Harold Scott MacDonald Coxeter
Encyclopedia
Harold Scott MacDonald "Donald" Coxeter, (February 9, 1907 – March 31, 2003) was a British-born Canadian geometer. Coxeter is regarded as one of the great geometers of the 20th century. He was born in London
but spent most of his life in Canada
.
He worked for 60 years at the University of Toronto
and published twelve books. He was most noted for his work on regular polytope
s and higher-dimensional geometries. He was a champion of the classical approach to geometry, in a period when the tendency was to approach geometry more and more via algebra.
Coxeter went up to Trinity College
, Cambridge
in 1926 to read mathematics. There he earned his BA in 1928, and his doctorate in 1931. 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
. Returning to Trinity for a year, he attended Ludwig Wittgenstein
's seminars on the philosophy of mathematics
. In 1934 he spent a further year at Princeton as a Procter Fellow.
In 1936 Coxeter moved to the University of Toronto, becoming a 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 Maurits Escher
and his work on geometric figures helped inspire some of Escher's works, particularly the Circle Limit series based on hyperbolic
tessellation
s. He also inspired some of the innovations of Buckminster Fuller
.
Coxeter, M. S. Longuet-Higgins and J. C. P. Miller
were the first to publish the full list of uniform polyhedra (1954).
Since 1978, the Canadian Mathematical Society
have awarded the Coxeter–James Prize
in his honor.
In 1990, he became a Foreign Member of the American Academy of Arts and Sciences
.
In 1997 he received Sylvester Medal
from the Royal Society
and was made a Companion of the Order of Canada
.
London
London is the capital city of :England and the :United Kingdom, the largest metropolitan area in the United Kingdom, and the largest urban zone in the European Union by most measures. Located on the River Thames, London has been a major settlement for two millennia, its history going back to its...
but spent most of his life in Canada
Canada
Canada is a North American country consisting of ten provinces and three territories. Located in the northern part of the continent, it extends from the Atlantic Ocean in the east to the Pacific Ocean in the west, and northward into the Arctic Ocean...
.
Biography
In his youth, Coxeter composed music and was an accomplished pianist at the age of 10. He felt that mathematics and music were intimately related, outlining his ideas in a 1962 article on "Mathematics and Music" in the Canadian Music Journal.He worked for 60 years at the University of Toronto
University of Toronto
The University of Toronto is a public research university in Toronto, Ontario, Canada, situated on the grounds that surround Queen's Park. It was founded by royal charter in 1827 as King's College, the first institution of higher learning in Upper Canada...
and published twelve books. He was most noted for his work on regular polytope
Regular polytope
In mathematics, a regular polytope is a polytope whose symmetry is transitive on its flags, thus giving it the highest degree of symmetry. All its elements or j-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of...
s and higher-dimensional geometries. He was a champion of the classical approach to geometry, in a period when the tendency was to approach geometry more and more via algebra.
Coxeter went up to Trinity College
Trinity College, Cambridge
Trinity College is a constituent college of the University of Cambridge. Trinity has more members than any other college in Cambridge or Oxford, with around 700 undergraduates, 430 graduates, and over 170 Fellows...
, Cambridge
University of Cambridge
The University of Cambridge is a public research university located in Cambridge, United Kingdom. It is the second-oldest university in both the United Kingdom and the English-speaking world , and the seventh-oldest globally...
in 1926 to read mathematics. There he earned his BA in 1928, and his doctorate in 1931. In 1932 he went to Princeton University
Princeton University
Princeton University is a private research university located in Princeton, New Jersey, United States. The school is one of the eight universities of the Ivy League, and is one of the nine Colonial Colleges founded before the American Revolution....
for a year as a Rockefeller Fellow, where he worked with Hermann Weyl
Hermann Weyl
Hermann Klaus Hugo Weyl was a German mathematician and theoretical physicist. Although much of his working life was spent in Zürich, Switzerland and then Princeton, he is associated with the University of Göttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski.His...
, Oswald Veblen
Oswald Veblen
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.-Life:...
, and Solomon Lefschetz
Solomon Lefschetz
Solomon Lefschetz was an American mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear ordinary differential equations.-Life:...
. Returning to Trinity for a year, he attended Ludwig Wittgenstein
Ludwig Wittgenstein
Ludwig Josef Johann Wittgenstein was an Austrian philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy of language. He was professor in philosophy at the University of Cambridge from 1939 until 1947...
