Bill Thurston 1946-2012

Bill Thurston died on the evening of August 21st. His son Dylan writes “He was surrounded by family, and went very peacefully, after a fight with melanoma since spring 2011. Please pass this on as appropriate.”

I knew Professor Thurston only through his writing, first in publication and recently on Mathoverflow. It was always a joy. I have avoided more routine obligations than I care to admit by reading and rereading his papers. He believed that mathematics was a fundamentally human task, and that his goal was not simply to provide the reader with a bulleted list of truths, but to provide a picture and an intuition that made them obvious.

I had the thought to organize a blogfest, where various math bloggers would write up expositions of some aspect of his work. But I ran into an obstacle: What subject did I think I could explain better than he did? So, instead, here are some of my favorites, for your pleasure and inspiration:

Simpler problems about planar graphs, solved by geometric insight

Rotation distance, triangulations, and hyperbolic geometry How hard is it to get from one triangulation of a polygon to another?

The absence of efficient dual pairs of spanning trees in planar graphs Non-obvious reasons that you might not be able to find low diameter dual pairs of spanning trees.

Shapes of polyhedra and triangulations of the sphere Can we describe all geodesic-dome-like grids?

Groups, tilings and finite state automata The height functions from Sections 4 and 6 are crucial tools in the study of statistical properties of dimer tilings; the other 9 sections are still, in my opinion, underexploited.

What upper bounds are known for the diameter of the minimum spanning tree of $n$ points in $[0,1]^2$?

Visualization and intuition

How to see $3$-manifolds I cannot improve on this title

Is there an underlying explanation for the magical powers of the Schwarzian derivative?

The human side of mathematics

What’s a mathematician to do? What is the role of mathematicians who do not revolutionize their field?

On proof and progress in mathematics One of the best essays ever on the subject of what it is we are trying to do