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

Born | London, England | February 9, 1907

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

Alma mater | University of Cambridge (B.A., 1929; Ph.D., 1931) |

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) Jeffery–Williams Prize (1973) CRM-Fields-PIMS prize (1995) Sylvester Medal (1997) |

Scientific career | |

Fields | Geometry |

Institutions | University of Toronto |

Doctoral advisor | H. F. Baker^{[1]} |

Doctoral students |

**Harold Scott MacDonald** "**Donald**" **Coxeter**, CC, FRS, FRSC (February 9, 1907 – March 31, 2003)^{[2]} was a British-born Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.^{[3]}

## Biography

Coxeter was born in Kensington to Harold Samuel Coxeter and Lucy (née 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 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.^{[4]}^{[5]}

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]}

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.^{[7]} 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.^{[6]}^{[8]} 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.^{[8]} 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.^{[8]}

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*,^{[9]} 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.^{[8]} Coxeter, M. S. Longuet-Higgins and J. C. P. Miller were the first to publish the full list of uniform polyhedra (1954).^{[10]}

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.^{[8]} In 1990, he became a Foreign Member of the American Academy of Arts and Sciences^{[11]} and in 1997 was made a Companion of the Order of Canada.^{[12]}

In 1973 he received the Jeffery–Williams Prize.^{[8]}

A 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.^{[13]}

## 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),^{[14]}(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*^{[15]} - 1957: (with W.O.J. Moser)
*Generators and Relations for Discrete Groups*^{[16]}1980: Second edition, Springer-Verlag ISBN 0-387-09212-9 - 1961:
*Introduction to Geometry*^{[17]}^{[18]} - 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.**^**https://www.oxforddnb.com/view/10.1093/ref:odnb/9780198614128.001.0001/odnb-9780198614128-e-89876?rskey=6NsgBZ&result=1**^**Biographical Memoirs of Fellows of the Royal Society, vol. 52, pp. 45-66, "Harold Scott MacDonald Coxeter", Siobhan Roberts and Asia Ivic Weiss, Royal Society, 2006- ^
^{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 **^**Biographical Memoirs of Fellows of the Royal Society, vol. 52, pp. 45-66, "Harold Scott MacDonald Coxeter", Siobhan Roberts and Asia Ivic Weiss, Royal Society, 2006- ^
^{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"**(PDF).*Bull. Amer. Math. Soc*.***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).*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

- Harold Scott MacDonald Coxeter archival papers held at the 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.