# Harold Scott MacDonald Coxeter

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 | Norman Johnson |

**Harold Scott MacDonald** "**Donald**" **Coxeter**, FRS, FRSC, CC (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. 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.^{[3]} 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.^{[4]}

## Biography

In his youth, Coxeter composed music and was an accomplished pianist at the age of 10.^{[5]} 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*.^{[5]}

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.^{[5]}^{[3]} 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.^{[3]} Returning to Trinity for a year, he attended Ludwig Wittgenstein's seminars on the philosophy of mathematics.^{[5]} In 1934 he spent a further year at Princeton as a Procter Fellow.^{[3]}

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*,^{[6]} 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 Maurits 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.^{[3]} Coxeter, M. S. Longuet-Higgins and J. C. P. Miller were the first to publish the full list of uniform polyhedra (1954).^{[7]}

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.^{[3]} In 1990, he became a Foreign Member of the American Academy of Arts and Sciences^{[8]} and in 1997 was made a Companion of the Order of Canada.^{[9]}

In 1973 he received the Jeffery–Williams Prize.^{[3]}

## 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),^{[10]}(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*^{[11]} - 1957: (with W.O.J. Moser)
*Generators and Relations for Discrete Groups*^{[12]}1980: Second edition, Springer-Verlag ISBN 0-387-09212-9 - 1961:
*Introduction to Geometry*^{[13]}^{[14]} - 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

- Boerdijk–Coxeter helix
- Coxeter functor
- Coxeter group
- Coxeter number
- Coxeter–Dynkin diagram
- Coxeter–James Prize
- Coxeter matroid
- Coxeter–Todd lattice
- Coxeter's loxodromic sequence of tangent circles
- Goldberg–Coxeter construction
- LCF notation
- Todd–Coxeter algorithm
- Tutte–Coxeter graph

## 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. - 1 2 3 4 5 6 7 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" - 1 2 3 4 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.