Wednesday, 16 December 2015

The Geometric Beauty of a Troubled Mind

Approaching the fifth anniversary of the death of a little known mathematical dilettante,  I wanted to celebrate the season with  beautiful geometry .

I first heard of George Phillips Odom, Jr. while reading King of Infinite Space, the biography of Donald Coxeter by Siobhan Roberts. He reappeared in her biography of John H. Conway and I determined that I needed to learn a little more. His story is both beautiful and sad, and I recommend both of these works, or a section of The Coxeter Legacy: Reflections and Projections by Doris Schattschneider to get more of his story.  Sadly, he died on December 18, 2010

As a grossly oversimple start to his history, Odom spent some forty years at the Hudson River Psychiatric Center in Poughkeepsie, suffering from depression. During that time he has communicated with a limited group of outstanding geometers, including Coxeter and Father Magnus J. Wenninger who may be the only person in the world who has made a paper model of all 75 uniform polyhedra. He was visited by Conway who used his work to improve on Euclid's construction of the dodecahedron. Odom, it seems was unfamiliar with Conway's work.

Among other things, Odom has found five different simple geometrical approaches to the golden ratio using equilateral triangles, and platonic solids. It is these five simple approaches that I want to celebrate here. If I didn't know them, I'm sure lots of teachers and students out there have not either, and they are too beautiful to be so unknown.

Shortly after his commitment, and during his early communication with Coxeter, he sent Coxeter one of the simplest and most beautiful constructions of the golden ratio ever. How it remained unknown for over 2000 years should inspire geometry students to believe there is still fresh stuff out there waiting for them. Coxeter turned the construction into a problem and posted it in the American Mathematical Monthly in 1983.In that fashion, I will leave each proof to the reader. 

Let A and B be midpoints of the sides EF and ED of an equilateral triangle DEF. Extend AB to meet the circumcircle of DEF at C. Show that B divides AC according to the golden section.
Odom has another, somewhat more involved golden ratio using an equilateral triangle.  In this one you first drop an altitude CD from vertex C of the triagle.  Then from the foot of the altitude, D, draw two quarter circles centered at D, one of length CD, and one of length DB as illustrated.  Now extend the altitude CD beyond D to intersect with the quarter circle from B at the point E.  Finally draw a ray EA intersecting the other arc at point F.  A divides EF in the golden section.

This beautiful illustration stumped Coxeter at first, according to the section by Schattschneider mentioned above.   This came to Odom while studying the vertices of a regular icosahedron, which coincide with the vertices of  three golden rectangles.

The three remaining constructions involve Platonic sections and their circumspheres.

The first involves a cube.

 If one connects the centers A and B of any two adjacent cube surfaces and extends the connecting line segment again, such that the extended line intersects the circumscribed sphere at C, then B will divide AC according to the golden section

Another uses the tetrahedron.  If the midpoints, A and B, of any two adjacent  edges of the tetrahedron are connected, and extended to intersect the circumsphere at C, then B divides AC in the golden section as well.  The following image of a tetrahedron sharing the vertices of a cube explain why these are the same.  Observe that the midpoint of any edge of the tetrahedron is the midpoint of the face of the cube. 
A third view of this same phenomenon is given by inscribing an cctahedron inside the cube with it's vertices at the center of the faces.  For any edge AB of the octahedron extended to the  circumcircle is the same as the segment ABC in the cube.

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