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

- 142 polytope
- Alternate names
- Coordinates
- Construction
- Projections
- Related polytopes and honeycombs
- Rectified 142 polytope
- Alternate names 2
- Construction 2
- Projections 2
- See also
- Notes
- References

Its Coxeter symbol is **1 _{42}**, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node sequences.

The **rectified 1 _{42}** 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 and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: .

1_{42} | |
---|---|

Type | Uniform 8-polytope |

Family | 1_{k2} polytope |

Schläfli symbol | {3,3^{4,2}} |

Coxeter symbol | 1_{42} |

Coxeter diagrams | |

7-faces | 2400: 240 1 _{32} 2160 1 _{41} |

6-faces | 106080: 6720 1 _{22} 30240 1 _{31} 69120 {3 ^{5}} |

5-faces | 725760: 60480 1 _{12} 181440 1 _{21} 483840 {3 ^{4}} |

4-faces | 2298240: 241920 1 _{02} 604800 1 _{11} 1451520 {3 ^{3}} |

Cells | 3628800: 1209600 1 _{01} 2419200 {3 ^{2}} |

Faces | 2419200 {3} |

Edges | 483840 |

Vertices | 17280 |

Vertex figure | t_{2}{3^{6}} |

Petrie polygon | 30-gon |

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

Properties | convex |

The **1 _{42}** is composed of 2400 facets: 240

This polytope, along with the demiocteract, can tessellate 8-dimensional space, represented by the symbol **1 _{52}**, and Coxeter-Dynkin diagram: .

- E. L. Elte (1912) excluded this polytope from his listing of semiregular polytopes, because it has more than two types of 6-faces, but under his naming scheme it would be called V
_{17280}for its 17280 vertices.^{ [1] } - Coxeter named it
**1**for its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node branch._{42} **Diacositetracont-dischiliahectohexaconta-zetton**(acronym*bif*) - 240-2160 facetted polyzetton (Jonathan Bowers)^{ [2] }

The 17280 vertices can be defined as sign and location permutations of:

All sign combinations (32): (280×32=8960 vertices)

- (4, 2, 2, 2, 2, 0, 0, 0)

Half of the sign combinations (128): ((1+8+56)×128=8320 vertices)

- (2, 2, 2, 2, 2, 2, 2, 2)
- (5, 1, 1, 1, 1, 1, 1, 1)
- (3, 3, 3, 1, 1, 1, 1, 1)

The edge length is 2√2 in this coordinate set, and the polytope radius is 4√2.

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

Removing the node on the end of the 4-length branch leaves the 1_{32}, .

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the birectified 7-simplex, 0_{42}, .

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 | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

A_{7} | ( ) | f_{0} | 17280 | 56 | 420 | 280 | 560 | 70 | 280 | 420 | 56 | 168 | 168 | 28 | 56 | 28 | 8 | 8 | 2r{3^{6}} | E_{8}/A_{7} = 192*10!/8! = 17280 | |

A_{4}A_{2}A_{1} | { } | f_{1} | 2 | 483840 | 15 | 15 | 30 | 5 | 30 | 30 | 10 | 30 | 15 | 10 | 15 | 3 | 5 | 3 | {3}x{3,3,3} | E_{8}/A_{4}A_{2}A_{1} = 192*10!/5!/2/2 = 483840 | |

A_{3}A_{2}A_{1} | {3} | f_{2} | 3 | 3 | 2419200 | 2 | 4 | 1 | 8 | 6 | 4 | 12 | 4 | 6 | 8 | 1 | 4 | 2 | {3.3}v{ } | E_{8}/A_{3}A_{2}A_{1} = 192*10!/4!/3!/2 = 2419200 | |

A_{3}A_{3} | 1_{10} | f_{3} | 4 | 6 | 4 | 1209600 | * | 1 | 4 | 0 | 4 | 6 | 0 | 6 | 4 | 0 | 4 | 1 | {3,3}v( ) | E_{8}/A_{3}A_{3} = 192*10!/4!/4! = 1209600 | |

A_{3}A_{2}A_{1} | 4 | 6 | 4 | * | 2419200 | 0 | 2 | 3 | 1 | 6 | 3 | 3 | 6 | 1 | 3 | 2 | {3}v{ } | E_{8}/A_{3}A_{2}A_{1} = 192*10!/4!/3!/2 = 2419200 | |||

