A Geometric Algebra Invitation to Space-Time Physics, Robotics and Molecular Geometry

Is there one method of multiplying vectors in spaces of any dimension or several? The discoveries of Grassmann, Hamilton, Pierce, and Clifford in the nineteenth century cast light on this question but it has taken a while for their theoretical constructions to reach scientists and engineers involved in the use of these concepts. At times the various schools of the mathematicians above have vigorously disagreed about which structure is best (the famous invention of “vector analysis” at the hands of J.W. Gibbs from the wreckage of quaternion analysis comes to mind as an instructive case). The book under review is an abbreviated introduction to Geometric Algebra and some of its uses.

The approach to this subject is seemingly modeled after that of David Hestenes. It follows the central code of abstract extension of subjects (“to create a virtue out of necessity”) but chooses to do so in a fairly direct way. There are only a few pictures to motivate the subject and the geometric product emerges from some prestidigitation with the Grassmann algebra. Both internal (vector decompositions, for instance) and external (symmetry groups of vector spaces) elements are present at the beginning but the transition back and forth might be a tad confusing for a beginning reader. If such a reader is an engineer with only vector calculus on 3-dimensional space as background, this will not be an easy ride. On the other hand, most practicing mathematicians will find the book an easy read with some startling results along the way. The geometric algebra of Clifford really does give a very quick route to a number of attractive topics (including the exponential mapping, Sylvester’s law of inertia, the Dirac representation). No tensors are used or needed in the exposition and the famous “debauch of indices” is nowhere in sight.

The applications featured are interesting examples of the utility of the operations. The Geometric Algebra approach to Minkowski Geometry is fairly standard and allows one to see a lot of interesting geometrical views of special relativity. Perhaps the most engaging application was the chapter devoted to molecular geometry. Here molecules are thought of as spheres that surround vertices in a graph. Distance geometry is modeled quite neatly using a geometric algebra approach and allows us to play with configurations that minimize total energy functions. At just about 120 pages this book offers a brisk and exciting view of the many roles of Geometric Algebra.

Jeff Ibbotson holds the Smith Teaching Chair in Mathematics at Phillips Exeter Academy.