The topic of geometric algebra is a new fascination for me. I first read about David Hestenes while reading Bret Victor’s kill maths project. Then got a copy of David Hestenes’ new foundations of classical mechanics book. One thing led to another and then I ended up reading an old out of print book by Clifford called common sense in exact sciences.
The basic premise of geometric algebra seems fascinating to me: a universal and simple mathematical theory for a wide variety of applications in Physics. The idea is simple: commonly defined vector addition has a geometric interpretation. You can see two vectors being added (say in 2 dimensions), look at the result and sort of nod the head saying, yeah that makes sense. But turn to vector dot product and something looks iffy. You are doing a multiplication of two vectors but get back a scalar quantity. Contrast this with multiplication of numbers. Number multiplication, when visualized, is intimately related to squares and rectangles. When multiplying two numbers a and b, we can think of taking a units (like marbles or beans of whatever) and swipping them over b. This idea is wonderfully expanded in Lara Alcock’s book Mathematics Rebooted. This comparison with number multiplication hints at one thing: there’s a probably a way to formulate vector multiplication to have similar geometric interpretation. Geometric algebra starts with the same premise: it defines a wedge/outer product of two vectors which has similar geometric visualization or interpretation. If you have two vectors A and B, imagine putting vector A at one end of vector B and then swip it across the length of B. What you get is a parallelogram representing wedge product of A and B. This result itself is not a scalar quantity. The magnitude of multiplication is related to the area of the parallelogram. But there’s a sense of direction in this operation depending upon how vector A and B are oriented. We can also think of this result as an oriented area.
Two important links on the introduction to geometric algebra are: this researcher’s webpage on geometric algebra and this beautiful video introduction to rotors by Marc ten Bosch
Now one may ask what does this kind of product buy you. Hestenes and other proponents of geometric algebra claim that using this formalism simplifies many areas of physics. They also highlight that in certain areas of physics (like relativity), you need a deep background in specific maths (tensor calculus), whereas you can use geometric algebra (and its extension geometric calculus) instead. I am yet to verify these claims for myself, but just the possibility of achieving this is a solid ground for studying geometric algebra further.