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."

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).

Dimensional analogy. Click to Enlarge.

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.


ACTIVITY

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MYSTERY

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.

Explore Further

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.

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