's seminars on the philosophy of mathematics
Philosophy of mathematics
The philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. The aim of the philosophy of mathematics is to provide an account of the nature and methodology of mathematics and to understand the place of...
. In 1934 he spent a further year at Princeton as a Procter Fellow.
In 1936 Coxeter moved to the University of Toronto, becoming a professor
Professor
A professor is a scholarly teacher; the precise meaning of the term varies by country. Literally, professor derives from Latin as a "person who professes" being usually an expert in arts or sciences; a teacher of high rank...
in 1948. He was elected a Fellow of the Royal Society of Canada
Royal Society of Canada
The Royal Society of Canada , may also operate under the more descriptive name RSC: The Academies of Arts, Humanities and Sciences of Canada , is the oldest association of scientists and scholars in Canada...
in 1948 and a Fellow of the Royal Society in 1950. He met Maurits Escher
M. C. Escher
Maurits Cornelis Escher , usually referred to as M. C. Escher , was a Dutch graphic artist. He is known for his often mathematically inspired woodcuts, lithographs, and mezzotints...
and his work on geometric figures helped inspire some of Escher's works, particularly the Circle Limit series based on hyperbolic
Hyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...
tessellation
Tessellation
A tessellation or tiling of the plane is a pattern of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of parts of the plane or of other surfaces. Generalizations to higher dimensions are also possible. Tessellations frequently appeared in the art...
s. He also inspired some of the innovations of Buckminster Fuller
Buckminster Fuller
Richard Buckminster “Bucky” Fuller was an American systems theorist, author, designer, inventor, futurist and second president of Mensa International, the high IQ society....
.
Coxeter, M. S. Longuet-Higgins and J. C. P. Miller
J. C. P. Miller
Jeffrey Charles Percy Miller was an English mathematician and computing pioneer. He worked in number theory and on geometry, particularly polyhedra, where Miller's monster refers to the great dirhombicosidodecahedron....
were the first to publish the full list of uniform polyhedra (1954).
Since 1978, the Canadian Mathematical Society
Canadian Mathematical Society
The Canadian Mathematical Society is an association of professional mathematicians dedicated to the interests of mathematical research and education in Canada.It was originally conceived in June 1945 as the Canadian Mathematical Congress...
have awarded the Coxeter–James Prize
Coxeter–James Prize
The Coxeter–James Prize is presented annually by the Canadian Mathematical Society. The award is presented to young mathematicians in recognition of outstanding contributions to mathematical research. The first award was presented in 1978. The prize was named in honor of the mathematicians Donald...
in his honor.
In 1990, he became a Foreign Member of the American Academy of Arts and Sciences
American Academy of Arts and Sciences
The American Academy of Arts and Sciences is an independent policy research center that conducts multidisciplinary studies of complex and emerging problems. The Academy’s elected members are leaders in the academic disciplines, the arts, business, and public affairs.James Bowdoin, John Adams, and...
.
In 1997 he received Sylvester Medal
Sylvester Medal
The Sylvester Medal is a bronze medal awarded by the Royal Society for the encouragement of mathematical research, and accompanied by a £1,000 prize...
from the Royal Society
Royal Society
The Royal Society of London for Improving Natural Knowledge, known simply as the Royal Society, is a learned society for science, and is possibly the oldest such society in existence. Founded in November 1660, it was granted a Royal Charter by King Charles II as the "Royal Society of London"...
and was made a Companion of the Order of Canada
Order of Canada
The Order of Canada is a Canadian national order, admission into which is, within the system of orders, decorations, and medals of Canada, the second highest honour for merit...
.
Works
- Coxeter, Longuet-Higgins, Miller, Uniform polyhedra, Phil. Trans. 1954, 246 A, 401–450.
- The Real Projective Plane (1949)
- Introduction to Geometry (1961)
- Regular Polytopes (1963), Macmillian Company
- Regular PolytopesRegular Polytopes (book)Regular Polytopes is a mathematical geometry book written by Canadian mathematician H.S.M. Coxeter. Originally written in 1947, the book was updated and republished in 1963 and 1973....
, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8
- Regular Polytopes
- Non-Euclidean Geometry (1965)
- Geometry Revisited (with S. L. GreitzerS. L. GreitzerSamuel L. Greitzer was an American mathematician, the founding chairman of the United States of America Mathematical Olympiad, and the publisher of the precollege mathematics journal Arbelos. He also wrote the textbook Geometry Revisited together with H. S. M...
