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 The University of Cambridge is a Public university, public collegiate university, collegiate research university in Cambridge, England. Founded in 1209, the University of Cambridge is the List of oldest universities in continuous operation, wo ...

, with student visits to

Princeton University Princeton University is a private university, private Ivy League research university in Princeton, New Jersey, United States. Founded in 1746 in Elizabeth, New Jersey, Elizabeth as the College of New Jersey, Princeton is the List of Colonial ...

. He worked for 60 years at the

University of Toronto The University of Toronto (UToronto or U of T) is a public university, public research university whose main campus is located on the grounds that surround Queen's Park (Toronto), Queen's Park in Toronto, Ontario, Canada. It was founded by ...

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 of Canada (RSC; , SRC), also known as the Academies of Arts, Humanities, and Sciences of Canada (French: ''Académies des arts, des lettres et des sciences du Canada''), is the senior national, bilingual council of distinguishe ...

, the

Royal Society The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a learned society and the United Kingdom's national academy of sciences. The society fulfils a number of roles: promoting science and its benefits, re ...

, and the

Order of Canada The Order of Canada () is a Canadian state order, national order and the second-highest Award, honour for merit in the system of orders, decorations, and medals of Canada, after the Order of Merit. To coincide with the Canadian Centennial, ce ...

. He was an author of 12 books, including ''

The Fifty-Nine Icosahedra ''The Fifty-Nine Icosahedra'' is a book written and illustrated by Harold Scott MacDonald Coxeter, H. S. M. Coxeter, Patrick du Val, P. Du Val, H. T. Flather and J. F. Petrie. It enumerates certain stellations of the regular convex or Platonic re ...

'' (1938) and ''

Regular Polytopes ''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 th ...

'' (1947). Many concepts in geometry and

group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...

are named after him, including the

Coxeter graph In the mathematics, mathematical field of graph theory, the Coxeter graph is a 3-regular graph with 28 vertices and 42 edges. It is one of the 13 known cubic graph, cubic distance-regular graphs. It is named after Harold Scott MacDonald Coxeter ...

Coxeter group 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 ref ...

Coxeter's loxodromic sequence of tangent circles In geometry, Coxeter's loxodromic sequence of tangent circles is an infinite sequence of circles arranged so that any four consecutive circles in the sequence are pairwise mutually tangent. This means that each circle in the sequence is tangent to ...

Coxeter–Dynkin diagram In geometry, a Harold Scott MacDonald Coxeter, Coxeter–Eugene Dynkin, Dynkin diagram (or Coxeter diagram, Coxeter graph) is a Graph (discrete mathematics), graph with numerically labeled edges (called branches) representing a Coxeter group or ...

s, and the

Todd–Coxeter algorithm In group theory, the Todd–Coxeter algorithm, created by J. A. Todd and H. S. M. Coxeter in 1936, is an algorithm for solving the coset enumeration problem. Given a presentation of a group ''G'' by generators and relations and a subgroup ''H'' ...

Biography

Coxeter was born in

Kensington Kensington is an area of London in the Royal Borough of Kensington and Chelsea, around west of Central London. The district's commercial heart is Kensington High Street, running on an east–west axis. The north-east is taken up by Kensingt ...

, 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 Nitrous oxide (dinitrogen oxide or dinitrogen monoxide), commonly known as laughing gas, nitrous, or factitious air, among others, is a chemical compound, an Nitrogen oxide, oxide of nitrogen with the Chemical formula, formula . At room te ...

), 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 The Royal Academy of Arts (RA) is an art institution based in Burlington House in Piccadilly London, England. Founded in 1768, it has a unique position as an independent, privately funded institution led by eminent artists and architects. Its ...

. A maternal cousin was the architect Sir

Giles Gilbert Scott Sir Giles Gilbert Scott (9 November 1880 – 8 February 1960) was a British architect known for his work on the New Bodleian Library, Cambridge University Library, Lady Margaret Hall, Oxford, Battersea Power Station, Liverpool Cathedral, and de ...

. In his youth, Coxeter composed music and was an accomplished pianist at the age of 10. Roberts, Siobhan, ''King of Infinite Space: Donald Coxeter, The Man Who Saved Geometry'', Walker & Company, 2006, He felt that

mathematics and music Music theory analyzes the pitch, timing, and structure of music. It uses mathematics to study elements of music such as tempo, chord progression, form, and meter. The attempt to structure and communicate new ways of composing and hearing music h ...

were intimately related, outlining his ideas in a 1962 article on "Music and Mathematics" in the ''Canadian Music Journal''. He was educated at

King Alfred School, London The King Alfred School is a co-educational independent day school in Golders Green in North West London. It was founded in London in 1898 by Charles E. Rice, a former teacher at Bedales School '. The school was considered "radical" for its er ...

