The Beauty Of Geometry Twelve Essays
"I’m a Platonist — a follower of Plato — who believes that one didn’t invent these sorts of things, that one discovers them. These early musings lead Coxeter to later discoveries about regular polytopes, geometric shapes that extend into the fourth dimension and far beyond. They finally changed the names around to Harold Scott Mac Donald Coxeter.Let me explain, first, what the Platonic solids are.To begin, consider something simpler: regular polygons.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 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. Soon after he recovered from the flu, Coxeter wrote a school essay on the idea of projecting geometric shapes into higher dimensions. Coxeter, though friends and relatives called him Donald. 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.Two thirteen-year-old boys are in side-by-side beds, recovering from the flu in their private school’s sickroom. 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. Here’s the explanation: At birth he was given the name Mac Donald Scott Coxeter, which led to his being called Donald for short. 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.We investigate a 3-dimensional analogue of the Penrose tiling, a class of 3-dimensional aperiodic tilings whose edge vectors are the vertex vectors of a regular icosahedron.It arises by an equivariant orthogonal projection of the unit lattice in euclidean 6-space with its natural representation of the icosahedral group, given by its action on the 6 icosahedral diagonals (with orientation).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. Ziegler, 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.