4 _{21} | 1 _{42} | 2 _{41} |

Rectified 4 _{21} | Rectified 1 _{42} | Rectified 2 _{41} |

Birectified 4 _{21} | Trirectified 4 _{21} | |

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

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

- 241 polytope
- Alternate names
- Coordinates
- Construction
- Images
- Related polytopes and honeycombs
- Rectified 2 41 polytope
- Alternate names 2
- Construction 2
- Visualizations
- See also
- Notes
- References

Its Coxeter symbol is **2 _{41}**, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node sequences.

The **rectified 2 _{41}** is constructed by points at the mid-edges of the

These polytopes are part of a family of 255 (2^{8} − 1) convex uniform polytopes in 8-dimensions, made of uniform polytope facets, defined by all permutations of rings in this Coxeter-Dynkin diagram: .

2_{41} polytope | |
---|---|

Type | Uniform 8-polytope |

Family | 2_{k1} polytope |

Schläfli symbol | {3,3,3^{4,1}} |

Coxeter symbol | 2_{41} |

Coxeter diagram | |

7-faces | 17520: 240 2 _{31} 17280 {3 ^{6}} |

6-faces | 144960: 6720 2 _{21} 138240 {3 ^{5}} |

5-faces | 544320: 60480 2 _{11} 483840 {3 ^{4}} |

4-faces | 1209600: 241920 {2 _{01} 967680 {3 ^{3}} |

Cells | 1209600 {3^{2}} |

Faces | 483840 {3} |

Edges | 69120 |

Vertices | 2160 |

Vertex figure | 1_{41} |

Petrie polygon | 30-gon |

Coxeter group | E_{8}, [3^{4,2,1}] |

Properties | convex |

The **2 _{41}** is composed of 17,520 facets (240 2

This polytope is a facet in the uniform tessellation, 2_{51} with Coxeter-Dynkin diagram:

- E. L. Elte named it V
_{2160}(for its 2160 vertices) in his 1912 listing of semiregular polytopes.^{ [1] } - It is named
**2**by Coxeter for its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node sequence._{41} **Diacositetracont-myriaheptachiliadiacosioctaconta-zetton**(Acronym Bay) - 240-17280 facetted polyzetton (Jonathan Bowers)^{ [2] }

The 2160 vertices can be defined as follows:

- 16 permutations of (±4,0,0,0,0,0,0,0) of (8-orthoplex)
- 1120 permutations of (±2,±2,±2,±2,0,0,0,0) of (trirectified 8-orthoplex)
- 1024 permutations of (±3,±1,±1,±1,±1,±1,±1,±1)
*with an odd number of minus-signs*

It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 8-dimensional space.

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

Removing the node on the short branch leaves the 7-simplex: . There are 17280 of these facets

Removing the node on the end of the 4-length branch leaves the 2_{31}, . There are 240 of these facets. They are centered at the positions of the 240 vertices in the 4_{21} polytope.

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the 7-demicube, 1_{41}, .

Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.^{ [3] }

E_{8} | k-face | f_{k} | f_{0} | f_{1} | f_{2} | f_{3} | f_{4} | f_{5} | f_{6} | f_{7} | k-figure | notes | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

D_{7} | ( ) | f_{0} | 2160 | 64 | 672 | 2240 | 560 | 2240 | 280 | 1344 | 84 | 448 | 14 | 64 | h{4,3,3,3,3,3} | E_{8}/D_{7} = 192*10!/64/7! = 2160 | |

A_{6}A_{1} | { } | f_{1} | 2 | 69120 | 21 | 105 | 35 | 140 | 35 | 105 | 21 | 42 | 7 | 7 | r{3,3,3,3,3} | E_{8}/A_{6}A_{1} = 192*10!/7!/2 = 69120 | |

A_{4}A_{2}A_{1} | {3} | f_{2} | 3 | 3 | 483840 | 10 | 5 | 20 | 10 | 20 | 10 | 10 | 5 | 2 | {}x{3,3,3} | E_{8}/A_{4}A_{2}A_{1} = 192*10!/5!/3!/2 = 483840 | |

A_{3}A_{3} | {3,3} | f_{3} | 4 | 6 | 4 | 1209600 | 1 | 4 | 4 | 6 | 6 | 4 | 4 | 1 | {3,3}V( ) | E_{8}/A_{3}A_{3} = 192*10!/4!/4! = 1209600 | |

A_{4}A_{3} | {3,3,3} | f_{4} | 5 | 10 | 10 | 5 | 241920 | * | 4 | 0 | 6 | 0 | 4 | 0 | {3,3} | E_{8}/A_{4}A_{3} = 192*10!/5!/4! = 241920 | |

A_{4}A_{2} | 5 | 10 | 10 | 5 | * | 967680 | 1 | 3 | 3 | 3 | 3 | 1 | {3}V( ) | E_{8}/A_{4}A_{2} = 192*10!/5!/3! = 967680 | |||

D_{5}A_{2} | {3,3,3^{1,1}} | f_{5} | 10 | 40 | 80 | 80 | 16 | 16 | 60480 | * | 3 | 0 | 3 | 0 | {3} | E_{8}/D_{5}A_{2} = 192*10!/16/5!/2 = 40480 | |

A_{5}A_{1} | {3,3,3,3} | 6 | 15 | 20 | 15 | 0 | 6 | * | 483840 | 1 | 2 | 2 | 1 | { }V( ) | E_{8}/A_{5}A_{1} = 192*10!/6!/2 = 483840 | ||

