Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian

geometer A geometer is a mathematician whose area of study is geometry. Some notable geometers and their main fields of work, chronologically listed, are: 1000 BCE to 1 BCE * Baudhayana (fl. c. 800 BC) – Euclidean geometry, geometric algebra * ...

. He is regarded as one of the greatest geometers of the 20th century.

Biography

Coxeter was born in

Kensington Kensington is a district in the Royal Borough of Kensington and Chelsea in the West End of London, 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 b ...

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 nos, is a chemical compound, an oxide of nitrogen with the formula . At room temperature, it is a colourless non-flammable gas, and has a ...

), 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 on Piccadilly in London. Founded in 1768, it has a unique position as an independent, privately funded institution led by eminent artists and architects. Its purpo ...

. 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 d ...

. 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 (music), pitch, timing, and structure of music. It uses mathematics to study elements of music such as tempo, chord progression, Musical form, form, and Meter (music), meter. The attempt to structure and communica ...

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 e ...

and St George's School, Harpenden, 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 is a constituent college of the University of Cambridge. Formally The King's College of Our Lady and Saint Nicholas in Cambridge, the college lies beside the River Cam and faces out onto King's Parade in the centre of the city ...

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

Cambridge Cambridge ( ) is a university city and the county town in Cambridgeshire, England. It is located on the River Cam approximately north of London. As of the 2021 United Kingdom census, the population of Cambridge was 145,700. Cambridge bec ...

in 1926 to read mathematics. There he earned his BA (as

Senior Wrangler The Senior Frog 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 a ...

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

Princeton University Princeton University is a private university, private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth, New Jersey, Elizabeth as the College of New Jersey, Princeton is the List of Colonial Colleges, fourth-oldest ins ...

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 second-oldest major philanthropic institution in America, after the Carneg ...

, where he worked with

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

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 lon ...

, and

Solomon Lefschetz Solomon Lefschetz (russian: Соломо́н Ле́фшец; 3 September 1884 – 5 October 1972) was an American mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear o ...

. Returning to Trinity for a year, he attended

Ludwig Wittgenstein Ludwig Josef Johann Wittgenstein ( ; ; 26 April 1889 – 29 April 1951) was an Austrian-British philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy of language. He is considere ...

's seminars on the

philosophy of mathematics The philosophy of mathematics is the branch of philosophy that studies the assumptions, foundations, and implications of mathematics. It aims to understand the nature and methods of mathematics, and find out the place of mathematics in people's ...

. 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, H.T. Flather, and

John Flinders Petrie 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 ...

published

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 ...

with

University of Toronto Press The University of Toronto Press is a Canadian university press founded in 1901. 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 calen ...

. 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 post-secondary education and research institutions in most countries. Literally, ''professor'' derives from Latin as a "person who pr ...

in 1948. Coxeter was elected a Fellow of the

Royal Society of Canada The Royal Society of Canada (RSC; french: Société royale du Canada, 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, bil ...

in 1948 and a

Fellow of the Royal Society Fellowship of the Royal Society (FRS, ForMemRS and HonFRS) is an award granted by the judges 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 mathematically inspired woodcuts, lithographs, and mezzotints. Despite wide popular interest, Escher was for most of his life neglected in t ...

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 is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry. The following phenomena are described as ''hyperbolic'' because they ...

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

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 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 c ...

(1954). He worked for 60 years at the

University of Toronto The University of Toronto (UToronto or U of T) is a public research university in Toronto, Ontario, Canada, located on the grounds that surround Queen's Park. It was founded by royal charter in 1827 as King's College, the first institution ...

and published twelve books.

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 the interests of mathematical research, outreach, scholarship and education in Canada. It serves the ...

have awarded the Coxeter–James Prize 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 (London) for the encouragement of mathematical research, and accompanied by a £1,000 prize. It was named in honour of James Joseph Sylvester, the Savilian Professor of Geometry a ...

. In 1990, he became a Foreign Member of the

American Academy of Arts and Sciences The American Academy of Arts and Sciences (abbreviation: AAA&S) 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 ...

and in 1997 was made a Companion of the

Order of Canada The Order of Canada (french: Ordre du Canada; abbreviated as OC) is a Canadian state order and the second-highest honour for merit in the system of orders, decorations, and medals of Canada, after the Order of Merit. To coincide with 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 presen ...

. 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

* 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. It was founded in 1918 and edited by Leon Lichtenstein together with Konrad Knopp, Erhard ...

46: 380-407, MR 2,10 * 1942
''Non-Euclidean Geometry''
(1st edition), (2nd ed, 1947), (3rd ed, 1957), (4th ed, 1961), (5th ed, 1965),

(6th ed, 1998), MAA. * 1954: (with

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

and J. C. P. Miller) "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 wide r ...

246: 401–50 * 1949: ''The Real Projective Plane'' * 1957: (with W.O.J. Moser) ''Generators and Relations for Discrete Groups'' 1980: Second edition, Springer-Verlag * 1961: ''Introduction to Geometry'' * 1963: ''Regular Polytopes'' (2nd edition), Macmillan Company * 1967: (with

S. L. Greitzer Samuel L. Greitzer (August 10, 1905 – February 22, 1988) was an American mathematician, the founding chairman of the United States of America Mathematical Olympiad, and the publisher of the precollege mathematics journal ''Arbelos''. Together wi ...

) ''Geometry Revisited'' * 1970
''Twisted honeycombs''
(American Mathematical Society, 1970, Regional conference series in mathematics Number 4, ) * 1973: ''

Regular Polytopes In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or -faces (for all , where is the dimension of the polytope) — cells, ...

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

Cambridge University Press Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing hou ...

* 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 was acquired by Harcourt, Brace & World in 1969. Reed Elsevier bought Harcourt in 2000, and Academic Press is now an imprint of Elsevier. Academic Press publishes reference ...

. * 1985
''Regular and Semi-Regular Polytopes II''
''Mathematische Zeitschrift'' 188: 559–591 * 1987
Projective Geometry
' (1987) * 1988
''Regular and Semi-Regular Polytopes III''
''Mathematische Zeitschrift'' 200: 3–45 * 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 founded in 1807 that focuses on academic publishing and instructional materials. The company produces books, journals, and encyclopedias, in p ...

* 1999: ''The Beauty of Geometry: Twelve Essays'', Dover Publications, ,

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 University of Toronto faculty 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

Awards

Works

See also

References

Further reading

External links