Gosset–Elte figures

File:E8Petrie.svg

The 4₂₁ polytope of 8-space
Template:CDD

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 k_i,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 k_i,j is (k − 1)_i,j, and each of its facets are represented by subtracting one from one of the nonzero subscripts, i.e. k_i − 1,j and k_i,j − 1.

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

History

Coxeter named these figures as k_i,j (or k_ij) in shorthand and gave credit of their discovery to Gosset and Elte:

• Thorold Gosset first published a list of regular and semi-regular figures in space of n dimensions 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. 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 k_ij 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

File:Simply Laced Dynkin Diagrams.svg

Simply-laced ADE groups

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

A finite polytope k_ij exists if

$\frac{1}{i+1}+\frac{1}{j+1}+\frac{1}{k+1}>1$

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

The Coxeter group [3^i,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 k_ij 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 0_ij 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]	Template:CDD = 000 File:1-simplex t0.svg
A2 [31]	Template:CDD = 010 File:2-simplex t0.svg
A3 [32]	Template:CDD = 020 File:3-simplex t0.svg	Template:CDD = 011 File:3-orthoplex.svg
A4 [33]	Template:CDD = 030 File:4-simplex t0.svg	Template:CDD = 021 File:4-simplex t1.svg
A5 [34]	Template:CDD = 040 File:5-simplex t0.svg	Template:CDD = 031 File:5-simplex t1.svg	Template:CDD = 022 File:5-simplex t2.svg
A6 [35]	Template:CDD = 050 File:6-simplex t0.svg	Template:CDD = 041 File:6-simplex t1.svg	Template:CDD = 032 File:6-simplex t2.svg
A7 [36]	Template:CDD = 060 File:7-simplex t0.svg	Template:CDD = 051 File:7-simplex t1.svg	Template:CDD = 042 File:7-simplex t2.svg	Template:CDD = 033 File:7-simplex t3.svg
A8 [37]	Template:CDD = 070 File:8-simplex t0.svg	Template:CDD = 061 File:8-simplex t1.svg	Template:CDD = 052 File:8-simplex t2.svg	Template:CDD = 043 File:8-simplex t3.svg
A9 [38]	Template:CDD = 080 File:9-simplex t0.svg	Template:CDD = 071 File:9-simplex t1.svg	Template:CDD = 062 File:9-simplex t2.svg	Template:CDD = 053 File:9-simplex t3.svg	Template:CDD = 044 File:9-simplex t4.svg
A10 [39]	Template:CDD = 090 File:10-simplex t0.svg	Template:CDD = 081 File:10-simplex t1.svg	Template:CDD = 072 File:10-simplex t2.svg	Template:CDD = 063 File:10-simplex t3.svg	Template:CDD = 054 File:10-simplex t4.svg
...	...

D-family [3^n−3,1,1] demihypercube

Each D_n group has two Gosset–Elte figures, the n-demihypercube as 1_k1, 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 0_k11.

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

E_n family [3^n−4,2,1]

Each E_n group from 4 to 8 has two or three Gosset–Elte figures, represented by one of the end-nodes ringed:k₂₁, 1_k2, 2_k1. A rectified 1_k2 series can also be represented as 0_k21.

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

Euclidean and hyperbolic honeycombs

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

Coxeter group	Honeycombs
${\tilde{E}}_6$ = [32,2,2]	Template:CDD = 222			Template:CDD = 0222
${\tilde{E}}_7$ = [33,3,1]	Template:CDD = 331	Template:CDD = 133		Template:CDD = 0331
${\tilde{E}}_8$ = [35,2,1]	Template:CDD = 251	Template:CDD = 152	Template:CDD = 521	Template:CDD = 0521

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

Coxeter group	Honeycombs
${\overline{T}}_7$ = [33,2,2]	Template:CDD = 322	Template:CDD = 232		Template:CDD = 0322
${\overline{T}}_8$ = [34,3,1]	Template:CDD = 431	Template:CDD = 341	Template:CDD = 143	Template:CDD = 0431
${\overline{T}}_9$ = [36,2,1]	Template:CDD = 261	Template:CDD = 162	Template:CDD = 621	Template:CDD = 0621

As a generalization more order-3 branches can also be expressed in this symbol. The 4-dimensional affine Coxeter group, ${\tilde{Q}}_4$, [3^1,1,1,1], has four order-3 branches, and can express one honeycomb, 1₁₁₁, Template:CDD, represents a lower symmetry form of the 16-cell honeycomb, and 0₁₁₁₁, Template:CDD for the rectified 16-cell honeycomb. The 5-dimensional hyperbolic Coxeter group, ${\overline{L}}_4$, [3^1,1,1,1,1], has five order-3 branches, and can express one honeycomb, 1₁₁₁₁, Template:CDD and its rectification as 0₁₁₁₁₁, Template:CDD.

References

• [1] [2]
• Coxeter, H.S.M. (3rd edition, 1973) Regular Polytopes, Dover edition, Template:ISBN
• Norman Johnson Uniform Polytopes, Manuscript (1991)
  • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966