Donald Coxeter“The Man Who Saved Geometry”

Nathan Cormier

April 10, 2007

Who is Donald Coxeter?

• Donald Coxeter was a classical geometer• As a classical geometer his goals were not just

about proving theorems but they were more aimed at finding “gemlike” geometric objects

• He explored and enumerated many different geometric configurations and showed how they related to each other through their symmetrical properties

Symmetry

• According to Coxeter Symmetry is “the unifying thread that runs through all his work”– “sym” – means together– “metry” – means measure

• There are 4 basic types of Symmetry1. Bilateral Symmetry

2. Rotational Symmetry

3. Translational Symmetry

4. Many different combinations of the above

Symmetry

• In geometry, an object is symmetrical if it looks the same after being subjected to a gemetric change such as a rotation or reflection.

• Eg. The Sphere

Symmetry

• This means the sphere is invariant or unchanging under an infinite number of symmetry operations

• Coxeter did not find these shapes interesting and preferred to work with shapes that had discrete symmetries

• A basic example of this is a square which has only 8 symmetries.– There are only 8 ways that its position can be changed

but will leave the square looking exactly the same.

8 Symmetries of a Square

• No action. • Rotate anticlockwise 90 degrees. • Rotate anticlockwise 180 degrees. • Rotate anticlockwise 270 degrees. • Reflect across the vertical (y) axis. • Reflect across the horizontal (x) axis. • Reflect across the diagonal y = x. • Reflect across the diagonal y = –x.

8 Symmetries of a Square

Group Theory

• A group in math is a set of actions that preserve an objects appearance

• For the square the group would be the set of 8 actions that preserve the square’s appearance

• These consist of the symmetry operations from the previous slide

The Coxeter Group

• Coxeter looked a much more complex shapes then a square

• He studied how the facets of a crystal align perfectly that makes it a highly symmetrical object

The Coxeter Group

• A Coxeter group is a finite group of symmetries• It consists of a finite number of rotations that will

preserve a crystals appearance

Polytopes

• Coxeter extended his work with symmetries into multiple dimensions

• In Hyperspace the shapes rotate and reflect upon themselves

• These shapes are called Polytopes meaning “many shapes”

• Coxeter was nicknamed “Mr. Polytope because he enjoyed working with them so much

Examples of Polytopes

4D Hypercube

6D Zonohedron

4D Simplex

5D Simplex 1 and 2

4D Polytope with 24 cells

Coxeter’s Polytopes

• Coxeter wrote a book on Polytopes titled “ Regular Polygons” which became a best seller

• Coxeter is often compared to Charles Darwin – Coxeter did for Polytopes what Darwin did for organic

beings – He classified and quantified their very existence

The Savior of Geometry

• While Coxeter was in the prime of his career geometry was slowly being taken over

• Algebra and Analysis were slowly becoming the popular mathematics

• According to E.T. Bell math was “all equations and no shapes, like prose without poetry”

The Savior of Geometry

• Walter Whiteley may have given Coxeter the ultimate compliment about his work when he said “should classical geometry become extinct there would be a geometry gap that would haunt us forever … Donald Coxeter did much to save us from such a loss”

The Savior of Geometry

• Coxeter became geometries “apostle” by the end of his career

• He ignored the “fad” fashions in math and continued to work with the shapes he loved to work with and preserved the classical traditions of geometry

The Savior of Geometry

• Because of his love for his work Coxeter preserved Geometry through its “lean” years and would not let it go extinct.

• Coxeter is a hero for many mathematicians around the world who may not have been able to study what they do if Donald Coxeter had not saved geometry.

Questions???