This is the late Bill Thurston, wrapping a tube around designer Dai Fujiwara at the Issey Miyake fashion show in Paris in the spring of 2010. A black and white version of this image was reproduced in Chapter 6 of *Mathematics without Apologies* and more recently on the Scientific American website.

Thurston was one of the all-time great geometers. He revolutionized the theory of foliations and brought hyperbolic geometry into the center of the study of 3-manifolds. His geometrization conjecture extended the Poincaré conjecture for 3-dimensional manifolds and, together with (Columbia colleague) Richard Hamilton’s ideas about the Ricci flow, provided a framework for Grigori Perelman’s successful solution of both problems. Among mathematicians of his stature, Thurston was also one of the most original thinkers about the meaning and the goals of the practice of mathematics; his article *On Proof and Progress in Mathematics* has influenced a generation of philosophers.

Here are a few photos of the Thurston/Issey Miyake show, among the first I ever took with a telephone (and it unfortunately shows).

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Michael BaranyYour choice of “revolutionize” for the verb for Thurston’s foliation work struck me as interesting in light of how he himself describes it in “Proof and Progress”. There, his verbs and verb phrases are “proved” “wrote” “built up a backlog”, and he puts in others’ voices “cleaning out” and “killing” (and also describes the process as “evacuation”: of the people working on foliations but also implicitly of the problems in the field). As you know, “revolution” is a very loaded term in the history of science (and, indeed, in history); mathematicians use it a lot, but in situations like this it’s not so clear to me what it means or implies.

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mathematicswithoutapologiesPost authorThurston was certainly being modest about his own contributions but you are right to flag the word “revolutionize.” I used the word uncritically; in other words, as a mathematician, not a critic, would use it, and just to enliven the text, not for its Kuhnian associations. I don’t actually know enough about the field of foliations to be able to say whether Thurston’s work represented anything like a new paradigm.

The lessons Thurston draws in “Proof and Progress” about the “evacuation” of the field are worth rereading — not least by me, grappling with what may or may not be an authentically new paradigm in my own speciality.

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