, and

St George's School, Harpenden St George's School, Harpenden (also known as St George's) is a non-selective state Day school, day and boarding school in Harpenden, Hertfordshire, England, educating students of both sexes between the ages of eleven and eighteen, with an emphasi ...

, where his best friend was John Flinders Petrie, later a mathematician for whom

Petrie polygon In geometry, a Petrie polygon for a regular polytope of dimensions is a skew polygon in which every consecutive sides (but no ) belongs to one of the facets. The Petrie polygon of a regular polygon is the regular polygon itself; that of a reg ...

s were named. He was accepted at

King's College, Cambridge King's College, formally The King's College of Our Lady and Saint Nicholas in Cambridge, is a List of colleges of the University of Cambridge, constituent college of the University of Cambridge. The college lies beside the River Cam and faces ...

, in 1925, but decided to spend a year studying in hopes of gaining admittance to

Trinity College Trinity College may refer to: Australia * Trinity Anglican College, an Anglican coeducational primary and secondary school in , New South Wales * Trinity Catholic College, Auburn, a coeducational school in the inner-western suburbs of Sydney, New ...

, where the standard of mathematics was higher. Coxeter won an entrance scholarship and went to Trinity in 1926 to read mathematics. There he earned his BA (as

Senior Wrangler The Senior Wrangler is the top mathematics undergraduate at the University of Cambridge in England, a position which has been described as "the greatest intellectual achievement attainable in Britain". Specifically, it is the person who achiev ...

) in 1928, and his doctorate in 1931. In 1932 he went to

for a year as a

Rockefeller Fellow The Rockefeller Foundation is an American private foundation and philanthropic medical research and arts funding organization based at 420 Fifth Avenue, New York City. The foundation was created by Standard Oil magnate John D. Rockefeller ("Seni ...

, where he worked with

Hermann Weyl Hermann Klaus Hugo Weyl (; ; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist, logician and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, ...

Oswald Veblen Oswald Veblen (June 24, 1880 – August 10, 1960) 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 lo ...

, and

Solomon Lefschetz Solomon Lefschetz (; 3 September 1884 – 5 October 1972) was a Russian-born American mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear ordinary differential equatio ...

. Returning to Trinity for a year, he attended

Ludwig Wittgenstein Ludwig Josef Johann Wittgenstein ( ; ; 26 April 1889 – 29 April 1951) was an Austrian philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy of language. From 1929 to 1947, Witt ...

's seminars on the

philosophy of mathematics Philosophy of mathematics is the branch of philosophy that deals with the nature of mathematics and its relationship to other areas of philosophy, particularly epistemology and metaphysics. Central questions posed include whether or not mathem ...

. In 1934 he spent a further year at Princeton as a Procter Fellow. In 1936 Coxeter moved to the University of Toronto. In 1938 he and

P. Du Val Patrick du Val (March 26, 1903 – January 22, 1987) was a British mathematician, known for his work on algebraic geometry, differential geometry, and general relativity. The concept of Du Val singularity of an algebraic surface is named afte ...

, H. T. Flather, and

John Flinders Petrie John Flinders Petrie (26 April 1907 – 1972) was an English mathematician. Petrie was the great grandson of the explorer and navigator, Matthew Flinders. He met the geometer Harold Scott MacDonald Coxeter as a student, beginning a lifelong fri ...

published ''

'' with

University of Toronto Press The University of Toronto Press is a Canadian university press. Although it was founded in 1901, the press did not actually publish any books until 1911. The press originally printed only examination books and the university calendar. Its first s ...

. In 1940 Coxeter edited the eleventh edition of ''Mathematical Recreations and Essays'', originally published by

W. W. Rouse Ball Walter William Rouse Ball (14 August 1850 – 4 April 1925), known as W. W. Rouse Ball, was a British mathematician, lawyer, and fellow at Trinity College, Cambridge, from 1878 to 1905. He was also a keen amateur magician, and the founding ...

in 1892. He was elevated to

professor Professor (commonly abbreviated as Prof.) is an Academy, academic rank at university, universities and other tertiary education, post-secondary education and research institutions in most countries. Literally, ''professor'' derives from Latin ...

in 1948. He was elected a Fellow of the

in 1948 and a

Fellow of the Royal Society Fellowship of the Royal Society (FRS, ForMemRS and HonFRS) is an award granted by the Fellows of the Royal Society of London to individuals who have made a "substantial contribution to the improvement of natural science, natural knowledge, incl ...

in 1950. He met

M. C. Escher Maurits Cornelis Escher (; ; 17 June 1898 – 27 March 1972) was a Dutch graphic artist who made woodcuts, lithography, lithographs, and mezzotints, many of which were Mathematics and art, inspired by mathematics. Despite wide popular int ...

