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

Born | London, England | February 9, 1907

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

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 and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.^{ [3] }

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.

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] }

- 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

**Jacques Tits** is a Belgium-born French mathematician who works on group theory and incidence geometry. He introduced Tits buildings, the Tits alternative, the Tits group, and the Tits metric.

**Walter William Rouse Ball**, 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 president of the Cambridge Pentacle Club in 1919, one of the world's oldest magic societies.

**Sir William Vallance Douglas Hodge** was a British mathematician, specifically a geometer.

**Oswald Veblen** 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 long considered the first rigorous proof of the theorem, many now also consider Camille Jordan's original proof rigorous.

**Edward Charles** "**Ted**" **Titchmarsh** was a leading British mathematician.

**Luther Pfahler Eisenhart** was an American mathematician, best known today for his contributions to semi-Riemannian geometry.

**Alicia Boole Stott** was an Irish-born English mathematician. Despite never holding an academic position, she made a number of valuable contributions to the field, receiving an honorary doctorate from the University of Groningen. She is best known for coining the term "polytope" for a convex solid in four dimensions, and having an impressive grasp of four-dimensional geometry from a very early age.

**Henry Seely White** was an American mathematician. He was born in Cazenovia, New York to parents Aaron White and Isadore Maria Haight. He matriculated at Wesleyan University in Connecticut and graduated with honors in 1882 at the age of twenty-one. White excelled at Wesleyan in astronomy, ethics, Latin, logic, mathematics, and philosophy. At the university, John Monroe Van Vleck taught White mathematics and astronomy. Later, Van Vleck persuaded White to continue to study mathematics at the graduate level. Subsequently, White studied at the University of Göttingen under Klein, and received his doctorate in 1891.

In five-dimensional geometry, a **five-dimensional polytope** or **5-polytope** is a 5-dimensional polytope, bounded by (4-polytope) facets. Each polyhedral cell being shared by exactly two 4-polytope facets.

In nine-dimensional geometry, a **nine-dimensional polytope** or **9-polytope** is a polytope contained by 8-polytope facets. Each 7-polytope ridge being shared by exactly two 8-polytope facets.

**Henry Frederick Baker** FRS FRSE was a British mathematician, working mainly in algebraic geometry, but also remembered for contributions to partial differential equations, and Lie groups.

* Regular Polytopes* is a geometry book on regular polytopes written by Harold Scott MacDonald Coxeter. It was originally published by Methuen in 1947 and by Pitman Publishing in 1948, with a second edition published by Macmillan in 1963 and a third edition by Dover Publications in 1973. The Basic Library List Committee of the Mathematical Association of America has recommended that it be included in undergraduate mathematics libraries.

In geometry, a **snub** is an operation applied to a polyhedron. The term originates from Kepler's names of two Archimedean solids, for the snub cube and snub dodecahedron. In general, snubs have chiral symmetry with two forms: with clockwise or counterclockwise orientation. By Kepler's names, a snub can be seen as an expansion of a regular polyhedron: moving the faces apart, twisting them about their centers, adding new polygons centered on the original vertices, and adding pairs of triangles fitting between the original edges.

In geometry, a **demienneract** or **9-demicube** is a uniform 9-polytope, constructed from the 9-cube, with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.

In ten-dimensional geometry, a 10-polytope is a 10-dimensional polytope whose boundary consists of 9-polytope facets, exactly two such facets meeting at each 8-polytope ridge.

In geometry, the **regular skew polyhedra** are generalizations to the set of regular polyhedra which include the possibility of nonplanar faces or vertex figures. Coxeter looked at skew vertex figures which created new 4-dimensional regular polyhedra, and much later Branko Grünbaum looked at regular skew faces.

In geometry, a **decagram** is a 10-point star polygon. There is one regular decagram, containing the vertices of a regular decagon, but connected by every third point. Its Schläfli symbol is {10/3}.

**Henry P. McKean, Jr.** is an American mathematician at New York University. He works in various areas of analysis. He obtained his PhD in 1955 from Princeton University under William Feller.

In geometry, a **hendecagrammic prism** is a star polyhedron made from two identical regular hendecagrams connected by squares. The related **hendecagrammic antiprisms** are made from two identical regular hendecagrams connected by equilateral triangles.

**Paul Joseph Kelly** was an American mathematician who worked in geometry and graph theory.

- ↑ 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. - ↑ "The Oxford Dictionary of National Biography" .
*Oxford Dictionary of National Biography*(online ed.). Oxford University Press. 2004. doi:10.1093/ref:odnb/89876.(Subscription or UK public library membership required.) - ↑ 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
- 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 - ↑ 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
- 1 2 3 4 5 6 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.

- 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.

- 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.

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