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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.

The Coxeter symbol for these figures has the form ki,j, where each letter represents a length of order-3 branches on a Coxeter–Dynkin diagram with a single ring on the end node of a k length sequence of branches. The vertex figure of ki,j is (k − 1)i,j, and each of its facets are represented by subtracting one from one of the nonzero subscripts, i.e. ki − 1,j and ki,j − 1.[1]

Rectified simplices are included in the list as limiting cases with k=0. Similarly 0i,j,k represents a bifurcated graph with a central node ringed.

History

Coxeter named these figures as ki,j (or kij) in shorthand and gave credit of their discovery to Gosset and Elte:[2]

Thorold Gosset first published a list of regular and semi-regular figures in space of n dimensions[3] in 1900, enumerating polytopes with one or more types of regular polytope faces. This included the rectified 5-cell 021 in 4-space, demipenteract 121 in 5-space, 221 in 6-space, 321 in 7-space, 421 in 8-space, and 521 infinite tessellation in 8-space.
E. L. Elte independently enumerated a different semiregular list in his 1912 book, The Semiregular Polytopes of the Hyperspaces.[4] He called them semiregular polytopes of the first kind, limiting his search to one or two types of regular or semiregular k-faces.

Elte's enumeration included all the kij polytopes except for the 142 which has 3 types of 6-faces.

The set of figures extend into honeycombs of (2,2,2), (3,3,1), and (5,4,1) families in 6,7,8 dimensional Euclidean spaces respectively. Gosset's list included the 521 honeycomb as the only semiregular one in his definition.
Definition
Simply-laced ADE groups

The polytopes and honeycombs in this family can be seen within ADE classification.

A finite polytope kij exists if

\( {\displaystyle {\frac {1}{i+1}}+{\frac {1}{j+1}}+{\frac {1}{k+1}}>1} \)

or equal for Euclidean honeycombs, and less for hyperbolic honeycombs.

The Coxeter group [3i,j,k] can generate up to 3 unique uniform Gosset–Elte figures with Coxeter–Dynkin diagrams with one end node ringed. By Coxeter's notation, each figure is represented by kij to mean the end-node on the k-length sequence is ringed.

The simplex family can be seen as a limiting case with k=0, and all rectified (single-ring) Coxeter–Dynkin diagrams.
A-family [3n] (rectified simplices)

The family of n-simplices contain Gosset–Elte figures of the form 0ij as all rectified forms of the n-simplex (i + j = n − 1).

They are listed below, along with their Coxeter–Dynkin diagram, with each dimensional family drawn as a graphic orthogonal projection in the plane of the Petrie polygon of the regular simplex.

Coxeter group Simplex Rectified Birectified Trirectified Quadrirectified
A1
[30]
CDel node 1.png = 000

