Gosset–Elte figures

The 4₂₁ polytope of 8-space

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.^[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:^[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 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

Simply-laced ADE groups

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

A finite polytope k_ij 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 [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 [3^0]	= 000
A2 [3^1]	= 010
A3 [3^2]	= 020	= 011
A4 [3^3]	= 030	= 021
A5 [3^4]	= 040	= 031	= 022
A6 [3^5]	= 050	= 041	= 032
A7 [3^6]	= 060	= 051	= 042	= 033
A8 [3^7]	= 070	= 061	= 052	= 043
A9 [3^8]	= 080	= 071	= 062	= 053	= 044
A10 [3^9]	= 090	= 081	= 072	= 063	= 054
...	...

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 [3^1,1,0]	= 110		= 0110
D4 [3^1,1,1]	= 111		= 0111
D5 [3^2,1,1]	= 121	= 211	= 0211
D6 [3^3,1,1]	= 131	= 311	= 0311
D7 [3^4,1,1]	= 141	= 411	= 0411
D8 [3^5,1,1]	= 151	= 511	= 0511
D9 [3^6,1,1]	= 161	= 611	= 0611
D10 [3^7,1,1]	= 171	= 711	= 0711
...	...	...
Dn [3^n−3,1,1]	... = 1n−3,1	... = (n−3)11	... = 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 [3^0,2,1]	= 201	= 120	= 021
E5 [3^1,2,1]	= 211	= 121	= 121	= 0211
E6 [3^2,2,1]	= 221	= 122	= 221	= 0221
E7 [3^3,2,1]	= 231	= 132	= 321	= 0321
E8 [3^4,2,1]	= 241	= 142	= 421	= 0421

Euclidean and hyperbolic honeycombs

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

Coxeter group	Honeycombs
${\tilde {E}}_{6}$ = [3^2,2,2]	= 222			= 0222
${\tilde {E}}_{7}$ = [3^3,3,1]	= 331	= 133		= 0331
${\tilde {E}}_{8}$ = [3^5,2,1]	= 251	= 152	= 521	= 0521

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

Coxeter group	Honeycombs
${\bar {T}}_{7}$ = [3^3,2,2]	= 322	= 232		= 0322
${\bar {T}}_{8}$ = [3^4,3,1]	= 431	= 341	= 143	= 0431
${\bar {T}}_{9}$ = [3^6,2,1]	= 261	= 162	= 621	= 0621

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

Notes

1. Coxeter 1973, p.201
2. Coxeter, 1973, p. 210 (11.x Historical remarks)
3. Gosset, 1900
4. E.L.Elte, 1912
5. 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
• 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