Harold Scott MacDonald " Donald" Coxeter, (February 9, 1907 – March 31, 2003) was a British and later also Canadian  geometer. He is regarded as one of the greatest geometers of the 20th century.
Coxeter was born in
Kensington to Harold Samuel Coxeter and Lucy ( 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 née 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. 
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 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. Coxeter won an entrance scholarship and went up to  Trinity College, Cambridge in 1926 to read mathematics. There he earned his BA (as Senior Wrangler) 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. In 1938 he and
P. Du Val, H.T. Flather, and John Flinders Petrie published The Fifty-Nine Icosahedra with University of Toronto Press. In 1940 Coxeter edited the eleventh edition of Mathematical Recreations and Essays, 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. Coxeter,  M. S. Longuet-Higgins and J. C. P. Miller were the first to publish the full list of uniform polyhedra (1954).
He worked for 60 years at the
University of Toronto and published twelve books.
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. In 1990, he became a Foreign Member of the  American Academy of Arts and Sciences and in 1997 was made a  Companion of the Order of Canada.
In 1973 he received the
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.
, Regular and Semi-Regular Polytopes I Mathematische Zeitschrift 46: 380-407, MR 2,10 doi: 10.1007/BF01181449 1942:
(1st edition), Non-Euclidean Geometry (2nd ed, 1947), (3rd ed, 1957), (4th ed, 1961), (5th ed, 1965),  University of Toronto Press (6th ed, 1998), MAA. 1954: (with
Michael S. Longuet-Higgins and J. C. P. Miller) "Uniform Polyhedra", Philosophical Transactions of the Royal Society A 246: 401-50 doi: 10.1098/rsta.1954.0003 1949:
The Real Projective Plane  1957: (with W.O.J. Moser)
Generators and Relations for Discrete Groups 1980: Second edition, Springer-Verlag  ISBN 0-387-09212-9 1961:
Introduction to Geometry   1963:
Regular Polytopes (2nd edition), Macmillan Company 1967: (with
S. L. Greitzer) Geometry Revisited 1970:
(American Mathematical Society, 1970, Regional conference series in mathematics Number 4, Twisted honeycombs ISBN 0-8218-1653-5) 1973:
, (3rd edition), Dover edition, Regular Polytopes ISBN 0-486-61480-8 1974:
Projective Geometry (2nd edition) 1974:
Regular Complex Polytopes, Cambridge University Press 1981: (with R. Frucht and D. L. Powers),
Zero-Symmetric Graphs, Academic Press. 1985:
, Regular and Semi-Regular Polytopes II Mathematische Zeitschrift 188: 559-591 1987
(1987) Projective Geometry ISBN 978-0-387-40623-7 1988:
, Regular and Semi-Regular Polytopes III Mathematische Zeitschrift 200: 3-45 1995: F. Arthur Sherk, Peter McMullen, Anthony C. Thompson and Asia Ivi? Weiss, editors:
Kaleidoscopes -- Selected Writings of H.S.M. Coxeter. John Wiley and Sons ISBN 0-471-01003-0 1999: The Beauty of Geometry: Twelve Essays, Dover Publications, LCCN 99-35678, ISBN 0-486-40919-8
Harold Scott MacDonald Coxeter at the Mathematics Genealogy Project
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Roberts, Siobhan; Ivi? Weiss, Asia (2006). Longair, Malcolm (ed.). "Harold Scott MacDonald Coxeter. 9 February 1907 -- 31 March 2003: Elected FRS 1950" (PDF). Biographical Memoirs of Fellows of the Royal Society. 52: 45-66. doi: . 10.1098/rsbm.2006.0004 ISSN 1748-8494.
"Geometry Revisited". Mathematical Association of America . Retrieved 2018.
. "The Oxford Dictionary of National Biography" (online ed.). Oxford University Press. 2004. Oxford Dictionary of National Biography doi: 10.1093/ref:odnb/89876. (Subscription or UK public library membership required.)
^ a b c d
Roberts, Siobhan, King of Infinite Space: Donald Coxeter, The Man Who Saved Geometry, Walker & Company, 2006, ISBN 0-8027-1499-4
^ a b c d e f
O'Connor, John J.; Robertson, Edmund F., "Harold Scott MacDonald Coxeter", , MacTutor History of Mathematics archive University of St Andrews
Frame, J. S. (1940). "Review: Mathematical Recreations and Essays, 11th edition, by W. W. Rouse Ball; revised by H. S. M. Coxeter" Bull. Amer. Math. Soc (PDF). . 45 (3): 211-213. doi: 10.1090/S0002-9904-1940-07170-8.
^ Harold Coxeter,
Michael S. Longuet-Higgins and J. C. P. Miller. "Uniform Polyhedra", Philosophical Transactions of the Royal Society A 246: 401-50 doi: 10.1098/rsta.1954.0003
Foreign Honorary Member elected 1990 2016 American Academy of Arts & Sciences
Office of the Governor General of Canada. Order of Canada citation. Queen's Printer for Canada. Retrieved 26 May 2010
Edge, W. L. (June 1983). "Review of . The Geometric Vein" Proceedings of the Edinburgh Mathematical Society. 26 (2): 284-285. doi: . 10.1017/s0013091500017016
Blumenthal, L. M. (1943). "Review: Non-euclidean geometry by H. S. M. Coxeter" (PDF). Bull. Amer. Math. Soc. 49 (9): 679-680. doi: . 10.1090/s0002-9904-1943-07977-3
DuVal, Patrick (1950). "Review: The real projective plane by H. S. M. Coxeter" (PDF). Bull. Amer. Math. Soc. 56 (4): 376-378. doi: . 10.1090/s0002-9904-1950-09414-2
Hall Jr., Marshall (1958). "Review: Generators and relations for discrete groups by H. S. M. Coxeter and W. O. J. Moser" (PDF). . 64, Part 1 (3): 106-108. Bulletin of the American Mathematical Society doi: . 10.1090/S0002-9904-1958-10178-0
Freudenthal, H. (1962). "Review: Introduction to geometry by H. S. M. Coxeter" (PDF). Bull. Amer. Math. Soc. 68 (2): 55-59. doi: . 10.1090/s0002-9904-1962-10714-9
Levi, H. (1963). "Review: Introduction to Geometry by H. S. M. Coxeter". The Journal of Philosophy. 60 (1): 19-21. doi: 10.2307/2023059. JSTOR 2023059.