, 1967) - Projective Geometry (2nd edition, 1974)
- Regular Complex Polytopes (1974), Cambridge University PressCambridge University PressCambridge University Press is the publishing business of the University of Cambridge. Granted letters patent by Henry VIII in 1534, it is the world's oldest publishing house, and the second largest university press in the world...
- The Beauty of Geometry: Twelve Essays (1999), Dover Publications, , ISBN 0-486-40919-8
- The Fifty-Nine IcosahedraThe fifty nine icosahedraThe Fifty Nine Icosahedra is a book written and illustrated by H. S. M. Coxeter, P. Du Val, H. T. Flather and J. F. Petrie. It enumerates the stellations of the regular convex or Platonic icosahedron, according to a set of rules put forward by J. C. P. Miller...
(with P. Du Val, H. T. Flather, J. F. Petrie) - Mathematical Recreations and Essays (with W. W. Rouse BallW. W. Rouse Ball-External links:*...
) - Twisted honeycombs (American Mathematical Society, 1970, Regional conference series in mathematics Number 4, ISBN 0-8218-1653-5)
- F. Arthur Sherk, Peter McMullen, Anthony C. Thompson and Asia Ivić Weiss, editors: Kaleidoscopes — Selected Writings of H.S.M. Coxeter. John Wiley, 1995, ISBN 0-471-01003-0
See also
- Boerdijk–Coxeter helixBoerdijk–Coxeter helixThe Boerdijk–Coxeter helix, named after H. S. M. Coxeter and A. H. Boerdijk, is a linear stacking of regular tetrahedra. There are two chiral forms, with either clockwise or counterclockwise windings. Contrary to any other stacking of Platonic solids, the Boerdijk–Coxeter helix is not rotationally...
- Coxeter functor
- Coxeter groupCoxeter groupIn mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...
- Coxeter numberCoxeter numberIn mathematics, the Coxeter number h is the order of a Coxeter element of an irreducible Coxeter group, hence also of a root system or its Weyl group. It is named after H.S.M. Coxeter.-Definitions:...
- Coxeter–Dynkin diagram
- Coxeter–James PrizeCoxeter–James PrizeThe Coxeter–James Prize is presented annually by the Canadian Mathematical Society. The award is presented to young mathematicians in recognition of outstanding contributions to mathematical research. The first award was presented in 1978. The prize was named in honor of the mathematicians Donald...
- Coxeter–Todd latticeCoxeter–Todd latticeIn mathematics, the Coxeter–Todd lattice K12, discovered by , is a the 12-dimensional even integral lattice of discriminant 36 with no norm-2 vectors...
- Coxeter's loxodromic sequence of tangent circlesCoxeter's loxodromic sequence of tangent circlesIn geometry, Coxeter's loxodromic sequence of tangent circles is an infinite sequence of circles arranged such that any four consecutive circles in the sequence are pairwise mutually tangent...
- LCF notationLCF notationIn combinatorial mathematics, LCF notation or LCF code is a notation devised by Joshua Lederberg, and extended by Coxeter and Frucht, for the representation of cubic graphs that are Hamiltonian. Since the graphs are Hamiltonian, the vertices can be arranged in a cycle, which accounts for two edges...
- Todd–Coxeter algorithm
- Tutte–Coxeter graph
External links
- H. S. M. Coxeter (1907–2003), Erich W. Ellers, Branko Grünbaum, Peter McMullen, Asia Ivic Weiss Notices of the AMS: Volume 50, Number 10.
- "The man who saved geometry: Crying `Death to Triangles!' a generation of mathematicians tried to eliminate geometry in favor of algebra. Were it not for Donald Coxeter, they might have succeeded", by Siobhan Roberts, The Boston GlobeThe Boston GlobeThe Boston Globe is an American daily newspaper based in Boston, Massachusetts. The Boston Globe has been owned by The New York Times Company since 1993...
, September 10, 2006. - Jaron's World: Shapes in Other Dimensions, Discover mag.Discover (magazine)Discover is an American science magazine that publishes articles about science for a general audience. The monthly magazine was launched in October 1980 by Time Inc. It was sold to Family Media, the owners of Health, in 1987. Walt Disney Company bought the magazine when Family Media went out of...
, Apr 2007 - The Mathematics in the Art of M.C. Escher video of a lecture by H.S.M. Coxeter, April 28, 2000.
- H.S.MACDONALD COXETER- Pakistan Mathematical Society