Coxeter symbol

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.

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

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.

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

A finite polytope k_ij exists if

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.

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