Gosset–Elte figures
# 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 $k_{i,j−1}.$

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 $0_{21}$ in 4-space, demipenteract $1_{21}$ in 5-space, $2_{21}$ in 6-space, $3_{21}$ in 7-space, $4_{21}$ in 8-space, and $5_{21}$ 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 $1_{42}$ 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.

## Definition

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 $[3^n]$ (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 |
| - | - | - | - | - | - |
| $A_1$ $[3^0]$ | = $0_{00}$ | | | | |
| $A_2$ $[3^1]$ | = $0_{10}$ | | | | |
| $A_3$ $[3^2]$ | = $0_{20}$ | = $0_{11}$ | | | |
| $A_4$ $[3^3]$ | = $0_{30}$ | = $0_{21}$ | | | |
| $A_5$ $[3^4]$ | = $0_{40}$ | = $0_{31}$ | = $0_{22}$ | | |
| $A_6$ $[3^5]$ | = $0_{50}$ | = $0_{41}$ | = $0_{32}$ | | |
| $A_7$ $[3^6]$ | = $0_{60}$ | = $0_{51}$ | = $0_{42}$ | = $0_{33}$ | |
| $A_8$ $[3^7]$ | = $0_{70}$ | = $0_{61}$ | = $0_{52}$ | = $0_{43}$ | |
| $A_9$ $[3^8]$ | = $0_{80}$ | = $0_{71}$ | = $0_{62}$ | = $0_{53}$ | = $0_{44}$ |
| $A_{10}$ $[3^9]$ | = $0_{90}$ | = $0_{81}$ | = $0_{72}$ | = $0_{63}$ | = $0_{54}$ |

## 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, $k_{11},$ 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 |
| - | - | - | - |
| $D_3$ $[3^{1,1,0}]$ | = $1_{10}$ | | = $0_{110}$ |
| $D_4$ $[3^{1,1,1}]$ | = $1_{11}$ | | = $0_{111}$ |
| $D_5$ $[3^{2,1,1}]$ | = $1_{21}$ | = $2_{11}$ | = $0_{211}$ |
| $D_6$ $[3^{3,1,1}]$ | = $1_{31}$ | = $3_{11}$ | = $0_{311}$ |
| $D_7$ $[3^{4,1,1}]$ | = $1_{41}$ | = $4_{11}$ | = $0_{411}$ |
| $D_8$ $[3^{5,1,1}]$ | = $1_{51}$ | = $5_{11}$ | = $0_{511}$ |
| $D_9$ $[3^{6,1,1}]$ | = $1_{61}$ | = $6_{11}$ | = $0_{611}$ |
| $D_{10}$ $[3^{7,1,1}]$ | = $1_{71}$ | = $7_{11}$ | = $0_{711}$ |
| $D_n$ $[3^{n−3,1,1}]$ | ... = $1_{n−3,1}$ | ... = $(n−3)_{11}$ | ... = $0_{n−3,1,1}$ |

## $E_n$ 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 $ringed:k_{21},$ $1_{k2},$ $2_{k1}.$ A rectified $1_{k2}$ series can also be represented as $0_{k21}.$

| | $2_{k1}$ | $1_{k2}$ | $k_{21}$ | $0_{k21}$ |
| - | - | - | - | - |
| $E_4$ $[3^{0,2,1}]$ | = $2_{01}$ | = $1_{20}$ | = $0_{21}$ | |
| $E_5$ $[3^{1,2,1}]$ | = $2_{11}$ | = $1_{21}$ | = $1_{21}$ | = $0_{211}$ |
| $E_6$ $[3^{2,2,1}]$ | = $2_{21}$ | = $1_{22}$ | = $2_{21}$ | = $0_{221}$ |
| $E_7$ $[3^{3,2,1}]$ | = $2_{31}$ | = $1_{32}$ | = $3_{21}$ | = $0_{321}$ |
| $E_8$ $[3^{4,2,1}]$ | = $2_{41}$ | = $1_{42}$ | = $4_{21}$ | = $0_{421}$ |

## Euclidean and hyperbolic honeycombs

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

| Coxeter group | Honeycombs | Honeycombs | Honeycombs | Honeycombs |
| - | - | - | - | - |
| ${\tilde {E}}_{6}$ = $[3^{2,2,2}]$ | = $2_{22}$ | | | = $0_{222}$ |
| ${\tilde {E}}_{7}$ = $[3^{3,3,1}]$ | = $3_{31}$ | = $1_{33}$ | | = $0_{331}$ |
| ${\tilde {E}}_{8}$ = $[3^{5,2,1}]$ | = $2_{51}$ | = $1_{52}$ | = $5_{21}$ | = $0_{521}$ |

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

| Coxeter group | Honeycombs | Honeycombs | Honeycombs | Honeycombs |
| - | - | - | - | - |
| ${\bar {T}}_{7}$ = $[3^{3,2,2}]$ | = $3_{22}$ | = $2_{32}$ | | = $0_{322}$ |
| ${\bar {T}}_{8}$ = $[3^{4,3,1}]$ | = $4_{31}$ | = $3_{41}$ | = $1_{43}$ | = $0_{431}$ |
| ${\bar {T}}_{9}$ = $[3^{6,2,1}]$ | = $2_{61}$ | = $1_{62}$ | = $6_{21}$ | = $0_{621}$ |

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, $1_{111},$ , represents a lower symmetry form of the 16-cell honeycomb, and $0_{1111},$ for the rectified 16-cell honeycomb. The 5-dimensional hyperbolic Coxeter group, ${\bar {L}}_{4}$, $[3^{1,1,1,1,1}],$ has five order-3 branches, and can express one honeycomb, $1_{1111},$ and its rectification as $0_{11111},$ .