1-simplex t0.svg
A2
[31]
CDel node 1.pngCDel 3.pngCDel node.png = 010
2-simplex t0.svg
A3
[32]
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png = 020
3-simplex t0.svg
CDel node 1.pngCDel split1.pngCDel nodes.png = 011
3-orthoplex.svg
A4
[33]
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png = 030
4-simplex t0.svg
CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3b.pngCDel nodeb.png = 021
4-simplex t1.svg
A5
[34]
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png = 040
5-simplex t0.svg
CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3b.pngCDel nodeb.pngCDel 3b.pngCDel nodeb.png = 031
5-simplex t1.svg
CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png = 022
5-simplex t2.svg
A6
[35]
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png = 050
6-simplex t0.svg
CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3b.pngCDel nodeb.pngCDel 3b.pngCDel nodeb.pngCDel 3b.pngCDel nodeb.png = 041
6-simplex t1.svg
CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3b.pngCDel nodeb.png = 032
6-simplex t2.svg
A7
[36]
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png = 060
7-simplex t0.svg
CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3b.pngCDel nodeb.pngCDel 3b.pngCDel nodeb.pngCDel 3b.pngCDel nodeb.pngCDel 3b.pngCDel nodeb.png = 051
7-simplex t1.svg
CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3b.pngCDel nodeb.pngCDel 3b.pngCDel nodeb.png = 042
7-simplex t2.svg
CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png = 033
7-simplex t3.svg
A8
[37]
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png = 070
8-simplex t0.svg
CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3b.pngCDel nodeb.pngCDel 3b.pngCDel nodeb.pngCDel 3b.pngCDel nodeb.pngCDel 3b.pngCDel nodeb.pngCDel 3b.pngCDel nodeb.png = 061
8-simplex t1.svg
CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3b.pngCDel nodeb.pngCDel 3b.pngCDel nodeb.pngCDel 3b.pngCDel nodeb.png = 052
8-simplex t2.svg
CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3b.pngCDel nodeb.png = 043
8-simplex t3.svg
A9
[38]
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png = 080
9-simplex t0.svg
CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3b.pngCDel nodeb.pngCDel 3b.pngCDel nodeb.pngCDel 3b.pngCDel nodeb.pngCDel 3b.pngCDel nodeb.pngCDel 3b.pngCDel nodeb.pngCDel 3b.pngCDel nodeb.png = 071
9-simplex t1.svg
CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3b.pngCDel nodeb.pngCDel 3b.pngCDel nodeb.pngCDel 3b.pngCDel nodeb.pngCDel 3b.pngCDel nodeb.png = 062
9-simplex t2.svg
CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3b.pngCDel nodeb.pngCDel 3b.pngCDel nodeb.png = 053
9-simplex t3.svg
CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png = 044
9-simplex t4.svg
A10
[39]
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png = 090
10-simplex t0.svg
CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3b.pngCDel nodeb.pngCDel 3b.pngCDel nodeb.pngCDel 3b.pngCDel nodeb.pngCDel 3b.pngCDel nodeb.pngCDel 3b.pngCDel nodeb.pngCDel 3b.pngCDel nodeb.pngCDel 3b.pngCDel nodeb.png = 081
10-simplex t1.svg
CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3b.pngCDel nodeb.pngCDel 3b.pngCDel nodeb.pngCDel 3b.pngCDel nodeb.pngCDel 3b.pngCDel nodeb.pngCDel 3b.pngCDel nodeb.png = 072
10-simplex t2.svg
CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3b.pngCDel nodeb.pngCDel 3b.pngCDel nodeb.pngCDel 3b.pngCDel nodeb.png = 063
10-simplex t3.svg
CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3b.pngCDel nodeb.png = 054
10-simplex t4.svg
... ...

D-family [3n−3,1,1] demihypercube

Each Dn group has two Gosset–Elte figures, the n-demihypercube as 1k1, and an alternated form of the n-orthoplex, k11, constructed with alternating simplex facets. Rectified n-demihypercubes, a lower symmetry form of a birectified n-cube, can also be represented as 0k11.

Class Demihypercubes Orthoplexes
(Regular)
Rectified demicubes
D3
[31,1,0]
CDel nodea 1.pngCDel 3a.pngCDel branch.png = 110
3-demicube.svg
  CDel nodea.pngCDel 3a.pngCDel branch 10.png = 0110
3-cube t2 B2.svg
D4
[31,1,1]
CDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.png = 111
4-demicube.svg
  CDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.png = 0111
4-cube t0 B3.svg
D5
[32,1,1]
CDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png = 121
5-demicube.svg
CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png = 211
5-orthoplex B4.svg
CDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png = 0211
5-cube t2 B4.svg
D6
[33,1,1]
CDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png = 131
6-demicube.svg
CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png = 311
6-orthoplex B5.svg
CDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png = 0311
6-cube t2 B5.svg
D7
[34,1,1]
CDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png = 141
7-demicube.svg
CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png = 411
7-orthoplex B6.svg
CDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png = 0411
7-cube t2 B6.svg
D8
[35,1,1]
CDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png = 151
8-demicube.svg
CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png = 511
8-orthoplex B7.svg
CDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png = 0511
8-cube t2 B7.svg
D9
[36,1,1]
CDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png = 161
9-demicube.svg
CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png = 611
9-orthoplex B8.svg
CDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png = 0611
9-cube t2 B8.svg
D10
[37,1,1]
CDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png = 171
10-demicube.svg
CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png = 711
10-orthoplex B9.svg
CDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png = 0711
10-cube t2 B9.svg
... ... ...
Dn
[3n−3,1,1]
CDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.png...CDel 3a.pngCDel nodea.png = 1n−3,1 CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.png...CDel 3a.pngCDel nodea 1.png = (n−3)11 CDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.png...CDel 3a.pngCDel nodea.png = 0n−3,1,1

En family [3n−4,2,1]