A_{4}A_{3} | 1_{20} | f_{4} | 5 | 10 | 10 | 5 | 0 | 241920 | * | * | 4 | 0 | 0 | 6 | 0 | 0 | 4 | 0 | {3,3} | E_{8}/A_{4}A_{3} = 192*10!/4!/4! = 241920 | |

D_{4}A_{2} | 1_{11} | 8 | 24 | 32 | 8 | 8 | * | 604800 | * | 1 | 3 | 0 | 3 | 3 | 0 | 3 | 1 | {3}v( ) | E_{8}/D_{4}A_{2} = 192*10!/8/4!/3! = 604800 | ||

A_{4}A_{1}A_{1} | 1_{20} | 5 | 10 | 10 | 0 | 5 | * | * | 1451520 | 0 | 2 | 2 | 1 | 4 | 1 | 2 | 2 | { }v{ } | E_{8}/A_{4}A_{1}A_{1} = 192*10!/5!/2/2 = 1451520 | ||

D_{5}A_{2} | 1_{21} | f_{5} | 16 | 80 | 160 | 80 | 40 | 16 | 10 | 0 | 60480 | * | * | 3 | 0 | 0 | 3 | 0 | {3} | E_{8}/D_{5}A_{2} = 192*10!/16/5!/3! = 40480 | |

D_{5}A_{1} | 16 | 80 | 160 | 40 | 80 | 0 | 10 | 16 | * | 181440 | * | 1 | 2 | 0 | 2 | 1 | { }v( ) | E_{8}/D_{5}A_{1} = 192*10!/16/5!/2 = 181440 | |||

A_{5}A_{1} | 1_{30} | 6 | 15 | 20 | 0 | 15 | 0 | 0 | 6 | * | * | 483840 | 0 | 2 | 1 | 1 | 2 | E_{8}/A_{5}A_{1} = 192*10!/6!/2 = 483840 | |||

E_{6}A_{1} | 1_{22} | f_{6} | 72 | 720 | 2160 | 1080 | 1080 | 216 | 270 | 216 | 27 | 27 | 0 | 6720 | * | * | 2 | 0 | { } | E_{8}/E_{6}A_{1} = 192*10!/72/6!/2 = 6720 | |

D_{6} | 1_{31} | 32 | 240 | 640 | 160 | 480 | 0 | 60 | 192 | 0 | 12 | 32 | * | 30240 | * | 1 | 1 | E_{8}/D_{6} = 192*10!/32/6! = 30240 | |||

A_{6}A_{1} | 1_{40} | 7 | 21 | 35 | 0 | 35 | 0 | 0 | 21 | 0 | 0 | 7 | * | * | 69120 | 0 | 2 | E_{8}/A_{6}A_{1} = 192*10!/7!/2 = 69120 | |||

E_{7} | 1_{32} | f_{7} | 576 | 10080 | 40320 | 20160 | 30240 | 4032 | 7560 | 12096 | 756 | 1512 | 2016 | 56 | 126 | 0 | 240 | * | ( ) | E_{8}/E_{7} = 192*10!/72/8! = 240 | |

D_{7} | 1_{41} | 64 | 672 | 2240 | 560 | 2240 | 0 | 280 | 1344 | 0 | 84 | 448 | 0 | 14 | 64 | * | 2160 | E_{8}/D_{7} = 192*10!/64/7! = 2160 |

Orthographic projections are shown for the sub-symmetries of E_{8}: E_{7}, E_{6}, B_{8}, B_{7}, B_{6}, B_{5}, B_{4}, B_{3}, B_{2}, A_{7}, and A_{5} Coxeter planes, as well as two more symmetry planes of order 20 and 24. Vertices are shown as circles, colored by their order of overlap in each projective plane.

E8 [30] | E7 [18] | E6 [12] |
---|---|---|

(1) | (1,3,6) | (8,16,24,32,48,64,96) |

[20] | [24] | [6] |

(1,2,3,4,5,6,7,8,10,11,12,14,16,18,19,20) |

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

(32,160,192,240,480,512,832,960) | (72,216,432,720,864,1080) | (8,16,24,32,48,64,96) |

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

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

1_{k2} 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 (order) | [3^{−1,2,1}] | [3^{0,2,1}] | [3^{1,2,1}] | [[3<sup>2,2,1</sup>]] | [3^{3,2,1}] | [3^{4,2,1}] | [3^{5,2,1}] | [3^{6,2,1}] | |||

