Harold Scott MacDonald Coxeter  

Born  London, England  9 February 1907
Died  31 March 2003 96) Toronto, Ontario, Canada  (aged
Alma mater  University of Cambridge (B.A., 1929; Ph.D., 1931) 
Known for  Coxeter element Coxeter functor Coxeter graph Coxeter group Coxeter matroid Coxeter notation Coxeter's loxodromic sequence of tangent circles Coxeter–Dynkin diagram Coxeter–Todd lattice Boerdijk–Coxeter helix Goldberg–Coxeter construction Todd–Coxeter algorithm* Tutte–Coxeter graph LCF notation 
Spouse(s)  Hendrina, died in 1999 
Children  a daughter, Susan Thomas, and a son, Edgar 
Awards 

Scientific career  
Fields  Geometry 
Institutions  University of Toronto 
Doctoral advisor  H. F. Baker ^{ [1] } 
Doctoral students 
Harold Scott MacDonald "Donald" Coxeter, CC , FRS , FRSC (9 February 1907 – 31 March 2003)^{ [2] } was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.^{ [3] }
Coxeter was born in Kensington to Harold Samuel Coxeter and Lucy ( née Gee). 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.^{ [4] }^{ [2] }
In his youth, Coxeter composed music and was an accomplished pianist at the age of 10.^{ [5] } 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.^{ [5] }
He was educated at King Alfred School, London and St George's School, Harpenden, where his best friend was John Flinders Petrie, later a mathematician for whom Petrie polygons were named. He was accepted at King's College, Cambridge in 1925, but decided to spend a year studying in hopes of gaining admittance to Trinity College, where the standard of mathematics was higher.^{ [2] } Coxeter won an entrance scholarship and went to Trinity College, Cambridge in 1926 to read mathematics. There he earned his BA (as Senior Wrangler) in 1928, and his doctorate in 1931.^{ [5] }^{ [6] } 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.^{ [6] } Returning to Trinity for a year, he attended Ludwig Wittgenstein's seminars on the philosophy of mathematics.^{ [5] } In 1934 he spent a further year at Princeton as a Procter Fellow.^{ [6] }
In 1936 Coxeter moved to the University of Toronto. In 1938 he and P. Du Val, H.T. Flather, and John Flinders Petrie published The FiftyNine Icosahedra with University of Toronto Press. In 1940 Coxeter edited the eleventh edition of Mathematical Recreations and Essays,^{ [7] } originally published by W. W. Rouse Ball in 1892. He was elevated to professor in 1948. Coxeter 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.^{ [6] } Coxeter, M. S. LonguetHiggins and J. C. P. Miller were the first to publish the full list of uniform polyhedra (1954).^{ [8] }
He worked for 60 years at the University of Toronto and published twelve books.
Coxeter was a vegetarian. He attributed his longevity to his vegetarian diet, daily exercise such as fifty pressups and standing on his head for fifteen minutes each morning, and consuming a nightly cocktail made from Kahlua, peach schnapps, and soy milk.^{ [4] }
Since 1978, the Canadian Mathematical Society have 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.^{ [6] } In 1990, he became a Foreign Member of the American Academy of Arts and Sciences ^{ [9] } and in 1997 was made a Companion of the Order of Canada.^{ [10] }
In 1973 he received the Jeffery–Williams Prize.^{ [6] }
A festschrift in his honour, The Geometric Vein, was published in 1982. It contained 41 essays on geometry, based on a symposium for Coxeter held at Toronto in 1979.^{ [11] } A second such volume, The Coxeter Legacy, was published in 2006 based on a Toronto Coxeter symposium held in 2004.^{ [12] }
Sir William Vallance Douglas Hodge was a British mathematician, specifically a geometer.
Oswald Veblen was an American mathematician, geometer and topologist, whose work found application in atomic physics and the theory of relativity. He proved the Jordan curve theorem in 1905; while this was long considered the first rigorous proof of the theorem, many now also consider Camille Jordan's original proof rigorous.
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In geometry, the great dirhombicosidodecahedron (or great snub disicosidisdodecahedron) is a nonconvex uniform polyhedron, indexed last as U_{75}. It has 124 faces (40 triangles, 60 squares, and 24 pentagrams), 240 edges, and 60 vertices.
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Alicia Boole Stott was an Irish mathematician. Despite never holding an academic position, she made a number of valuable contributions to the field, receiving an honorary doctorate from the University of Groningen. She is best known for introducing the term "polytope" for a convex solid in four dimensions, and having an impressive grasp of fourdimensional geometry from a very early age.
In geometry, the Wythoff symbol is a notation representing a Wythoff construction of a uniform polyhedron or plane tiling within a Schwarz triangle. It was first used by Coxeter, LonguetHiggins and Miller in their enumeration of the uniform polyhedra. Later the Coxeter diagram was developed to mark uniform polytopes and honeycombs in ndimensional space within a fundamental simplex.
In geometry, a polytope or a tiling is isotoxal or edgetransitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given two edges, there is a translation, rotation, and/or reflection that will move one edge to the other, while leaving the region occupied by the object unchanged.
In geometry, a fivedimensional polytope is a polytope in fivedimensional space, bounded by (4polytope) facets, pairs of which share a polyhedral cell.
Regular Polytopes is a geometry book on regular polytopes written by Harold Scott MacDonald Coxeter. It was originally published by Methuen in 1947 and by Pitman Publishing in 1948, with a second edition published by Macmillan in 1963 and a third edition by Dover Publications in 1973. The Basic Library List Committee of the Mathematical Association of America has recommended that it be included in undergraduate mathematics libraries.
In geometry, a snub is an operation applied to a polyhedron. The term originates from Kepler's names of two Archimedean solids, for the snub cube and snub dodecahedron. In general, snubs have chiral symmetry with two forms: with clockwise or counterclockwise orientation. By Kepler's names, a snub can be seen as an expansion of a regular polyhedron: moving the faces apart, twisting them about their centers, adding new polygons centered on the original vertices, and adding pairs of triangles fitting between the original edges.
In geometry, a decagram is a 10point star polygon. There is one regular decagram, containing the vertices of a regular decagon, but connected by every third point. Its Schläfli symbol is {10/3}.
In Euclidean geometry, a eutactic star is a geometrical figure in a Euclidean space. A star is a figure consisting of any number of opposing pairs of vectors issuing from a central origin. A star is eutactic if it is the orthogonal projection of plus and minus the set of standard basis vectors from a higherdimensional space onto a subspace. Such stars were called "eutactic" – meaning "wellsituated" or "wellarranged" – by Schläfli because, for a common scalar multiple, their vectors are projections of an orthonormal basis.
In geometry, the small complex icosidodecahedron is a degenerate uniform star polyhedron. Its edges are doubled, making it degenerate. The star has 32 faces, 60 (doubled) edges and 12 vertices and 4 sharing faces. The faces in it are considered as two overlapping edges as topological polyhedron.
In sixdimensional geometry, a sixdimensional polytope or 6polytope is a polytope, bounded by 5polytope facets.
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In geometry, a hendecagrammic prism is a star polyhedron made from two identical regular hendecagrams connected by squares. The related hendecagrammic antiprisms are made from two identical regular hendecagrams connected by equilateral triangles.
Paul Joseph Kelly was an American mathematician who worked in geometry and graph theory.
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