The Beauty Of Geometry Twelve Essays Pdf

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Geometry Loses One of Its Most Eloquent Expositors

By Eric W. Weisstein

April 2, 2003--Noted geometer and author H. S. M. Coxeter passed away peacefully at his home in Toronto on March 31. He was 96. Known to friends and colleagues as "Donald," Harold Scott MacDonald Coxeter was originally to be named "MacDonald Scott" until a godparent suggested that his father's name "Harold" be added as his first name. At this point, another relative pointed out that "Harold MacDonald Scott" would give him the same initials as "Her Majesty's Ship" (as in, for example, Gilbert and Sullivan's well-known H. M. S. Pinafore), and hence "Harold MacDonald Scott" became "Harold Scott MacDonald."

Coxeter was born on February 9, 1907, in London. He was artistically gifted, especially in music, but decided to become a mathematician as a result of his love of the beauty of symmetry, an interest that was reflected in his work throughout his life. Coxeter received his B.A. in mathematics at Cambridge University in 1929, quickly followed by his Ph.D. in 1931. After briefly working as a fellow at Cambridge University and a visiting researcher at Princeton University, Coxeter accepted a position at the University of Toronto in 1936, where he taught and continued his research until his death this year.

Coxeter edited the Canadian Journal of Mathematics between 1949 and 1958. He received many awards and honors throughout his life and also served as vice president of the Mathematical Association of America.

Coxeter did fundamental work in a number of areas of geometry, group theory, graph theory, and discrete groups, particularly in projective geometry, non-Euclidean geometry, and the geometry of higher-dimensional analogs of the regular Platonic solids known as polytopes.

With M. S. Longuet-Higgins and J. C. P. Miller, Coxeter published in 1954 the definitive paper enumerating and describing the properties of the so-called uniform polyhedra, which are generalizations of the Platonic and Archimedean solids that have symmetrically equivalent vertices but may have less overall symmetry than the better-known Platonics and Archimedeans.

Coxeter is perhaps best known for his introduction of Coxeter groups and Coxeter-Dynkin diagrams, but a large number of other interesting mathematical objects bear his name, including the Coxeter graph and the Tutte-Coxeter graph (now more commonly known as the Levi graph).

In addition to his great talents in geometric research, Coxeter was also an eloquent expositor in all areas of geometry. His books Introduction to Geometry (sadly now out of print) and Geometry Revisited (coauthored with Samuel Greitzer) remain masterpieces of didactic exposition. On a personal note, the author of this website began his geometric education by reading Coxeter's books.

Coxeter's textbooks in a number of areas of geometry are modern classics. Coxeter also wrote eloquently on topics in recreational mathematics, revising Ball's classic treatise Mathematical Recreations and Essays. This book is a treasure trove of information on both mathematics and recreational mathematics and is required reading for any recreational mathematician. Coxeter is one of the few authors (Subramanyan Chandrasekhar being another) to be honored by having three of his books reprinted by Dover Publications. Another testament to Coxeter's prodigious output is the fact that he is cited more than four hundred times on the MathWorld website.

Donald Coxeter is survived by his daughter Susan Thomas and his son Edgar.

Coxeter's mathematical legacy lives on through his work and his many writings, but the world has lost one of its finest and most eloquent geometers.


Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, 1987.

Bland, J. (Chair, Department of Mathematics, University of Toronto). "Donald Coxeter."

Coxeter, H. S. M. The Beauty of Geometry: Twelve Essays. New York: Dover, 1999.

Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, 1989.

Coxeter, H. S. M. Non-Euclidean Geometry, 6th ed. Washington, DC: Math. Assoc. Amer., 1998.

Coxeter, H. S. M. Projective Geometry, 2nd ed. New York: Springer-Verlag, 1987.

Coxeter, H. S. M. The Real Projective Plane, 3rd ed. with an Appendix for Macintosh. New York: Springer-Verlag, 1992.

Coxeter, H. S. M. Regular Complex Polytopes, 2nd ed. Cambridge, England: Cambridge University Press, 1991.

Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, 1973.

Coxeter, H. S. M; Du Val, P.; Flather, H. T.; and Petrie, J. F. The Fifty-Nine Icosahedra, 3rd ed. Stradbroke, England: Tarquin Publications, 1999.

Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., 1967.

Coxeter, H. S. M.; Longuet-Higgins, M. S.; and Miller, J. C. P. "Uniform Polyhedra." Philos. Trans. Roy. Soc. London Ser. A246, 401-450, 1954.

Mathematical Association of America. "In Memoriam H. S. M. Coxeter, 1907-2003."

O'Connor, J. J. and Robertson, E. F. "Harold Scott MacDonald Coxeter"

Sherk, F. A.; McMullen, P.; Thompson, A. C.; and Weiss, A. I. Kaleidoscopes: Selected Writings of H. S. M. Coxeter. New York: Wiley, 1995.

Tesseractic honeycomb honeycomb
(No image)
TypeHyperbolic regular honeycomb
Schläfli symbol{4,3,3,4,3}
Coxeter diagram

Cell figure{3}
Face figure{4,3}
Edge figure{3,4,3}
Vertex figure{3,3,4,3}
DualOrder-4 24-cell honeycomb honeycomb
Coxeter groupR5, [3,4,3,3,4]

In the geometry of hyperbolic 5-space, the tesseractic honeycomb honeycomb is one of five paracompact regular space-filling tessellations (or honeycombs). It is called paracompact because it has infinite vertex figures, with all vertices as ideal points at infinity. With Schläfli symbol {4,3,3,4,3}, it has three tesseractic honeycombs around each cell. It is dual to the order-4 24-cell honeycomb honeycomb.

Related honeycombs[edit]

It is related to the regular Euclidean 4-space tesseractic honeycomb, {4,3,3,4}.

It is analogous to the paracompact cubic honeycomb honeycomb, {4,3,4,3}, in 4-dimensional hyperbolic space, square tiling honeycomb, {4,4,3}, in 3-dimensional hyperbolic space, and the order-3 apeirogonal tiling, {∞,3} of 2-dimensional hyperbolic space, each with hypercube honeycomb facets.

See also[edit]


  • Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
  • Coxeter, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999 ISBN 0-486-40919-8 (Chapter 10: Regular honeycombs in hyperbolic space, Summary tables II,III,IV,V, p212-213)


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