E_{6}A_{1} | {3,3,3^{2,1}} | f_{6} | 27 | 216 | 720 | 1080 | 216 | 432 | 27 | 72 | 6720 | * | 2 | 0 | { } | E_{8}/E_{6}A_{1} = 192*10!/72/6! = 6720 | |

A_{6} | {3,3,3,3,3} | 7 | 21 | 35 | 35 | 0 | 21 | 0 | 7 | * | 138240 | 1 | 1 | E_{8}/A_{6} = 192*10!/7! = 138240 | |||

E_{7} | {3,3,3^{3,1}} | f_{7} | 126 | 2016 | 10080 | 20160 | 4032 | 12096 | 756 | 4032 | 56 | 576 | 240 | * | ( ) | E_{8}/E_{7} = 192*10!/72!/8! = 240 | |

A_{7} | {3,3,3,3,3,3} | 8 | 28 | 56 | 70 | 0 | 56 | 0 | 28 | 0 | 8 | * | 17280 | E_{8}/A_{7} = 192*10!/8! = 17280 |

Petrie polygon projections can be 12, 18, or 30-sided based on the E6, E7, and E8 symmetries. The 2160 vertices are all displayed, but lower symmetry forms have projected positions overlapping, shown as different colored vertices. For comparison, a B6 coxeter group is also shown.

E8 [30] | [20] | [24] |
---|---|---|

(1) | ||

E7 [18] | E6 [12] | [6] |

(1,8,24,32) |

D3 / B2 / A3 [4] | D4 / B3 / A2 [6] | D5 / B4 [8] |
---|---|---|

D6 / B5 / A4 [10] | D7 / B6 [12] | D8 / B7 / A6 [14] |

(1,3,9,12,18,21,36) | ||

B8 [16/2] | A5 [6] | A7 [8] |

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^{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 | 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} |

Rectified 2_{41} polytope | |
---|---|

Type | Uniform 8-polytope |

Schläfli symbol | t_{1}{3,3,3^{4,1}} |

Coxeter symbol | t_{1}(2_{41}) |

Coxeter diagram | |

7-faces | 19680 total: 240 t |

6-faces | 313440 |

5-faces | 1693440 |

4-faces | 4717440 |

Cells | 7257600 |

Faces | 5322240 |

Edges | 19680 |

Vertices | 69120 |

Vertex figure | rectified 6-simplex prism |

Petrie polygon | 30-gon |

Coxeter group | E_{8}, [3^{4,2,1}] |

Properties | convex |

The **rectified 2 _{41}** is a rectification of the 2

- Rectified Diacositetracont-myriaheptachiliadiacosioctaconta-zetton for rectified 240-17280 facetted polyzetton (known as robay for short)
^{ [4] }^{ [5] }

It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 8-dimensional space, defined by root vectors of the E_{8} Coxeter group.

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

Removing the node on the short branch leaves the rectified 7-simplex: .

Removing the node on the end of the 4-length branch leaves the rectified 2_{31}, .

Removing the node on the end of the 2-length branch leaves the 7-demicube, 1_{41}.

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the rectified 6-simplex prism, .

Petrie polygon projections can be 12, 18, or 30-sided based on the E6, E7, and E8 symmetries. The 2160 vertices are all displayed, but lower symmetry forms have projected positions overlapping, shown as different colored vertices. For comparison, a B6 coxeter group is also shown.

E8 [30] | [20] | [24] |
---|---|---|

(1) | ||

E7 [18] | E6 [12] | [6] |

(1,8,24,32) |

D3 / B2 / A3 [4] | D4 / B3 / A2 [6] | D5 / B4 [8] |
---|---|---|

D6 / B5 / A4 [10] | D7 / B6 [12] | D8 / B7 / A6 [14] |

(1,3,9,12,18,21,36) | ||

B8 [16/2] | A5 [6] | A7 [8] |

In six-dimensional geometry, a **uniform polypeton** is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform 5-polytopes.

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

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 6-dimensional geometry, the **2 _{21}** polytope is a uniform 6-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 7-dimensional geometry, the **3 _{31} honeycomb** is a uniform honeycomb, also given by Schläfli symbol {3,3,3,3

In 7-dimensional geometry, **1 _{33}** is a uniform honeycomb, also given by Schläfli symbol {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 seven-dimensional geometry, a **rectified 7-simplex** is a convex uniform 7-polytope, being a rectification of the regular 7-simplex.

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

In eight-dimensional geometry, a **heptellated 8-simplex** is a convex uniform 8-polytope, including 7th-order truncations (heptellation) from the regular 8-simplex.

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

In seven-dimensional Euclidean geometry, the **quarter 7-cubic honeycomb** is a uniform space-filling tessellation. It has half the vertices of the 7-demicubic honeycomb, and a quarter of the vertices of a 7-cube honeycomb. Its facets are 7-demicubes, pentellated 7-demicubes, and {3^{1,1,1}}×{3,3} duoprisms.

- Elte, E. L. (1912),
*The Semiregular Polytopes of the Hyperspaces*, Groningen: University of Groningen - 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, ISBN 978-0-471-01003-6- (Paper 24) H.S.M. Coxeter,
*Regular and Semi-Regular Polytopes III*, [Math. Zeit. 200 (1988) 3-45]

- (Paper 24) H.S.M. Coxeter,
- Klitzing, Richard. "8D Uniform polyzetta". x3o3o3o *c3o3o3o3o - bay, o3x3o3o *c3o3o3o3o - robay

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