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.
History
Coxeter named these figures as k_{i,j} (or k_{ij}) in shorthand and gave credit of their discovery to Gosset and Elte:[2]
- Thorold Gosset first published a list of regular and semi-regular figures in space of n dimensions[3] 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.[4] 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
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 |
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A_{1} [3^{0}] |
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A_{2} [3^{1}] |
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A_{3} [3^{2}] |
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A_{4} [3^{3}] |
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A_{5} [3^{4}] |
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A_{6} [3^{5}] |
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A_{7} [3^{6}] |
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A_{8} [3^{7}] |
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A_{9} [3^{8}] |
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A_{10} [3^{9}] |
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... | ... |
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 |
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D_{3} [3^{1,1,0}] |
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D_{4} [3^{1,1,1}] |
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D_{5} [3^{2,1,1}] |
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D_{6} [3^{3,1,1}] |
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D_{7} [3^{4,1,1}] |
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D_{8} [3^{5,1,1}] |
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D_{9} [3^{6,1,1}] |
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D_{10} [3^{7,1,1}] |
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... | ... | ... | |
D_{n} [3^{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} | |
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E_{4} [3^{0,2,1}] |
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E_{5} [3^{1,2,1}] |
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E_{6} [3^{2,2,1}] |
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E_{7} [3^{3,2,1}] |
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E_{8} [3^{4,2,1}] |
Euclidean and hyperbolic honeycombs
There are three Euclidean (affine) Coxeter groups in dimensions 6, 7, and 8:[5]
Coxeter group | Honeycombs | |||
---|---|---|---|---|
= [3^{2,2,2}] | ||||
= [3^{3,3,1}] | ||||
= [3^{5,2,1}] |
There are three hyperbolic (paracompact) Coxeter groups in dimensions 7, 8, and 9:
Coxeter group | Honeycombs | |||
---|---|---|---|---|
= [3^{3,2,2}] | ||||
= [3^{4,3,1}] | ||||
= [3^{6,2,1}] |
As a generalization more order-3 branches can also be expressed in this symbol. The 4-dimensional affine Coxeter group, , [3^{1,1,1,1}], has four order-3 branches, and can express one honeycomb, 1_{111},
Notes
- Coxeter 1973, p.201
- Coxeter, 1973, p. 210 (11.x Historical remarks)
- Gosset, 1900
- E.L.Elte, 1912
- Coxeter 1973, pp.202-204, 11.8 Gosset's figures in six, seven, and eight dimensions.
References
- Gosset, Thorold (1900). "On the regular and semi-regular figures in space of n dimensions". Messenger of Mathematics. 29: 43–48.
- Elte, E. L. (1912), The Semiregular Polytopes of the Hyperspaces, Groningen: University of Groningen, ISBN 1-4181-7968-X
- Coxeter, H.S.M. (3rd edition, 1973) Regular Polytopes, Dover edition, ISBN 0-486-61480-8
- Norman Johnson Uniform Polytopes, Manuscript (1991)
- N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966