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.
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.
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.
or equal for Euclidean honeycombs, and less for hyperbolic honeycombs.
The Coxeter group [3i,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 kij to mean the end-node on the k-length sequence is ringed.
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.