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.
Coxeter named these figures as ki,j (or kij) in shorthand and gave credit of their discovery to Gosset and Elte:[2]
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.
The polytopes and honeycombs in this family can be seen within ADE classification.
A finite polytope kij exists if
| 1 | + | |
| i+1 |
| 1 | + | |
| j+1 |
| 1 | |
| k+1 |
>1
The Coxeter group [3<sup>i,j,k</sup>] 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.
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.
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.
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.
There are three Euclidean (affine) Coxeter groups in dimensions 6, 7, and 8:[5]
| Coxeter group | Honeycombs | ||||
|---|---|---|---|---|---|
{\tilde{E}}6 | = 222 | = 0222 | |||
{\tilde{E}}7 | = 331 | = 133 | = 0331 | ||
{\tilde{E}}8 | = 251 | = 152 | = 521 | = 0521 | |
There are three hyperbolic (paracompact) Coxeter groups in dimensions 7, 8, and 9:
| Coxeter group | Honeycombs | ||||
|---|---|---|---|---|---|
{\bar{T}}7 | = 322 | = 232 | = 0322 | ||
{\bar{T}}8 | = 431 | = 341 | = 143 | = 0431 | |
{\bar{T}}9 | = 261 | = 162 | = 621 | = 0621 | |
As a generalization more order-3 branches can also be expressed in this symbol. The 4-dimensional affine Coxeter group,
{\tilde{Q}}4
{\bar{L}}4