My friend and colleague William Thurston, who died on August 21, 2012, was one of the most original and powerful mathematicians of his time. Accounts of his scientific work and influence will appear elsewhere; what I'd like to do in this column is give an impression of what it was like to interact with him as a fellow mathematician or as a student.

Talking to Bill was being able, for a short time, to experience the world of mathematics as a visible and almost palpable reality. His own perceptions of mathematical objects and procedures were so vivid that they would come to almost physical life in the space around him. The video segments embedded in this page may give an idea of this phenomenon.

I first met Bill in 1971, when
he was a graduate student at Berkeley. We were in fairly regular
contact for the next twenty-odd years. In 1992 he was back at
Berkeley, and during one of my visits there I asked him to
contribute to a series of video recordings I was making:
a mathematician would explain some geometrical aspect of knot theory.
Bill agreed, and suggested that we illustrate
how different knots determine different
branched coverings of 3-space.

Imagine that a branching of order
two occurs around each strand of a knot: go around once and you're
in a different world; go around twice and you're back where you
started. If we're dealing with a simple closed loop ("the un-knot"), that's
the end of the story. It is completely analogous to the one-sided
property of the Möbius strip: go around once and you're on
"the other side," twice and you're back where you began. But if the
loop is knotted, the topology of the knot makes the process more
complex and much
more interesting.

The next day Bill had a class scheduled, and invited me to record him
explaining the same process to his students. The video segment above
was assembled from several takes, and edited for the most efficient
delivery of the narrative. In class this was not possible. The camera
was in one place and ran uninterruptedly for sixteen minutes. (To
conform with YouTube size restrictions, the record has been split into
two).

In the *New York Times*
obituary Bill's son Dylan is quoted as
saying: "Math was always very fun for him." I hope these video records
show how much fun it was for those around him.