Learning Geometric Algebra

[PatrickPowers]

I read/skimmed through maybe six geometric algebra books but wasn't getting it. The first problem was that all the books were the same, copies of some original written in the author's own words. The second problem was that I felt like trying to swim while dragging an anchor. Eventually I figured out what at least part of the problem was. They all relied on showing how familiar things like complex numbers could be done in GA. Well, if I already know complex numbers then what's the point of doing it some superficially different way? It's just clutter. What I want to know is what GA can do that familiar systems can't do, and I wasn't getting that. I wasn't even getting any understanding of the very foundation of GA, the geometric product. Most authors concentrated on familiar cases and ignored the general case. One guy tackled it and came up with a formula half a page long. There must be something simpler.

What I want is a text that starts with a blank slate, ignoring the past and building everything from scratch. If GA is superior then the result will be better, right? So forget about vectors, forget about complex numbers, forget quaternions. Show me your stuff direct. No one else is going to do that so if I want this I'm just going to have to do it myself.

The very terminology of GA is rooted in the past. Bivectors, trivectors. That's not what it's really about. Forget about vectors. Forget about the basis of the space, that just complicates things. The essence of geometric algebra is subspaces. Lines, planes, and volumes. Infinite closed subspaces. That's the place to start. You are better off not even choosing a space to work in. Don't fix a N dimensions for your space. Instead think of everything as a subspace. Instead of a vector you have a 1-space, a line. Instead of a bivector you have a 2-space that is a plane, and so forth. So what happens when you multiply two spaces? Not one to stand on the sidelines and throw stones I got a geometric algebra interpreter and let it give me the results. The geometric product of two subspaces is like the symmetric difference of those subspaces, or the exclusive or of logic. It can be explained as follows.

First an imprecise version.
------------
Let s₁ and s₂ be two subspaces.
Let S be the smallest space that contains both subspaces.
Let s be the intersection of the two subspaces.
The geometric product of the two subspaces, s₁*s₂, is the largest subspace of S that excludes s.

Next in the more exact jargon of vector algebra
-----------
Let S be the span of s₁ and s₂. S = span(s₁,s₂)
Let s be the largest subspace that is a subspace of both s₁ and s₂.
s₁*s₂ is the dual of s in S.

OK, so then what use is the scalar attached to each subspace? Subspaces are infinite, so it can't be a size. There are two uses. The first is as a scaling factor when the subspace is used as an operator. For example, a 2-space is often used to rotate other subspaces. The scalar tells you how much to rotate whatever. The second use of the scalar is to specify a specific point in a 1-space. But quite often you don't care about a specific point, you just want the 1-space. So instead of a vector space it is better to think of GA as a space space. It's a ring, so one may call it a space ring. Each space has an inverse, one of the main motivations behind GA. But the sum of spaces usually doesn't have an inverse. So it is not a division ring. Some elements have inverses, others don't.

The scalars are also useful for specifying spaces by adding together basis spaces. But one of the advantages of GA is that you don't have to think about the basis. Just let a computer do it for you. That was a big gripe I had about the texts. They are preoccupied with basis vectors, heavily focused on them. That's shooting yourself in the foot, thinks I. I didn't get far until I got out of this frame of mind. The basis matters only when you are done and want to draw a figure on a 2D screen or otherwise relate to our real world. I think that's largely where the "dragging a boat anchor" feeling came from.

So in GA just what is a vector? There is a 1-space, a distance, and a sign. The 1-space is a line through the origin. The distance specifies an n-sphere centered at the origin with radius of that distance. The intersection of the n-sphere with the line is two points. The sign tells you which point to choose. That’s a vector.

A very nice feature of GA is that you don't need to worry about signs. There is no right hand rule or anything like that. Signs take care of themselves.

Now a 1-space together with a scalar, isn't that a two dimensional space? Indeed it is. But we are stuck with this.

One author mentioned that the scalars could be complex numbers. Why introduce that when complex numbers are a subset of GA? That inspired some silly speculation on my part. The ostensible scalars could be another full fledged GA. That "scalar" GA could then be have yet another GA as its scalars and so forth ad infinitum. As they say, "it's turtles all the way down."

[mr_e_man]

The geometric product of two subspaces is like the symmetric difference of those subspaces, or the exclusive or of logic. It can be explained as follows.

[...]

That generally works only if the subspaces are parallel or perpendicular, not at an arbitrary angle.

