"Through the Looking Glass," by Dana Mackenzie. The Sciences, May/June1997, pages 32-37.
This article explores the notion of "noncommutativity" as it arises inmathematics, physics, and the everyday world. Two operations---such as "put onyour socks" and "put on your watch"---are commutative if it makes no differencein which order they are performed. Two operations are noncommutative---"put onyour socks" and "put on your shoes"---if changing the order in which they areperformed yields a different result. The article brings together vivid,accessible examples from mathematics that illustrate the importance of thisnotion. One example is mathematician John H. Conway's method for bringingorder to a knotted tangle of two lengths of string---a method that ingeniouslytranslates noncommutative operations on the strings into noncommutativeoperations with numbers. The article also touches on the recent work ofmathematician Alain Connes, which holds the potential for using noncommutativestructures to unify the fundamental forces of physics.
--- Allyn Jackson