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Donald Coxeter “The Man Who Saved Geometry” Nathan Cormier April 10, 2007

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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

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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 Symmetry 1.Bilateral Symmetry 2.Rotational Symmetry 3.Translational Symmetry 4.Many different combinations of the above

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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

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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.

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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.

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8 Symmetries of a Square

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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

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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

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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

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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

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Examples of Polytopes

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4D Hypercube

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6D Zonohedron

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4D Simplex

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5D Simplex 1 and 2

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4D Polytope with 24 cells

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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

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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”

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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”

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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

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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.

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2.4 –Symmetry. Line of Symmetry: A line that folds a shape in half that is a mirror image.

2.4 –Symmetry. Line of Symmetry: A line that folds a shape in half that is a mirror image.

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