{6}/A_{4} = 51840/120 = 432 | |

A_{4}A_{1} | 5 | 10 | 10 | 5 | * | 216 | 0 | 2 | E_{6}/A_{4}A_{1} = 51840/120/2 = 216 | ||||

A_{5} | {3,3,3,3} | f_{5} | 6 | 15 | 20 | 15 | 6 | 0 | 72 | * | ( ) | E_{6}/A_{5} = 51840/720 = 72 | |

D_{5} | {3,3,3,4} | 10 | 40 | 80 | 80 | 16 | 16 | * | 27 | E_{6}/D_{5} = 51840/1920 = 27 |

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.

E6 [12] | D5 [8] | D4 / A2 [6] | B6 [12/2] |
---|---|---|---|

(1,3) | (1,3) | (3,9) | (1,3) |

A5 [6] | A4 [5] | A3 / D3 [4] | |

(1,3) | (1,2) | (1,4,7) |

The **2 _{21}** is related to the 24-cell by a geometric folding of the E6/F4 Coxeter-Dynkin diagrams. This can be seen in the Coxeter plane projections. The 24 vertices of the 24-cell are projected in the same two rings as seen in the 2

E_{6} | F_{4} |

2_{21} | 24-cell |

This polytope can tessellate Euclidean 6-space, forming the ** 2 _{22} ** honeycomb with this Coxeter-Dynkin diagram: .

The regular complex polygon _{3}{3}_{3}{3}_{3}, , in has a real representation as the *2 _{21}* polytope, , in 4-dimensional space. It is called a Hessian polyhedron after Edmund Hess. It has 27 vertices, 72 3-edges, and 27 3{3}3 faces. Its complex reflection group is

The 2_{21} is fourth in a dimensional series of semiregular polytopes. Each progressive uniform polytope is constructed vertex figure of the previous polytope. Thorold Gosset identified this series in 1900 as containing all regular polytope facets, containing all simplexes and orthoplexes.

k_{21} figures in n dimensional | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Space | Finite | Euclidean | Hyperbolic | ||||||||

E_{n} | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |||

Coxeter group | E_{3}=A_{2}A_{1} | E_{4}=A_{4} | E_{5}=D_{5} | E_{6} | E_{7} | E_{8} | E_{9} = = E_{8}^{+} | E_{10} = = E_{8}^{++} | |||

Coxeter diagram | |||||||||||

Symmetry | [3^{−1,2,1}] | [3^{0,2,1}] | [3^{1,2,1}] | [3^{2,2,1}] | [3^{3,2,1}] | [3^{4,2,1}] | [3^{5,2,1}] | [3^{6,2,1}] | |||

Order | 12 | 120 | 1,920 | 51,840 | 2,903,040 | 696,729,600 | ∞ | ||||

Graph | - | - | |||||||||

Name | −1_{21} | 0_{21} | 1_{21} | 2_{21} | 3_{21} | 4_{21} | 5_{21} | 6_{21} |

The 2_{21} polytope is fourth in dimensional series 2_{k2}.

2_{k1} figures in n dimensions | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Space | Finite | Euclidean | Hyperbolic | ||||||||

n | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |||

Coxeter group | E_{3}=A_{2}A_{1} | E_{4}=A_{4} | E_{5}=D_{5} | E_{6} | E_{7} | E_{8} | E_{9} = = E_{8}^{+} | E_{10} = = E_{8}^{++} | |||

Coxeter diagram | |||||||||||

Symmetry | [3^{−1,2,1}] | [3^{0,2,1}] | [[3<sup>1,2,1</sup>]] | [3^{2,2,1}] | [3^{3,2,1}] | [3^{4,2,1}] | [3^{5,2,1}] | [3^{6,2,1}] | |||

Order | 12 | 120 | 384 | 51,840 | 2,903,040 | 696,729,600 | ∞ | ||||

Graph | - | - | |||||||||

Name | 2_{−1,1} | 2_{01} | 2_{11} | 2_{21} | 2_{31} | 2_{41} | 2_{51} | 2_{61} |

The 2_{21} polytope is second in dimensional series 2_{2k}.

Space | Finite | Euclidean | Hyperbolic | ||
---|---|---|---|---|---|

n | 4 | 5 | 6 | 7 | 8 |

Coxeter group | A_{2}A_{2} | A_{5} | E_{6} | =E_{6}^{+} | E_{6}^{++} |

Coxeter diagram | |||||

Graph | ∞ | ∞ | |||

Name | 2_{2,-1} | 2_{20} | 2_{21} | 2_{22} | 2_{23} |

Rectified 2_{21} polytope | |
---|---|

Type | Uniform 6-polytope |

Schläfli symbol | t_{1}{3,3,3^{2,1}} |

Coxeter symbol | t_{1}(2_{21}) |

Coxeter-Dynkin diagram | or |

5-faces | 126 total: 72 t |

4-faces | 1350 |

Cells | 4320 |

Faces | 5040 |

Edges | 2160 |

Vertices | 216 |

Vertex figure | rectified 5-cell prism |

Coxeter group | E_{6}, [3^{2,2,1}], order 51840 |

Properties | convex |

The **rectified 2 _{21}** has 216 vertices, and 126 facets: 72 rectified 5-simplices, and 27 rectified 5-orthoplexes and 27 5-demicubes . Its vertex figure is a rectified 5-cell prism.

- Rectified icosihepta-heptacontidi-peton as a rectified 27-72 facetted polypeton (acronym rojak) (Jonathan Bowers)
^{ [6] }

Its construction is based on the E_{6} 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, .