Order | 12 | 120 | 1,920 | 103,680 | 2,903,040 | 696,729,600 | ∞ | ||||

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

Name | 1_{−1,2} | 1_{02} | 1_{12} | 1_{22} | 1_{32} | 1_{42} | 1_{52} | 1_{62} |

Rectified 1_{42} | |
---|---|

Type | Uniform 8-polytope |

Schläfli symbol | t_{1}{3,3^{4,2}} |

Coxeter symbol | 0_{421} |

Coxeter diagrams | |

7-faces | 19680 |

6-faces | 382560 |

5-faces | 2661120 |

4-faces | 9072000 |

Cells | 16934400 |

Faces | 16934400 |

Edges | 7257600 |

Vertices | 483840 |

Vertex figure | {3,3,3}×{3}×{} |

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

Properties | convex |

The **rectified 1 _{42}** is named from being a rectification of the 1

- 0
_{421}polytope - Birectified 2
_{41}polytope - Quadrirectified 4
_{21}polytope - Rectified diacositetracont-dischiliahectohexaconta-zetton as a rectified 240-2160 facetted polyzetton (acronym
*buffy*) (Jonathan Bowers)^{ [4] }

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 end of the 1-length branch leaves the birectified 7-simplex,

Removing the node on the end of the 2-length branch leaves the birectified 7-cube, .

Removing the node on the end of the 3-length branch leaves the rectified 1_{32}, .

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the 5-cell-triangle duoprism prism, .

Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.^{ [5] }

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

A_{4}A_{2}A_{1} | ( ) | f_{0} | 483840 | 30 | 30 | 15 | 60 | 10 | 15 | 60 | 30 | 60 | 5 | 20 | 30 | 60 | 30 | 30 | 10 | 20 | 30 | 30 | 15 | 6 | 10 | 10 | 15 | 6 | 3 | 5 | 2 | 3 | {3,3,3}x{3,3}x{} | |

A_{3}A_{1}A_{1} | { } | f_{1} | 2 | 7257600 | 2 | 1 | 4 | 1 | 2 | 8 | 4 | 6 | 1 | 4 | 8 | 12 | 6 | 4 | 4 | 6 | 12 | 8 | 4 | 1 | 6 | 4 | 8 | 2 | 1 | 4 | 1 | 2 | ||

A_{3}A_{2} | {3} | f_{2} | 3 | 3 | 4838400 | * | * | 1 | 1 | 4 | 0 | 0 | 1 | 4 | 4 | 6 | 0 | 0 | 4 | 6 | 6 | 4 | 0 | 0 | 6 | 4 | 4 | 1 | 0 | 4 | 1 | 1 | ||

A_{3}A_{2}A_{1} | 3 | 3 | * | 2419200 | * | 0 | 2 | 0 | 4 | 0 | 1 | 0 | 8 | 0 | 6 | 0 | 4 | 0 | 12 | 0 | 4 | 0 | 6 | 0 | 8 | 0 | 1 | 4 | 0 | 2 | ||||

A_{2}A_{2}A_{1} | 3 | 3 | * | * | 9676800 | 0 | 0 | 2 | 1 | 3 | 0 | 1 | 2 | 6 | 3 | 3 | 1 | 3 | 6 | 6 | 3 | 1 | 3 | 3 | 6 | 2 | 1 | 3 | 1 | 2 | ||||

A_{3}A_{3} | 0_{200} | f_{3} | 4 | 6 | 4 | 0 | 0 | 1209600 | * | * | * | * | 1 | 4 | 0 | 0 | 0 | 0 | 4 | 6 | 0 | 0 | 0 | 0 | 6 | 4 | 0 | 0 | 0 | 4 | 1 | 0 | ||

0_{110} | 6 | 12 | 4 | 4 | 0 | * | 1209600 | * | * | * | 1 | 0 | 4 | 0 | 0 | 0 | 4 | 0 | 6 | 0 | 0 | 0 | 6 | 0 | 4 | 0 | 0 | 4 | 0 | 1 | ||||

A_{3}A_{2} | 6 | 12 | 4 | 0 | 4 | * | * | 4838400 | * | * | 0 | 1 | 1 | 3 | 0 | 0 | 1 | 3 | 3 | 3 | 0 | 0 | 3 | 3 | 3 | 1 | 0 | 3 | 1 | 1 | ||||

A_{3}A_{2}A_{1} | 6 | 12 | 0 | 4 | 4 | * | * | * | 2419200 | * | 0 | 0 | 2 | 0 | 3 | 0 | 1 | 0 | 6 | 0 | 3 | 0 | 3 | 0 | 6 | 0 | 1 | 3 | 0 | 2 | ||||

