# Harold Scott MacDonald Coxeter

**Harold Scott MacDonald** "**Donald**" **Coxeter**, CC, FRS, FRSC (February 9, 1907 – March 31, 2003)[2] was a British-born Canadian geometer. Coxeter is regarded as one of the greatest geometers of the 20th century.[3] He was born in London, received his BA (1929) and PhD (1931) from Cambridge, but lived in Canada from age 29. He was always called Donald, from his third name MacDonald.[4] He was most noted for his work on regular polytopes 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.[5]

Harold Scott MacDonald Coxeter | |
---|---|

Born | London, England | February 9, 1907

Died | March 31, 2003 96) Toronto, Ontario, Canada | (aged

Residence | Toronto, Ontario, Canada |

Known for | study of geometry and mathematics |

Spouse(s) | Hendrina, died in 1999 |

Children | Susan Thomas, and a son, Edgar |

Awards | Smith's Prize (1931) Henry Marshall Tory Medal (1949) CRM-Fields-PIMS prize (1995) Sylvester Medal (1997) |

Scientific career | |

Fields | Geometry |

Institutions | University of Toronto |

Doctoral advisor | H. F. Baker[1] |

Doctoral students |

## Biography

In his youth, Coxeter composed music and was an accomplished pianist at the age of 10.[6] 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*.[6]

Coxeter 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.[6][4] 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.[4] Returning to Trinity for a year, he attended Ludwig Wittgenstein's seminars on the philosophy of mathematics.[6] In 1934 he spent a further year at Princeton as a Procter Fellow.[4]

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*,[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.[4] Coxeter, M. S. Longuet-Higgins 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.

## Awards

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.[4] 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.[4]

## Works

- 1940:
*Regular and Semi-Regular Polytopes I*, Mathematische Zeitschrift 46: 380-407, MR 2,10 doi:10.1007/BF01181449 - 1942:
*Non-Euclidean Geometry*(1st edition),[11] (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*[12] - 1957: (with W.O.J. Moser)
*Generators and Relations for Discrete Groups*[13] 1980: Second edition, Springer-Verlag ISBN 0-387-09212-9 - 1961:
*Introduction to Geometry*[14][15] - 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, ISBN 0-8218-1653-5) - 1973:
*Regular Polytopes*, (3rd edition), Dover edition, 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
*Projective Geometry*(1987) 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 Ivić 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

## See also

## References

- Harold Scott MacDonald Coxeter at the Mathematics Genealogy Project
- Roberts, S.; Ivic Weiss, A. (2006). "Harold Scott MacDonald Coxeter. 9 February 1907 – 31 March 2003: Elected FRS 1950".
*Biographical Memoirs of Fellows of the Royal Society*.**52**: 45–66. doi:10.1098/rsbm.2006.0004. - "Geometry Revisited".
*Mathematical Association of America*. Retrieved 25 December 2018. - O'Connor, John J.; Robertson, Edmund F., "Harold Scott MacDonald Coxeter",
*MacTutor History of Mathematics archive*, University of St Andrews. - The Boston Globe (September 10, 2006) "Review:
*The Man Who Saved Geometry*by Siobhan Roberts. "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" - Roberts, Siobhan,
*King of Infinite Space: Donald Coxeter, The Man Who Saved Geometry*, Walker & Company, 2006, ISBN 0-8027-1499-4 - Frame, J. S. (1940). "Review:
*Mathematical Recreations and Essays, 11th edition, by W. W. Rouse Ball; revised by H. S. M. Coxeter"*(PDF).*Bull. Amer. Math. Soc*.**45**(3): 211–213. doi:10.1090/S0002-9904-1940-07170-8. - Coxeter 1954
- 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
- 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).*Bulletin of the American Mathematical Society*. 64, Part 1 (3): 106–108. 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.

## Further reading

- Davis, Chandler; Ellers, Erich W, eds. (2006).
*The Coxeter Legacy: Reflections and Projections*. Providence, R.I.: American Mathematical Society. ISBN 978-0-8218-3722-1. OCLC 62282754. - Roberts, Siobhan (2006).
*King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry*. New York: Walker & Company. ISBN 978-0-8027-1499-2. OCLC 71436884.

## External links

- Archival papers held at University of Toronto Archives and Records Management Services
- Harold Scott MacDonald Coxeter at the Mathematics Genealogy Project
- 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.