One math to rule them all, one division ring to find them, one algebra to bring them all and with notation bind them.

David Hestenes, a professor of physics at Arizona State University, received the 2002 Oersted medal from the American Association of Physics Teachers for his rediscovery and application of Clifford or Geometric Algebra (GA) to putting physics in a coherent and clear form. (Richard Feynman was particularly proud of receiving the Oersted, and many of the greatest names in physics also received the medal.)

Geometric algebra gives a single notation for classical mechanics, relativity, and quantum mechanics while also giving a visualizable geometric interpretation to all of these, replacing imaginary numbers, matrices and tensors with more understandable equivalents and greatly simplifying equations and calculations. For instance, the relativistic version of Maxwell's equations of electromagnetism can be usefully condensed into just four symbols.

GA has applications in computer vision, computer graphics, robotic path planning, signal processing, neural computing, AI, structural engineering, advanced physics and more. Black hole physics, the Dirac equation, quantum uncertainty, and particle scattering have become much easier to handle using GA. The recent "amplitudihedron" discovery (which allows pencil and paper calculation of particle scattering properties which previously could not be done on a supercomputer using Feynman-diagram based methods) is closely related to to GA.

Geometric algebra is a generalization of vector algebra (oriented lengths) to higher-dimensional objects (oriented planes, volumes, etc.) Perhaps the heart of its power is its ability to cleanly represent any sequence of rotations of any sort of object in any number of dimensions and any sort of space (mixture of positive and negative-square dimensions). All sorts of geometries can be represented - Euclidean, hyperbolic, spherical, projective and conformal.

I think any kid who is ready for algebra is capable of GA, and learning GA will lead to much greater ability both in applied and pure math.

A good introduction may be found here:
http://www.science.uva.nl/ga/publications/index.html
Scroll down to: "Geometric Algebra: a computational framework for geometrical applications" (2 parts)

And software for calulation and visualization, along with tutorials are here:
http://www.science.uva.nl/ga/viewer/index.html

The Cambridge GA group's papers are also especially worth reading: www.mrao.cam.ac.uk/~clifford/‎

Jaap Suter's GA Primer is also a good reference:
http://www.jaapsuter.com/geometric-algebra/
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"Reeling and Writhing, of course, to begin with," the Mock Turtle replied...-- Lewis Carroll