Removing the ring on the end of the other 2-length branch leaves the rectified 5-orthoplex 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 is determined by removing the ringed ring and ringing the neighboring ring. This makes rectified 5-cell prism, t_{1}{3,3,3}x{}, .

Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow.

E6 [12] | D5 [8] | D4 / A2 [6] | B6 [12/2] |
---|---|---|---|

A5 [6] | A4 [5] | A3 / D3 [4] | |

Truncated 2_{21} polytope | |
---|---|

Type | Uniform 6-polytope |

Schläfli symbol | t{3,3,3^{2,1}} |

Coxeter symbol | t(2_{21}) |

Coxeter-Dynkin diagram | or |

5-faces | 72+27+27 |

4-faces | 432+216+432+270 |

Cells | 1080+2160+1080 |

Faces | 720+4320 |

Edges | 216+2160 |

Vertices | 432 |

Vertex figure | ( ) v r{3,3,3} |

Coxeter group | E_{6}, [3^{2,2,1}], order 51840 |

Properties | convex |

The **truncated 2 _{21}** has 432 vertices, 5040 edges, 4320 faces, 1350 cells, and 126 4-faces. Its vertex figure is a rectified 5-cell pyramid.

Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow, green, cyan, blue, purple.

E6 [12] | D5 [8] | D4 / A2 [6] | B6 [12/2] |
---|---|---|---|

A5 [6] | A4 [5] | A3 / D3 [4] | |

- ↑ Gosset, 1900
- ↑ Coxeter, H.S.M. (1940). "The Polytope 2
_{21}Whose Twenty-Seven Vertices Correspond to the Lines on the General Cubic Surface".*Amer. J. Math*.**62**(1): 457–486. doi:10.2307/2371466. JSTOR 2371466. - ↑ Elte, 1912
- ↑ Klitzing, (x3o3o3o3o *c3o - jak)
- ↑ Coxeter, Regular Polytopes, 11.8 Gossett figures in six, seven, and eight dimensions, p. 202-203
- ↑ Klitzing, (o3x3o3o3o *c3o - rojak)

In geometry, **demihypercubes** are a class of n-polytopes constructed from alternation of an n-hypercube, labeled as *hγ _{n}* for being

In five-dimensional geometry, a **demipenteract** or **5-demicube** is a semiregular 5-polytope, constructed from a *5-hypercube* (penteract) with alternated vertices removed.

In geometry, the **Gosset–Elte figures**, named by Coxeter after Thorold Gosset and E. L. Elte, are a group of uniform polytopes which are not regular, generated by a Wythoff construction with mirrors all related by order-2 and order-3 dihedral angles. They can be seen as *one-end-ringed* Coxeter–Dynkin diagrams.

In geometry, a **uniform k_{21} polytope** is a polytope in

In ten-dimensional geometry, a 10-polytope is a 10-dimensional polytope whose boundary consists of 9-polytope facets, exactly two such facets meeting at each 8-polytope ridge.

In five-dimensional geometry, a **rectified 5-simplex** is a convex uniform 5-polytope, being a rectification of the regular 5-simplex.

In geometry, **2 _{k1} polytope** is a uniform polytope in

In 7-dimensional geometry, **2 _{31}** is a uniform polytope, constructed from the E7 group.

In 6-dimensional geometry, the **1 _{22}** polytope is a uniform polytope, constructed from the E

In 7-dimensional geometry, **1 _{32}** is a uniform polytope, constructed from the E7 group.

In 8-dimensional geometry, the **1 _{42}** is a uniform 8-polytope, constructed within the symmetry of the E

In 8-dimensional geometry, the **2 _{41}** is a uniform 8-polytope, constructed within the symmetry of the E

In 7-dimensional geometry, the **3 _{21}** polytope is a uniform 7-polytope, constructed within the symmetry of the E

In 8-dimensional geometry, the **4 _{21}** is a semiregular uniform 8-polytope, constructed within the symmetry of the E

In geometry, the **2 _{22} honeycomb** is a uniform tessellation of the six-dimensional Euclidean space. It can be represented by the Schläfli symbol {3,3,3

In six-dimensional geometry, a **pentellated 6-simplex** is a convex uniform 6-polytope with 5th order truncations of the regular 6-simplex.

In six-dimensional geometry, a **rectified 6-simplex** is a convex uniform 6-polytope, being a rectification of the regular 6-simplex.

In geometry, the **5 _{21} honeycomb** is a uniform tessellation of 8-dimensional Euclidean space. The symbol 5

In eight-dimensional geometry, a **rectified 8-simplex** is a convex uniform 8-polytope, being a rectification of the regular 8-simplex.

In geometry, an **E _{9} honeycomb** is a tessellation of uniform polytopes in hyperbolic 9-dimensional space. , also (E

- T. Gosset:
*On the Regular and Semi-Regular Figures in Space of n Dimensions*, Messenger of Mathematics, Macmillan, 1900 - Elte, E. L. (1912),
*The Semiregular Polytopes of the Hyperspaces*, Groningen: University of Groningen **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, ISBN 978-0-471-01003-6- (Paper 17) Coxeter,
*The Evolution of Coxeter-Dynkin diagrams*, [Nieuw Archief voor Wiskunde 9 (1991) 233-248] See figure 1: (p. 232) (Node-edge graph of polytope)

- (Paper 17) Coxeter,
- Klitzing, Richard. "6D uniform polytopes (polypeta)". x3o3o3o3o *c3o - jak, o3x3o3o3o *c3o - rojak

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