A_{3}A_{1}A_{1} | 0_{200} | 4 | 6 | 0 | 0 | 4 | * | * | * | * | 7257600 | 0 | 0 | 0 | 2 | 1 | 2 | 0 | 1 | 2 | 4 | 2 | 1 | 1 | 2 | 4 | 2 | 1 | 2 | 1 | 2 | |||

A_{4}A_{3} | 0_{210} | f_{4} | 10 | 30 | 20 | 10 | 0 | 5 | 5 | 0 | 0 | 0 | 241920 | * | * | * | * | * | 4 | 0 | 0 | 0 | 0 | 0 | 6 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | ||

A_{4}A_{2} | 10 | 30 | 20 | 0 | 10 | 5 | 0 | 5 | 0 | 0 | * | 967680 | * | * | * | * | 1 | 3 | 0 | 0 | 0 | 0 | 3 | 3 | 0 | 0 | 0 | 3 | 1 | 0 | ||||

D_{4}A_{2} | 0_{111} | 24 | 96 | 32 | 32 | 32 | 0 | 8 | 8 | 8 | 0 | * | * | 604800 | * | * | * | 1 | 0 | 3 | 0 | 0 | 0 | 3 | 0 | 3 | 0 | 0 | 3 | 0 | 1 | |||

A_{4}A_{1} | 0_{210} | 10 | 30 | 10 | 0 | 20 | 0 | 0 | 5 | 0 | 5 | * | * | * | 2903040 | * | * | 0 | 1 | 1 | 2 | 0 | 0 | 1 | 2 | 2 | 1 | 0 | 2 | 1 | 1 | |||

A_{4}A_{1}A_{1} | 10 | 30 | 0 | 10 | 20 | 0 | 0 | 0 | 5 | 5 | * | * | * | * | 1451520 | * | 0 | 0 | 2 | 0 | 2 | 0 | 1 | 0 | 4 | 0 | 1 | 2 | 0 | 2 | ||||

A_{4}A_{1} | 0_{300} | 5 | 10 | 0 | 0 | 10 | 0 | 0 | 0 | 0 | 5 | * | * | * | * | * | 2903040 | 0 | 0 | 0 | 2 | 1 | 1 | 0 | 1 | 2 | 2 | 1 | 1 | 1 | 2 | |||

D_{5}A_{2} | 0_{211} | f_{5} | 80 | 480 | 320 | 160 | 160 | 80 | 80 | 80 | 40 | 0 | 16 | 16 | 10 | 0 | 0 | 0 | 60480 | * | * | * | * | * | 3 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | {3} | |

A_{5}A_{1} | 0_{220} | 20 | 90 | 60 | 0 | 60 | 15 | 0 | 30 | 0 | 15 | 0 | 6 | 0 | 6 | 0 | 0 | * | 483840 | * | * | * | * | 1 | 2 | 0 | 0 | 0 | 2 | 1 | 0 | { }v() | ||

D_{5}A_{1} | 0_{211} | 80 | 480 | 160 | 160 | 320 | 0 | 40 | 80 | 80 | 80 | 0 | 0 | 10 | 16 | 16 | 0 | * | * | 181440 | * | * | * | 1 | 0 | 2 | 0 | 0 | 2 | 0 | 1 | |||

A_{5} | 0_{310} | 15 | 60 | 20 | 0 | 60 | 0 | 0 | 15 | 0 | 30 | 0 | 0 | 0 | 6 | 0 | 6 | * | * | * | 967680 | * | * | 0 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | ( )v( )v() | ||

A_{5}A_{1} | 15 | 60 | 0 | 20 | 60 | 0 | 0 | 0 | 15 | 30 | 0 | 0 | 0 | 0 | 6 | 6 | * | * | * | * | 483840 | * | 0 | 0 | 2 | 0 | 1 | 1 | 0 | 2 | { }v() | |||

0_{400} | 6 | 15 | 0 | 0 | 20 | 0 | 0 | 0 | 0 | 15 | 0 | 0 | 0 | 0 | 0 | 6 | * | * | * | * | * | 483840 | 0 | 0 | 0 | 2 | 1 | 0 | 1 | 2 | ||||

E_{6}A_{1} | 0_{221} | f_{6} | 720 | 6480 | 4320 | 2160 | 4320 | 1080 | 1080 | 2160 | 1080 | 1080 | 216 | 432 | 270 | 432 | 216 | 0 | 27 | 72 | 27 | 0 | 0 | 0 | 6720 | * | * | * | * | 2 | 0 | 0 | { } | |

