Introducing geometric algebra

Transcript

Geometric algebra has been making headlines lately. More and more people are picking it up as a tool for computer graphics or as a framework for understanding physics more intuitively. It's an extremely powerful and flexible set of concepts and we will spend quite a long series of videos exploring it. If I had to summarize geometric algebra in only a few sentences, it would probably go something like this. We will construct and discover a set of tools that allows us to manipulate vectors, but also lines and planes and volumes and other geometric shapes and even entire spaces.

These new tools will always have a very clear and intuitive geometric purpose such as rotating or reflecting objects. But at the same time, these tools will also have simple elegant algebraic expressions. By calculating with those expressions, we automatically get the correct geometric answers. We combine geometry and algebra in an extremely useful way. I hope that already sounds cool enough to get you interested.

If I had a little bit more time to summarize geometric algebra, this is what I would say. We are going to invent a brand new product between vectors. We want our product to be linear, which is just a fancy way of saying that we want it to distribute over sums and linear combinations. That's great because it allows us to pull algebraic expressions apart into smaller pieces which will all be composed from a small set of basis vectors. We can then just focus on how the new product behaves on those basis vectors.

And that is also pretty easy because there are really only two rules. When we multiply a basis vector with itself, the result is one. More generally, whenever two vectors are parallel to each other, their product contracts to a real number. But when we multiply two different basis vectors, we get a new kind of object called a bvector. More generally, whenever two vectors are perpendicular to each other, they multiply to a bi vector, which you can think of as a little square shape.

For now, those are the only rules. Parallel vectors get treated one way and orthogonal ones get treated a different way. The geometric concepts of being parallel and being orthogonal are dual to each other. And we will be able to express this duality in very precise terms. When our two vectors are not parallel and also not orthogonal but a mix of both their product will naturally be a mix of scalers and bctors and this is where most of the power of this new product comes from.

It will allow us to detect orthogonality and parallelity and it will often treat both cases very differently. So when you have a plane and a vector that sticks out of the plane, you can easily split it into the part that lies inside the plane, parallel to the plane, and the part orthogonal to it. The geometric product then treats these two parts differently in exactly the way that you need. It may, for example, rotate the parallel part while leaving the orthogonal part alone. And that's precisely what we will need in order to rotate the original vector parallel to the plane.

Geometric algebra handles this distinction perfectly, giving us an incredibly general purpose and yet very simple formula for rotations, reflections, translations, and more. Okay, so maybe the shortest summary would go like this. We can pull vectors apart into pieces, treat those pieces differently according to our needs, and then compose the results back together into the final answer. To make your mouth water with anticipation, let me just name a couple of things that geometric algebra will do for us. It unifies the algebra of matrices, tensors, spinners, complex numbers, quaternians, differential forms, poly matrices, drack matrices, homogeneous coordinates, and many other domains.

Not only does it unify these, it also clarifies them. The multiplication rules for complex numbers or quatronians follow very naturally from the properties of our new geometric product. The new product is complex multiplication in 2D and it is quatron multiplication in 3D. So every time we turn a corner, we will bump into something familiar but in a new form, a much simpler form. It makes you realize that something you already knew and understood has an additional structure or meaning or layer or purpose that you hadn't noticed yet.

Everything is connected to everything else. I absolutely love these kinds of connections and I hope you will too. And the good news is that it's not extremely difficult to understand. You only have to know a few things. basically the rules I mentioned earlier and then the math takes care of everything else.

Geometric algebra has many advantages when compared to the alternatives. It combines geometry which is more intuitive and more visual with algebra which is more automatic. The calculations can be performed by a computer and they always give the correct answers. This combination of algebra and geometry allows us to calculate directly on shapes rather than on specific measurements of those shapes. Instead of measuring the angle between a plane and a vector, you just perform algebraic operations directly on the plane and the vector themselves.

This means that you don't need coordinates. Many of the amazing results of geometric algebra are coordinate free. They depend only on the shapes themselves and not on their coordinates or other measurements in a specific basis. There's a large number of operators that allow you to algebraically calculate the parallelogram spanned by two vectors or projections, reflections, rotations or the line through two points. the point where two lines intersect and so on.

Another cool feature of geometric algebra, one that we will explore in depth, is that it makes no distinction between shapes and operations on those shapes. A vector can serve as an arrow that you can rotate, but it can also serve as a reflection, flipping other objects around its line. But the main advantage for me at least is that geometric algebra unifies and clarifies so many things. The cross productduct is an excellent example. This is one of the weirdest inventions of vector calculus and we will discover a much better alternative.

It will reveal precisely why the cross productduct is so weird, why it doesn't generalize to higher dimensions and more. Ever since we announced that we would be covering geometric algebra, many of you have expressed how much you were looking forward to these videos, some of the comments have been quite um passionate, complete with exclamation marks and all caps. And some of you have proclaimed that linear algebra is out of date and that we should be teaching nothing but geometric algebra from now on. Personally, I have a more nuanced opinion. Geometric algebra is amazing.

It's beautiful and worth exploring. I have already mentioned many of its big advantages. I'm a huge fan. But it also has a steep learning curve. That's not just my own opinion.

Here is Alan Macdonald, one of the most famous promoters of geometric algebra. At first, you will likely find the novelty and scope of the mathematics presented here overwhelming. This is to be expected. It takes years of serious study to understand the standard approaches to the mathematics discussed here. But after some study, I hope that you will find with me great unity, simplicity, and elegance in geometric algebra.

Well, if it takes years of serious study to get familiar with the ins and outs of geometric algebra, according to one of its biggest proponents, that makes it less suitable for math education in schools and even in higher education. Engineers can definitely get by with linear algebra and calculus without ever hearing about the geometric product, the contraction axiom, verse. The many amazing applications of linear algebra such as the singular value decomposition or value analysis aren't all going to get translated into geometric algebra and they don't have to be. They can be fully studied and understood and appreciated in their own terms. One thing I would recommend for physics education is replacing the cross productduct with the wedge product which is superior in every possible way.

But even that doesn't require a full transition to geometric algebra. The wedge product can be understood by itself as evidenced by the fact that we have already published a video about it. Anyway, these are just my personal opinions. Of course, make of them what you will. Don't forget that you can already binge the entire geometric algebra series, all 15 videos, right now on Patreon.

Please consider supporting us. And now, welcome to the marvelous world of geometric algebra.