Greatest classical geometer of the 20th century
"I’m a Platonist — a follower of Plato — who believes that one didn’t invent these sorts of things, that one discovers them. In a sense, all these mathematical facts are right there waiting to be discovered."
The aroma of antiseptic and crisp sheets mingles with the sooty smell of a small coal-burning fireplace at the end of the infirmary room. Two thirteen-year-old boys are in side-by-side beds, recovering from the flu in their private school’s sickroom.
“Coxeter, how do you imagine time travel would work?” asks John Petrie, one of the boys.
“You mean as in H. G. Wells?” says Donald Coxeter, the other boy. H. G. Wells’ classic science fiction book, The Time Machine, is a popular topic of conversation. Both boys believe time travel will eventually be possible. After a few seconds, Coxeter says, “I suppose one might find it necessary to pass into the fourth dimension.” That is the moment when he began forming ideas about hyperdimensional geometries.
Both boys were very bright. They started using the books and games by their beds to play around with ideas of higher dimensional space — spaces and dimensions that go beyond the ordinary three dimensions of natural space as we see it. These early musings lead Coxeter to later discoveries about regular polytopes, geometric shapes that extend into the fourth dimension and far beyond.
Soon after he recovered from the flu, Coxeter wrote a school essay on the idea of projecting geometric shapes into higher dimensions. Impressed by his son’s geometrical talents and wishing to help the boy’s mind develop, his father took him to visit Bertrand Russell, the brilliant English philosopher, educator and peace activist. Russell helped the Coxeters find an excellent math tutor who worked with Coxeter, enabling him to enter Cambridge University.
Coxeter was known as H. S. M. Coxeter, though friends and relatives called him Donald. Here’s the explanation: At birth he was given the name MacDonald Scott Coxeter, which led to his being called Donald for short. But a godparent suggested that his father’s name should be added, so Harold was added at the front. Then somebody noticed that H. M. S. Coxeter sounded like the name of a ship. They finally changed the names around to Harold Scott MacDonald Coxeter.
At 19, in 1926, before Coxeter had a university degree, he discovered a new regular polyhedron, a shape having six hexagonal faces at each vertex. He went on to study the mathematics of kaleidoscopes, which are instruments that use mirrors and bits of glass to create an endlessly changing pattern of repeating reflections. By 1933 he had counted and specified the n-dimensional kaleidoscopes (“n-dimensions” means one-dimensional, two-dimensional, three-dimensional, et cetera, up to any number [n] dimensions). This branch of mathematics determines how shapes will behave and how many symmetries they generate when repeatedly reflected in a kaleidoscope.
In 1936 Coxeter received a completely unexpected invitation from Sam Beatty at the University of Toronto, offering him an assistant professorship there. Coxeter’s father, foreseeing World War II, advised his son to accept the offer. As Coxeter said, “Rien and I were married and we sailed off to begin our life together in the safe country of Canada.”
Coxeter lived to the age of 96, working and lecturing right until his death. He attributed his long life to a strict vegetarian diet and he did 50 push-ups every day. He said, “I am never bored.”
Geometry is a branch of mathematics that deals with points, lines, angles, surfaces and solids. One of Coxeter’s major contributions to geometry was in the area of dimensional analogy, the process of stretching geometrical shapes into higher dimensions. He is also famous for “Coxeter groups,” the inversive distance between two disjoint circles (or spheres).
1. Line: A line is a one-dimensional shape. You can only move along a line in one direction — forward or back. If you sweep a line sidewise in the second dimension you create a square.
2. Square: Four lines at right angles make a square. A square has a two-dimensional surface. You can move in two directions — forward or backward and right or left.
3. Cube: If you pull a square upwards, you are moving into the third dimension. The result is a cube. Six squares make a cube. Inside a cube you can move forward and backwards, right and left, or up and down — three directions, or three dimensions.
4. Hypercube: If you pull a cube into the fourth dimension you get a hypercube. Eight cubes make a hypercube. The figure you see here cannot exist in the real world, which only has three-dimensional space. It is a projection of a four-dimensional object onto two dimensions, just as the cube before it is a projection from three-dimensional space to the two-dimensional flat surface of the paper.
5. Regular polytope: If you keep pulling the hypercube into higher and higher dimensions you get a polytope. Coxeter is famous for his work on regular polytopes. When they involve coordinates made of complex numbers they are called complex polytopes.
Coxeter always hoped that somebody would come up with a better proof for the four-colour-map problem, which simply says that if you have any map in two dimensions and the countries are any shape, you need only four colours for the countries so that two countries of the same colour never touch each other. Though it can be demonstrated easily with some paper and coloured pencils, nobody has ever proved (or disproved) this idea with pure geometry and math. A controversial computer proof was completed in the 1970s that claims to prove the four-colour-map problem. The computer tested millions of different maps, but to Coxeter this proof was not satisfying because it is too huge and complicated to be checked by a human being. In recent years, mathematicians have reduced the number of computer-generated maps but this new proof still requires the use of a computer and is impractical for humans to check alone.
Coxeter never felt the computer proof of the four-colour-map theorem was elegant. Such a proof would be easily understood and would use only mathematical or geometrical ideas presented in a logical way with nothing more than a pencil and paper. So, to Coxeter and many other mathematicians, the four-colour-map theorem is still an open problem.
Martin Aigner and Gunter M. Ziegler, Proofs from The Book, Springer-Verlag Telos, 2000.
W. W. R. Ball and H. S. M. Coxeter, Mathematical Recreations and Essays, 13th edition, Dover, 1987.
H. S. M. Coxeter, The Beauty of Geometry: Twelve Essays, Dover, 1999.
H. S. M. Coxeter, Introduction to Geometry, second edition, Wiley, 1989.
H. S. M. Coxeter, Non-Euclidean Geometry, sixth edition, Math. Assoc. Amer., 1998.
Siobhan Roberts, King of Infinite Space: The Story of Donald Coxeter, the Man Who Saved Geometry, House of Anansi/Groundwood Books, 2006.
Robin Wilson, Four Colors Suffice: How the Map Problem Was Solved, Princeton University Press, 2003.
Coxeter's Obituary at Wolfram Research.
So You Want to Be a Geometer
Coxeter loved his work and once said of his career, “I am extremely fortunate for being paid for what I would have done anyway.” His advice to young people thinking about a career in mathematics: “If you are keen on mathematics, you have to love it, dream about it all the time.”
Careers that involve mathematics with a specialty in geometry include architecture, cryptography (secret codes), crystallography, networks, map making, ballistics, astronomy, engineering, physics, computer visualization and computer gaming — any work that involves the visualization or manipulation of things or ideas in multiple dimensions.