A_{6} | 0_{320} | 35 | 210 | 140 | 0 | 210 | 35 | 0 | 105 | 0 | 105 | 0 | 21 | 0 | 42 | 0 | 21 | 0 | 7 | 0 | 7 | 0 | 0 | * | 138240 | * | * | * | 1 | 1 | 0 | |||

D_{6} | 0_{311} | 240 | 1920 | 640 | 640 | 1920 | 0 | 160 | 480 | 480 | 960 | 0 | 0 | 60 | 192 | 192 | 192 | 0 | 0 | 12 | 32 | 32 | 0 | * | * | 30240 | * | * | 1 | 0 | 1 | |||

A_{6} | 0_{410} | 21 | 105 | 35 | 0 | 140 | 0 | 0 | 35 | 0 | 105 | 0 | 0 | 0 | 21 | 0 | 42 | 0 | 0 | 0 | 7 | 0 | 7 | * | * | * | 138240 | * | 0 | 1 | 1 | |||

A_{6}A_{1} | 21 | 105 | 0 | 35 | 140 | 0 | 0 | 0 | 35 | 105 | 0 | 0 | 0 | 0 | 21 | 42 | 0 | 0 | 0 | 0 | 7 | 7 | * | * | * | * | 69120 | 0 | 0 | 2 | ||||

E_{7} | 0_{321} | f_{7} | 10080 | 120960 | 80640 | 40320 | 120960 | 20160 | 20160 | 60480 | 30240 | 60480 | 4032 | 12096 | 7560 | 24192 | 12096 | 12096 | 756 | 4032 | 1512 | 4032 | 2016 | 0 | 56 | 576 | 126 | 0 | 0 | 240 | * | * | ( ) | |

A_{7} | 0_{420} | 56 | 420 | 280 | 0 | 560 | 70 | 0 | 280 | 0 | 420 | 0 | 56 | 0 | 168 | 0 | 168 | 0 | 28 | 0 | 56 | 0 | 28 | 0 | 8 | 0 | 8 | 0 | * | 17280 | * | |||

D_{7} | 0_{411} | 672 | 6720 | 2240 | 2240 | 8960 | 0 | 560 | 2240 | 2240 | 6720 | 0 | 0 | 280 | 1344 | 1344 | 2688 | 0 | 0 | 84 | 448 | 448 | 448 | 0 | 0 | 14 | 64 | 64 | * | * | 2160 |

Orthographic projections are shown for the sub-symmetries of B_{6}, B_{5}, B_{4}, B_{3}, B_{2}, A_{7}, and A_{5} Coxeter planes. Vertices are shown as circles, colored by their order of overlap in each projective plane.

(Planes for E_{8}: E_{7}, E_{6}, B_{8}, B_{7}, [24] are not shown for being too large to display.)

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

D6 / B5 / A4 [10] | D7 / B6 [12] | [6] |

A5 [6] | A7 [8] | [20] |

- ↑ Elte, E. L. (1912),
*The Semiregular Polytopes of the Hyperspaces*, Groningen: University of Groningen - ↑ Klitzing, (o3o3o3x *c3o3o3o3o - bif)
- ↑ Coxeter, Regular Polytopes, 11.8 Gossett figures in six, seven, and eight dimensions, p. 202-203
- ↑ Klitzing, (o3o3o3x *c3o3o3o3o - buffy)
- ↑ Coxeter, Regular Polytopes, 11.8 Gossett figures in six, seven, and eight dimensions, p. 202-203

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 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 **2 _{41}** 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 geometry, the **1 _{52} honeycomb** is a uniform tessellation of 8-dimensional Euclidean space. It contains

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

In geometry, the **simplectic honeycomb** is a dimensional infinite series of honeycombs, based on the affine Coxeter group symmetry. It is given a Schläfli symbol {3^{[n+1]}}, and is represented by a Coxeter-Dynkin diagram as a cyclic graph of *n+1* nodes with one node ringed. It is composed of n-simplex facets, along with all rectified n-simplices. It can be thought of as an n-dimensional hypercubic honeycomb that has been subdivided along all hyperplanes , then stretched along its main diagonal until the simplices on the ends of the hypercubes become regular. The vertex figure of an *n-simplex honeycomb* is an expanded n-simplex.

- 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". o3o3o3x *c3o3o3o3o - bif, o3o3o3x *c3o3o3o3o - buffy

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