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Harold Scott MacDonald "Donald" Coxeter CC FRS FRSC (9 February 1907 – 31 March 2003) [2] was a British-Canadian geometer and mathematician. He is regarded as one of the greatest geometers of the 20th century. [3]Coxeter was born in England and educated at the University of Cambridge, with student visits to Princeton University.He worked for 60 years at the University of Toronto in Canada ...

In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; for example, the symmetry group of each regular polyhedron is a finite Coxeter group.

H.S.M. Coxeter was a British-born Canadian geometer, who was a leader in the understanding of non-Euclidean geometries, reflection patterns, and polytopes (higher-dimensional analogs of three-dimensional polyhedra). Coxeter’s work served as an inspiration for R. Buckminster Fuller’s concept of the

A Coxeter group is called indecomposable if it is not a direct product of two non-trivial standard subgroups; this is equivalent to the connectedness of its Coxeter graph. All finite (or affine) Coxeter groups are direct products of indecomposable Coxeter groups of the same type; all hyperbolic Coxeter groups are indecomposable.

Coxeter–Dynkin diagrams for the fundamental finite Coxeter groups Coxeter–Dynkin diagrams for the fundamental affine Coxeter groups. In geometry, a Coxeter–Dynkin diagram (or Coxeter diagram, Coxeter graph) is a graph with numerically labeled edges (called branches) representing a Coxeter group or sometimes a uniform polytope or uniform tiling constructed from the group.

Coxeter 群是一类群, 它由一些反射生成, 各个反射之间满足一定的关系 (定义 1.1).. Coxeter 群可以用来描述高度对称的几何体 (例如正多胞体) 的对称性 (命题 2.3), 也可以作为 Weyl 群的抽象与推广, 来研究半单 Lie 代数 (更一般的, Kac–Moody 代数) 的性质.

Donald Coxeter is the greatest living classical geometer. His work has had signiflcant impact in the worlds of chemistry, physics, com-puter programming and medical research. Buckminster Fuller’s iconic geodesic dome design was in°uenced by Coxeter, and M.C. Escher re-lied heavily on Coxeter’s theories for his famous Circle Limit drawings.

Learn about the life and achievements of Donald Coxeter, a prominent geometer who made contributions to polytopes, non-Euclidean geometry, group theory and combinatorics. Find out how he overcame a difficult childhood, studied with Bertrand Russell and Alan Robson, and became a professor at Toronto.

H.S.M. Coxeter was born and educated in England, but his professional connections with North America began early. Shortly after finishing his doctoral studies at Cambridge University, and while he was a research fellow there, he spent two years as a research visitor at Princeton University. In 1936 he joined the Faculty of the University of ...

Coxeter used the notation [3^(p,q,r)] for the Coxeter group generated by the nodes of a Y-shaped Coxeter-Dynkin diagram whose three arms have p, q, and r graph edges. A Coxeter group of this form is finite iff 1/(p+1)+1/(q+1)+1/(r+1)>1. TOPICS. Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics Foundations of ...

The Coxeter groups are those that arise as part of a Coxeter system. The cardinality of \(S\) is called the rank of the Coxeter system. Exercises [fg12] Up to isomorphism, the only Coxeter system of rank \(1\) is \((C_{2},\{s\})\). [fg13] The Coxeter systems of rank \(2\) are indexed by \(m(s,t)\geq2\).

To summerize: Coxeter diagram and Coxeter matrix are a tool to encode the presentation of the Coxeter group. Each Coxeter group has such a special representation. Each Coxeter group can be realized geometrically as a group generated by reflection of "something".

In geometry, Coxeter notation (also Coxeter symbol) is a system of classifying symmetry groups, describing the angles between fundamental reflections of a Coxeter group in a bracketed notation expressing the structure of a Coxeter-Dynkin diagram, with modifiers to indicate certain subgroups.The notation is named after H. S. M. Coxeter, and has been more comprehensively defined by Norman Johnson.

Coxeter, H. S. M. (Harold Scott Macdonald), 1907-2003. Publication date 1969 Topics Geometry Publisher New York, Wiley Collection internetarchivebooks; printdisabled Contributor Internet Archive Language English Item Size 875.7M . xvi, 469 p. 24 cm Bibliography: p. 415-417

3. Finite Coxeter groups Definition 3.1 (Coxeter groups). Lemma 3.2 (Reflection ⇒ Coxeter). Every finite reflection group is a finite Coxeter group. In particular, if W is a finite reflection group and S is a set of simple reflections in W, then the Coxeter presentation generated by S whose relations

Boerdijk-Coxeter Helix (with A.H. Boerdijk) Goldberg-Coxeter Construction; Tutte-Coxeter Graph (with William Thomas Tutte) LCF Notation (with Joshua Lederberg and Robert Wertheimer Frucht) Coxeter's Loxodromic Sequence of Tangent Circles; Todd-Coxeter Algorithm (with John Arthur Todd) Results named for Harold Scott MacDonald Coxeter can be ...

In particular I am intersted in the study of the combinatorics of Coxeter groups, hyperplane arrangements, oriented matroids, root systems, and generalized associahedra and permutahedra. Publications. Conjugacy Class Growth in Virtually Abelian Groups, with Alex Evetts in Groups, Complexity, Cryptology 17 (1) (2025) - arXiv link;

3D model of a tetrahemihexahedron. In geometry, the tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U 4.It has 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices. [1] Its vertex figure is a crossed quadrilateral.Its Coxeter–Dynkin diagram is (although this is a double covering of the tetrahemihexahedron).. The tetrahemihexahedron is the only non ...

Algebraic linearization of dynamics of Calogero type for any Coxeter group. J. Math. Phys. (July 2000) Spectroscopic evidence for high symmetry in (benzene) 13. J. Chem. Phys. (March 1991) Thermal Decomposition of Gallium Arsenide. J. Appl. Phys. (January 1965) Online ISSN 1089-7550; Print ISSN 0021-8979; Resources.

A zero-dimensional Hausdorff space is necessarily totally disconnected, but the converse fails.However, a locally compact Hausdorff space is zero-dimensional if and only if it is totally disconnected. (See (Arhangel'skii & Tkachenko 2008, Proposition 3.1.7, p.136) for the non-trivial direction.)Zero-dimensional Polish spaces are a particularly convenient setting for descriptive set theory.