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 Hyperbolic may refer to: * of or pertaining to a hyperbola, a type of smooth curve lying in a plane in mathematics ** Hyperbolic geometry, a non-Euclidean geometry ** Hyperbolic functions, analogues of ordinary trigonometric functions, defined u ...

tessellation A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety ...

s. He also inspired some of the innovations of

Buckminster Fuller Richard Buckminster Fuller (; July 12, 1895 – July 1, 1983) was an American architect, systems theorist, writer, designer, inventor, philosopher, and futurist. He styled his name as R. Buckminster Fuller in his writings, publishing more t ...

. Coxeter, M. S. Longuet-Higgins and

J. C. P. Miller Jeffrey Charles Percy Miller (31 August 1906 – 24 April 1981) was an English mathematician and computing pioneer. He worked in number theory and on geometry, particularly polyhedra, where Miller's monster is a nickname of the great dirhombic ...

were the first to publish the full

list of uniform polyhedra In geometry, a uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive ( transitive on its vertices, isogonal, i.e. there is an isometry mapping any vertex onto any other). It follows that all vertices are ...

(1954). He worked for 60 years at the

and published twelve books.

Personal life

Coxeter was a

vegetarian Vegetarianism is the practice of abstaining from the Eating, consumption of meat (red meat, poultry, seafood, insects as food, insects, and the flesh of any other animal). It may also include abstaining from eating all by-products of animal slau ...

. He attributed his longevity to his vegetarian diet, daily exercise such as fifty press-ups and standing on his head for fifteen minutes each morning, and consuming a nightly cocktail made from

Kahlúa Kahlúa () is a brand of coffee liqueur owned by the Pernod Ricard company and produced in Veracruz, Mexico. The drink contains rum, sugar, and arabica coffee. History Pedro Domecq began producing Kahlúa in 1936. It was named Kahlúa, m ...

(a coffee liqueur),

peach schnapps Schnapps ( or ) or schnaps is a type of alcoholic beverage that may take several forms, including distilled fruit brandies, herbal liqueurs, infusions, and "flavored liqueurs" made by adding fruit syrups, spices, or artificial flavorings to neu ...

, and

soy milk Soy milk (or soymilk), also known as soya milk, is a plant-based milk produced by soaking and grinding soybeans, boiling the mixture, and filtering out remaining particulates. It is a stable emulsion of oil, water, and protein. Its original ...

Awards

Since 1978, the

Canadian Mathematical Society The Canadian Mathematical Society (CMS; French: ''Société mathématique du Canada'') is an association of professional mathematicians dedicated to advancing mathematical research, outreach, scholarship and education in Canada. The Society se ...

have awarded the

Coxeter–James Prize The Coxeter-James Prize is a mathematics award given by the Canadian Mathematical Society (CMS) to recognize outstanding contributions to mathematics by young mathematicians in Canada. First presented in 1978, the prize is named after two renowned ...

in his honor. He was made a

in 1950 and in 1997 he was awarded their

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. It was named in honour of James Joseph Sylvester, the Savilian chair of geometry, Savilian Prof ...

. In 1990, he became a Foreign Member of the

American Academy of Arts and Sciences The American Academy of Arts and Sciences (The Academy) is one of the oldest learned societies in the United States. It was founded in 1780 during the American Revolution by John Adams, John Hancock, James Bowdoin, Andrew Oliver, and other ...

and in 1997 was made a

Companion Companion may refer to: Relationships Currently * Any of several interpersonal relationships such as friend or acquaintance * A domestic partner, akin to a spouse * Sober companion, an addiction treatment coach * Companion (caregiving), a caregive ...

of the

. In 1973 he received the

Jeffery–Williams Prize The Jeffery–Williams Prize is a mathematics award presented annually by the Canadian Mathematical Society. The award is presented to individuals in recognition of outstanding contributions to mathematical research. The first award was present ...

. A

festschrift In academia, a ''Festschrift'' (; plural, ''Festschriften'' ) is a book honoring a respected person, especially an academic, and presented during their lifetime. It generally takes the form of an edited volume, containing contributions from the h ...

