Harold Scott MacDonald Coxeter

Harold Scott MacDonald "Donald" Coxeter (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician. He is regarded as one of the greatest geometers of the 20th century.

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, from 1936 until his retirement in 1996, becoming a full professor there in 1948. His many honours included membership in the Royal Society of Canada, the Royal Society, and the Order of Canada.

He was an author of 12 books, including The Fifty-Nine Icosahedra (1938) and Regular Polytopes (1947). Many concepts in geometry and group theory are named after him, including the Coxeter graph, Coxeter groups, Coxeter's loxodromic sequence of tangent circles, Coxeter–Dynkin diagrams, and the Todd–Coxeter algorithm.

Biography Coxeter was born in Kensington, England, to Harold Samuel Coxeter and Lucy . His father had taken over the family business of Coxeter & Son, manufacturers of surgical instruments and compressed gases (including a mechanism for anaesthetising surgical patients with nitrous oxide), but was able to retire early and focus on sculpting and baritone singing; Lucy Coxeter was a portrait and landscape painter who had attended the Royal Academy of Arts. A maternal cousin was the architect Sir Giles Gilbert Scott. He felt that mathematics and music were intimately related, outlining his ideas in a 1962 article on "Music and Mathematics" in the Canadian Music Journal. In 1932 he went to Princeton University for a year as a Rockefeller Fellow, where he worked with Hermann Weyl, Oswald Veblen, and Solomon Lefschetz. In 1934 he spent a further year at Princeton as a Procter Fellow. originally published by W. W. Rouse Ball in 1892. He was elevated to professor in 1948. He was elected a Fellow of the Royal Society of Canada in 1948 and a Fellow of the Royal Society in 1950. He met M. C. Escher in 1954 and the two became lifelong friends; his work on geometric figures helped inspire some of Escher's works, particularly the Circle Limit series based on hyperbolic tessellations. He also inspired some of the innovations of Buckminster Fuller.

Coxeter classified the finite Coxeter groups in 1935. These groups were associated with diagrams consisting of dots and segment lines connecting them. These diagrams are known as the Dynkin diagram, named for Soviet–American mathematician Eugene Dynkin, describing the families of finite-dimensional simple Lie algebras of root systems , , , , , , , , , ,. The amalgamation is now known as Coxeter–Dynkin diagram or Coxeter graph. Coxeter denotes these groups and their diagram structures in bracket notations, named as Coxeter notation or Coxeter symbol.

Polyhedra and polytopes

The book The Fifty-Nine Icosahedra by Coxeter, Patrick du Val, H. T. Flather, and John Flinders Petrie, published by the University of Toronto in 1938, describes the 59 stellations of a regular icosahedron, made by extending its faces to form a star-shaped solid.

Coxeter's Regular Polytopes was published in numerous editions: by Methuen in 1947, by Pitman Publishing in 1948, the second edition by Macmillan in 1963, and the third edition by Dover Publications in 1973.

English mathematician John Flinders Petrie, in 1926, took the concept of regular skew polygons—where the vertices are not in the same plane—to three-dimensional polyhedra. He found two regular skew apeirohedra, constructed of cubes and truncated octahedra by removing the shared square faces of adjacent polyhedra to form a tunnels. Coxeter discovered the third regular skew polyhedron, constructed of multiple truncated tetrahedra under Petrie's procedure. Respectively, these polyhedra were later named by English mathematician John Conway as mucube, muoctahedron, and mutetrahedron.

Michael Goldberg introduced the Goldberg polyhedron family, whose faces are pentagons and hexagons. The dual of such a polyhedron is geodesic polyhedron, described as an almost spherical shape with triangular faces, which Buckminster Fuller called "geodesic domes". These polyhedra were extended and applied in Donald Caspar and Aaron Klug's article on the geometry of viral capsids. Coxeter covered much of the information without referring to biological explanations. Caspar and Klug were the first to publish the most general correct construction of a geodesic polyhedron, making the name "Goldberg–Coxeter construction" an instance of Stigler's law of eponymy.

Spirals and helices

Coxeter studied the infinite circles tangent to each other arranged loxodromically, so that any four consecutive circles in a sequence are pairwise mutually tangent. This was later generalized to the case of spheres and hyperspheres in higher-dimensional space.

Coxeter and constructed an assemblage of stacked regular tetrahedra linearly, such that the edges form three helices, thereby what is nowadays called Boerdijk–Coxeter helix.

Graph theory

Coxeter in 1983's paper My Graph showed a 3-regular graph with 28 vertices and 42 edges. A graph with 30 vertices and 45 edges was discovered by William Thomas Tutte, and the study of its connection to geometric configurations was investigated by Tutte and Coxeter, now known as Tutte–Coxeter graph.

Coxeter, along with Robert Frucht, extended LCF notation devised by Joshua Lederberg for the representation of cubic graphs that contain a Hamiltonian cycle.

Awards Since 1978, the Canadian Mathematical Society has awarded the Coxeter–James Prize in his honor.

He was made a Fellow of the Royal Society in 1950 and in 1997 he was awarded their Sylvester Medal. and in 1997 was made a Companion of the Order of Canada.

In 1973 he received the Jeffery–Williams Prize. A second such volume, The Coxeter Legacy, was published in 2006 based on a Toronto Coxeter symposium held in 2004.

Works Coxeter published twelve books, and over 200 articles:

Books

Selected papers