Each En group from 4 to 8 has two or three Gosset–Elte figures, represented by one of the end-nodes ringed:k21, 1k2, 2k1. A rectified 1k2 series can also be represented as 0k21.
2k1 1k2 k21 0k21

2k1 1k2 k21 0k21
E4
[30,2,1]
CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.png = 201
4-simplex t0.svg
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01l.png = 120
4-simplex t0.svg
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.png = 021
4-simplex t1.svg
E5
[31,2,1]
CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.png = 211
5-orthoplex B4.svg
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.png = 121
5-demicube.svg
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea 1.png = 121
5-demicube.svg
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.png = 0211
5-cube t2 B4.svg
E6
[32,2,1]
CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png = 221
E6 graph.svg
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png = 122
Gosset 1 22 polytope.png
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png = 221
E6 graph.svg
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png = 0221
Up 1 22 t1 E6.svg
E7
[33,2,1]
CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png = 231
Gosset 2 31 polytope.svg
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png = 132
Up2 1 32 t0 E7.svg
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png = 321
E7 graph.svg
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png = 0321
Up2 1 32 t1 E7.svg
E8
[34,2,1]
CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png = 241
2 41 polytope petrie.svg
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png = 142
Gosset 1 42 polytope petrie.svg
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png = 421
Gosset 4 21 polytope petrie.svg
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png = 0421

Euclidean and hyperbolic honeycombs

There are three Euclidean (affine) Coxeter groups in dimensions 6, 7, and 8:[5]

Coxeter group Honeycombs
\( {\tilde {E}}_{6} \) = [32,2,2] CDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png = 222     CDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png = 0222
\( {\tilde {E}}_{7} \) = [33,3,1] CDel nodes 10r.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png = 331 CDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.png = 133   CDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.png = 0331
\( {\tilde {E}}_{8} \) = [35,2,1] CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png = 251 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png = 152 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png = 521 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png = 0521

There are three hyperbolic (paracompact) Coxeter groups in dimensions 7, 8, and 9:

Coxeter group Honeycombs
\( {\bar {T}}_{7} \) = [33,2,2] CDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png = 322 CDel nodes 10r.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png = 232   CDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png = 0322
\( {\bar {T}}_{8} \) = [34,3,1] CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png = 431 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png = 341 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png = 143 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png = 0431
\( {\bar {T}}_{9} \) = [36,2,1] CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png = 261 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png = 162 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png = 621 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png = 0621

As a generalization more order-3 branches can also be expressed in this symbol. The 4-dimensional affine Coxeter group, \( {\displaystyle {\tilde {Q}}_{4}} , \) [31,1,1,1], has four order-3 branches, and can express one honeycomb, 1111, CDel node 1.pngCDel 3.pngCDel node.pngCDel splitsplit1.pngCDel branch3.pngCDel node.png, represents a lower symmetry form of the 16-cell honeycomb, and 01111, CDel nodes.pngCDel split2.pngCDel node 1.pngCDel split1.pngCDel nodes.png for the rectified 16-cell honeycomb. The 5-dimensional hyperbolic Coxeter group, \( {\displaystyle {\bar {L}}_{4}}, \) [31,1,1,1,1], has five order-3 branches, and can express one honeycomb, 11111, CDel nodes.pngCDel split2.pngCDel node.pngCDel splitsplit1.pngCDel branch3.pngCDel node 1.png and its rectification as 011111, CDel nodes.pngCDel split2.pngCDel node 1.pngCDel splitsplit1.pngCDel branch3.pngCDel node.png.

Notes

Coxeter 1973, p.201
Coxeter, 1973, p. 210 (11.x Historical remarks)
Gosset, 1900
E.L.Elte, 1912

Coxeter 1973, pp.202-204, 11.8 Gosset's figures in six, seven, and eight dimensions.

References

Gosset, Thorold (1900). "On the regular and semi-regular figures in space of n dimensions". Messenger of Mathematics. 29: 43–48.
Elte, E. L. (1912), The Semiregular Polytopes of the Hyperspaces, Groningen: University of Groningen, ISBN 1-4181-7968-X [1] [2]
Coxeter, H.S.M. (3rd edition, 1973) Regular Polytopes, Dover edition, ISBN 0-486-61480-8
Norman Johnson Uniform Polytopes, Manuscript (1991)
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966

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