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. A second such volume, ''The Coxeter Legacy'', was published in 2006 based on a Toronto Coxeter symposium held in 2004.

Works

Books

* 1942
''Non-Euclidean Geometry''
(1st edition), (2nd ed, 1947), (3rd ed, 1957), (4th ed, 1961), (5th ed, 1965),

(6th ed, 1998), MAA, . * 1949: ''The Real Projective Plane'' * 1961: ''Introduction to Geometry'', (2nd paperback edition 1989, .) * 1963: ''Regular Polytopes'' (2nd edition), Macmillan Company * 1967: (with S. L. Greitzer) ''Geometry Revisited'' * 1970
''Twisted honeycombs''
(American Mathematical Society, 1970, Regional conference series in mathematics Number 4, ) * 1973: ''

'', (3rd edition), Dover edition, * 1974: ''Projective Geometry'' (2nd edition) * 1974: ''Regular Complex Polytopes'',

Cambridge University Press Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessme ...

, . * 1981: (with R. Frucht and D. L. Powers), ''Zero-Symmetric Graphs'',

Academic Press Academic Press (AP) is an academic book publisher founded in 1941. It launched a British division in the 1950s. Academic Press was acquired by Harcourt, Brace & World in 1969. Reed Elsevier said in 2000 it would buy Harcourt, a deal complete ...

, . * 1987
Projective Geometry
' (1987) * 1995: F. Arthur Sherk, Peter McMullen, Anthony C. Thompson and Asia Ivić Weiss, editors: ''Kaleidoscopes — Selected Writings of H. S. M. Coxeter''.

John Wiley and Sons John Wiley & Sons, Inc., commonly known as Wiley (), is an American multinational publishing company that focuses on academic publishing and instructional materials. The company was founded in 1807 and produces books, journals, and encyclop ...

, . * 1999: ''The Beauty of Geometry: Twelve Essays'', Dover Publications, , * 2011: The Fifty-Nine Icosahedra, Tarquin Group,

Selected Papers

* 1940
"Regular and Semi-Regular Polytopes I"
''

Mathematische Zeitschrift ''Mathematische Zeitschrift'' ( German for ''Mathematical Journal'') is a mathematical journal for pure and applied mathematics published by Springer Verlag. History The journal was founded in 1917, with its first issue appearing in 1918. It wa ...

'' 46: 380–407, MR 2,10 * 1954: (with

Michael S. Longuet-Higgins Michael Selwyn Longuet-Higgins FRS (8 December 1925 – 26 February 2016) was a British mathematician and oceanographer at the Department of Applied Mathematics and Theoretical Physics (DAMTP), Cambridge University, England and Institute fo ...

and

) "Uniform Polyhedra", ''

Philosophical Transactions of the Royal Society A ''Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences'' is a fortnightly peer-reviewed scientific journal published by the Royal Society. It publishes original research and review content in a w ...

'' 246: 401–50 * 1957: (with W. O. J. Moser) ''Generators and Relations for Discrete Groups'' 1980: Second edition, Springer-Verlag * * 1985
"Regular and Semi-Regular Polytopes II"
''Mathematische Zeitschrift'' 188: 559–591 * 1988
"Regular and Semi-Regular Polytopes III"
''Mathematische Zeitschrift'' 200: 3–45

References

External links

Harold Scott MacDonald Coxeter archival papers
held at th
University of Toronto Archives and Records Management Services
*
H. S. M. Coxeter (1907–2003), Erich W. Ellers, Branko Grünbaum, Peter McMullen, Asia Ivic Weiss
Notices of the AMS: Volume 50, Number 10.
www.donaldcoxeter.com www.math.yorku.ca/dcoxeter
webpages dedicated to him (in development)
Jaron's World: Shapes in Other Dimensions
Discover mag., Apr 2007
The Mathematics in the Art of M.C. Escher
video of a lecture by H.S.M. Coxeter, April 28, 2000. {{DEFAULTSORT:Coxeter, H.S.M. 1907 births 2003 deaths Companions of the Order of Canada Fellows of the Royal Society Fellows of the Royal Society of Canada British geometers Chirality Polytopes 20th-century English mathematicians Recreational mathematicians Alumni of Trinity College, Cambridge Academic staff of the University of Toronto Canadian mathematicians 20th-century Canadian mathematicians People from Harpenden Academics of the University of East Anglia Senior Wranglers Presidents of the Canadian Mathematical Society Canadian fellows of the Royal Society

Biography

Personal life

Awards

Works

Books

Selected Papers

See also

References

Further reading

External links