# Gosset–Elte figures

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.

Coxeter named these figures as *k*_{i,j} (or *k*_{ij}) in shorthand and gave credit of their discovery to Gosset and Elte:^{[2]}

Elte's enumeration included all the *k*_{ij} polytopes except for the *1*_{42} 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 *5*_{21} honeycomb as the only semiregular one in his definition.

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

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**to mean the end-node on the

_{ij}*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 **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.

Each D_{n} group has two Gosset–Elte figures, the *n*-demihypercube as **1 _{k1}**, and an alternated form of the

*n*-orthoplex,

**k**, constructed with alternating simplex facets. Rectified

_{11}*n*-demihypercubes, a lower symmetry form of a birectified

*n*-cube, can also be represented as

**0**.

_{k11}Each E_{n} group from 4 to 8 has two or three Gosset–Elte figures, represented by one of the end-nodes ringed:**k _{21}**,

**1**,

_{k2}**2**. A rectified

_{k1}**1**series can also be represented as

_{k2}**0**.

_{k21}There are three Euclidean (affine) Coxeter groups in dimensions 6, 7, and 8:^{[5]}

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