Stand up and look around. Walk in a circle. Jump in the air. Wave your arms. You are a collection of particles moving about within a small region of a 3-manifold--a three-dimensional space--that extends in all directions for many billions of light-years.

Manifolds are mathematical constructs. The triumph of physics since the time of Galileo and Kepler has been the successful description of reality by mathematics of one flavor or another, such as the mathematics of manifolds. According to physics, everything that happens takes place against the backdrop of three-dimensional space (leaving aside the speculations of string theorists that there are tiny dimensions in addition to the three that are manifest) [see "The Theory Formerly Known as Strings," by Michael J. Duff; Scientific American, February 1998]. Three dimensions means that three numbers are needed to specify the location of a particle. Near Earth, for instance, the three numbers could be latitude, longitude and altitude.

   
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