One author mentioned that the scalars could be complex numbers. Why introduce that when complex numbers are a subset of GA? That inspired some silly speculation on my part. The ostensible scalars could be another full fledged GA. That "scalar" GA could then be have yet another GA as its scalars and so forth ad infinitum. As they say, "it's turtles all the way down."

The scalars need to commute with everything in the algebra. In particular, they need to commute with each other. I think they also need to be invertible (except 0); linear algebra over a commutative ring without invertibility doesn't always give a well-defined dimension. (The maximal size of a linearly independent set may be smaller than the minimal size of a spanning set.)

[quickfur]

I also struggled with GA a lot the first time I encountered it. I was looking for a definition of the geometric product so crucial to the entirety of GA, but could not find one that satisfactorily explain what the geometric product was.

Eventually, the light bulb went on, and I realized that the reason I was having trouble with it, was because I was looking at it from the wrong angle. I was unconsciously looking for a definition in terms of, perhaps, highschool arithmetic involving +, -, *, /, and so forth, or perhaps some calculus or some obscure discrete mathematical operation, but none of the definitions I found had that. So I was frustrated by the abstractness of it. What was this elusive geometric product that's so central to GA, yet nobody seemed to present a working definition of, that I could use to actually calculate some of the results by hand to see how it worked? As it turned out, that totally misses the point of GA.

The whole premise of GA is to postulate the geometric product as an axiom. That is, we first start by assuming the existence of an operation called the geometric product, that satisfied certain axiomatic properties. Then, by making use of these properties, we may deduce all sorts of interesting identities involving the geometric product -- the totality of which is GA. The postulated geometric product is abstract -- we don't define what it is, besides the GA axioms that it satisfies; because these axioms are its definition!!!. The geometric product is the (abstract) operation that satisfies the axioms of GA, and by manipulating these axioms, we may derive all of the concrete properties of the geometric product, and indeed, the entirety of GA. That's the beauty of it. You assume the existence of this product and the axioms it satisfies, and "magically" everything else in GA "falls out" of it -- the dot product, the cross product, vectors, bivectors, polyvectors, etc., the entire structure of GA unfolds from this one starting point: the abstract operation we call the geometric product and its associated set of axioms. Even how you calculate specific instances of the geometric product may be derived as a natural consequence of these axioms. So in a sense, it "defines itself" once you accept its defining axiomatic system. Everything is self-consistent and coherent, and produces a rich structure of interactions that may be used to compute all sorts of things about geometry, without any weird edge cases like cross products existing only in 3D. Instead, you have bivectors that accurately models all the properties of the cross product in 3D, yet trivially generalizable to all dimensions, thus allowing easy dimensional analogy of lower-dimensional systems like 3D physics to 4D and beyond. Marvelous.

[PatrickPowers]

The geometric product of two subspaces is like the symmetric difference of those subspaces, or the exclusive or of logic. It can be explained as follows.

[...]

That generally works only if the subspaces are parallel or perpendicular, not at an arbitrary angle.

It took a month, but I finally managed to get this intuition into a satisfactory (I hope) form, which I just "published" as an answer on math stackexchange.

https://math.stackexchange.com/questions/444988/looking-for-a-clear-definition-of-the-geometric-product/4102677#4102677

One author mentioned that the scalars could be complex numbers. Why introduce that when complex numbers are a subset of GA? That inspired some silly speculation on my part. The ostensible scalars could be another full fledged GA. That "scalar" GA could then be have yet another GA as its scalars and so forth ad infinitum. As they say, "it's turtles all the way down."

The scalars need to commute with everything in the algebra. In particular, they need to commute with each other. I think they also need to be invertible (except 0); linear algebra over a commutative ring without invertibility doesn't always give a well-defined dimension. (The maximal size of a linearly independent set may be smaller than the minimal size of a spanning set.)

My "silly speculation" was a joke, but it was based on a sensible idea. The scalars can be any field. I have seen it suggested that such scalars could be complex numbers. Well if GA is better it seems to me that instead of complex numbers one should use the complex number subset of GA as scalars. Or any subset of GA that is a field.

[quickfur]

Why limit yourself to complex numbers? Why not use the surreal numbers instead?

I betcha that GA over the surreal numbers would allow us to define and manipulate infinite-dimensional polytopes in a consistent way. The last time I tried to define infinite-dimensional polytopes, I ran into problems with scalars being finite (the outradius of an infinite-dimensional hypercube is infinite, so rotating one of its vertices onto a coordinate axis is not possible). But with surreal scalars, there is no longer such a problem, and we can perform closed computations with infinite-dimensional polytopes. Then GA would allow us to define infinite-dimensional physics.