Geometric Algebra

Hmm... Geometric Algebra.

"It contains, in a fully integrated manner, linear algebra, vector calculus, differential geometry, complex number and quaternions as real geometric entities, and lots more. This powerful language is based in CliffordAlgebra."

It looks like a very promising tool for programming related to physics, geometry, and graphics. It would be interesting to hear from people who've applied it to real-world programming problems.

An overwhelmingly compelling motivation for switching the geometrical foundations of mathematical physics to GeometricAlgebra/CliffordAlgebra is provided in the manifesto by DavidHestenesOerstedMedalLecture 2002: "Reforming the Mathematical Language of Physics"; the following is a key excerpt showing nine areas that are thus unified. -- DougMerritt

"Limitations of mathematics are evident in the fact that the analytic geometry that provides the foundation for classical mechanics is insufficient for General Relativity. This should alert one to the possibility of other conceptual limits in the mathematics used by physicists. Since Newton's day a variety of different symbolic systems have been invented to address problems in different contexts. Figure 1 [bullet list below] lists nine such systems in use by physicists today. Few physicists are proficient with all of them, but each system has advantages over the others in some application domain. For example, for applications to rotations, quaternions are demonstrably more efficient than the vectorial and matrix methods taught in standard physics courses. The difference hardly matters in the world of academic exercises, but in the aerospace industry, for instance, where rotations are bread and butter, engineers opt for quaternions."

"Each of the mathematical systems in Fig. 1 incorporates some aspect of geometry. Taken together, they constitute a highly redundant system of multiple representations for geometric concepts that are essential in physics. As a mathematical language for physics, this Babel of mathematical tongues has the following defects:
  1. Limited access. The ideas, methods and results of theoretical physics are distributed broadly across these diverse mathematical systems. Since most physicists are proficient with only a few of the systems, their access to knowledge formulated in other systems is limited or denied. Of course, this language barrier is even greater for students.
  2. Wasteful redundancy. In many cases, the same information is represented in several different systems, but one of them is invariably better suited than the others for a given application. For example, Goldstein's textbook on mechanics12 gives three different ways to represent rotations: coordinate matrices, vectors and Pauli spin matrices. The costs in time and effort for translation between these representations are considerable.
  3. Deficient integration. The collection of systems in Fig. 1 is not an integrated mathematical structure. This is especially awkward in problems that call for the special features of two or more systems. For, example, vector algebra and matrices are often awkwardly combined in rigid body mechanics, while Pauli matrices are used to express equivalent relations in quantum mechanics.
  4. Hidden structure. Relations among physical concepts represented in different symbolic systems are difficult to recognize and exploit.
  5. Reduced information density. The density of information about physics is reduced by distributing it over several different symbolic systems."

Some relevant links have been moved to CliffordAlgebraResources.

See also CliffordAlgebra, in particular for discussion of why the algebra has two names. See also QuaternionMathematics ComplexNumbers MatrixAnalysis

CategoryMath

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