"Through the Looking Glass," by Dana Mackenzie. The Sciences, May/June 1997, pages 32-37.

This article explores the notion of "noncommutativity" as it arises in mathematics, physics, and the everyday world. Two operations---such as "put on your socks" and "put on your watch"---are commutative if it makes no difference in which order they are performed. Two operations are noncommutative---"put on your socks" and "put on your shoes"---if changing the order in which they are performed yields a different result. The article brings together vivid, accessible examples from mathematics that illustrate the importance of this notion. One example is mathematician John H. Conway's method for bringing order to a knotted tangle of two lengths of string---a method that ingeniously translates noncommutative operations on the strings into noncommutative operations with numbers. The article also touches on the recent work of mathematician Alain Connes, which holds the potential for using noncommutative structures to unify the fundamental forces of physics.

--- Allyn Jackson

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