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Geometric intuition
One day, perhaps, Cliords algebra will be taught routinely to students in place of vector analysis.
Richard Feynman once made a statementto the eect that the history of mathematics is largely the history of improvements in notation the progressive invention of ever more efficient means for describing logical relationships and making them easier to grasp and manipulate. The Romans were stymied in their eorts to advance mathematics by the clumsiness of Roman numerals for arithmetic calculations. Aer Euclid, geometry stagnated for nearly2,000 years until Descartes invented a new notation with his coordinates, which made it easy to represent points and lines in spacealgebraically.
Feynman himself, of course, introduced into physics a profound change in notation with his spacetime diagrams for quantum eld theory. Previously, writing out the terms in an innite series for a probability amplitude involved a laborious algebraic procedure, which Feynman replacedwith simple pictures and explicit rulesto translate them into mathematical expressions. This was an advance in housekeeping, if you will, but also among the most important advances in twentieth-century mathematical physics.
However, one of the most important and elegant advances in mathematical notation has perhaps not yet achieved the wide recognition it deserves. In 1873, the English mathematician and philosopher William Cliord invented a deceptively simple algebraic system unifying Cartesian coordinates with complex numbers, and oering a compact representation of lines, areas and volumes, as well as rotations,in 3-space. In more advanced physics, Cliords algebra he called it geometric algebra is now well recognized as the natural algebra for describing physicsin 3-space, but it hasnt yet caught on in engineering, or...