Donald (H. S. M.) Coxeter

Pure and Applied Mathematics

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.

As a young scientist ...

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.”

The Science