Some quick summary / commentary to motivate reading the link:
- By the 19th century, the complex numbers had become a generally acceptable concept, for example with lots of attempts to prove the fundamental theorem of algebra. The idea that the complex numbers could be thought of as 1–1 mapped to a two-dimensional plane was widely known. Complex numbers used as an algebraic proxy for points in the plane were useful for solving geometry problems that Cartesian analytic geometry couldn’t manage.
- Many thinkers were trying to extend these ideas to three-dimensional space. Hermann Grassmann, who invented GA in the 1840s, was one of many, but in many ways he got things “right”. Another thinker who worked extensively on this problem was Hamilton, who devoted decades to the quaternions (a sort of generalization of the complex numbers with three imaginary units i, j, k instead of one) which can be quite useful for characterizing rotations and are still used in niche physics applications even though they’ve been otherwise marginalized by matrix approaches.
- It turns out that there is a quite natural relationship between imaginary numbers, the vector part of a quaternion, and geometric algebra. As Hestenes describes in the linked essay, the imaginary unit, a “bivector”, can be thought of either as an operator which performs a 90° rotation on vectors in the plane, or as a “directed area” with unit magnitude. A quaternion’s vector part is just a combination of three unit bivector components of geometric algebra in 3-dimensional space. This gives complex numbers and quaternions a nice intuitive geometrical interpretation, and makes them a lot less foreign seeming to students.
- The traditional/mainstream conception of vector algebra (which ended up winning the 19th century fight among tools for the algebra of space) uses a dot product and a cross product, which are defined to be, respectively, a scalar and a vector. Cross products constantly come up in the physics of rotations, and in trying to compute areas and volumes. Under GA, the “outer product”, mathematically dual to the cross product, is seen to be a bivector. This is a much better description because the behavior of vectors produced by cross products is quite different than other vectors, which means that considering them to be the same type of object results in a great deal of confusion.
- Geometric algebra has a natural relationship with projective geometry (homogeneous coordinates, Plücker coordinates). Such mathematics is used constantly in applications like computer graphics.
- Physicist David Hestenes has spent most of his career since the 1960s elaborating and advocating GA. In exploring various areas of physics, he found that in almost every case GA was at least as convenient as the alternative mathematical formulations, and often quite insightful. For instance, some quantum phenomena which were previously considered “counterintuitive” or strange become quite obviously necessary and natural conclusions of their mathematical context in GA.
- GA is still far from mainstream. As far as I can tell, it is taught to almost no undergraduates even though it is pedagogically and operationally superior to the alternatives. It mainly comes up in very technical advanced physics courses and extremely abstract mathematical proofs. Reading some of these, one would never realize how broadly applicable and powerfully connective its ideas are.
- Geometric Algebra is (I firmly believe) the future of geometric education. It should be taught to all students of mathematics, physics, engineering, computer graphics, computer vision, and similar fields, perhaps starting in high school. It simplifies both understanding and computation; for instance, many problems can be done coordinate free, making their solutions easier to tackle and more intuitively relevant.