A Swift Introduction to Geometric Algebra

This video is an introduction to geometric algebra, a severely underrated mathematical language that can be used to describe almost all of physics. This video was made as a presentation for my lab that I work in. While I had the people there foremost in my mind when making this, I realized that this might be useful to the general public, so I also tried to make this useful to others as well.

Several things in this video were incorrectly simplified. Please watch the video before reading the rest of the description.

Before I get into the things that were incorrectly simplified, I need to state a few definitions. The grade of a k-vector is defined to be k. A blade is defined to be the outer product of vectors. A k-blade is a blade of grade k. Some of the terminology between linear algebra and geometric algebra can be confusing, such as the use of the terms vector space and dimension. Because multivectors form their own vector space, but we don't consider all of them to be vectors, some people use the term linear space instead of vector space. While we think of dimension in terms of spatial degrees of freedom, the mathematical definition means that the 3D geometric algebra shown here is actually eight-dimensional. This is part of the reason for the distinction between the terms grade and dimension.

The following are the things that I purposefully got wrong to make things simpler:

The biggest thing I glossed over is the distinction between blades and vectors. A lot of the ways that I described k-vectors intuitively actually only applied to k-blades, not k-vectors. For example, in four or more dimensions, all 2-blades can be represented as an oriented area, but not all 2-vectors can. I didn't mention this fact for two reasons: first, for a first look, the distinction is not that important. Second, the distinction doesn't even matter until you reach four or more dimensions. In three dimensions, all k-vectors are k-blades.

My use of the term basis was not quite right, and I should have used the term "orthonormal basis". I wanted this video to be understood by people who had learned about vectors in a physics class where they don't go into too much detail about the exact definitions a linear algebra class would give you, so I didn't want to use the term orthonormal. I tried to mitigate this somewhat by always using terms like "the basis" and assuming that it was known that I meant the standard orthonormal basis. Hopefully this doesn't cause any confusion.

I mentioned that the inner product can get complicated when generalized to generic blades. What I didn't mention is that there is a large amount of disagreement on what this generalization is. I've seen at least five different extensions of the inner product to higher-grade blades. I didn't mention this because it was not pertinent to the discussion. However, even though I didn't talk much about the inner and outer products in this video, they are essential to the usage of geometric algebra in many applications.

This wasn't necessarily wrong, but I assumed that space is Euclidean throughout the video. In some applications, especially in relativity, space is not Euclidean, and a few things aren't the same.

In general, the pseudoscalar for a geometric algebra is the highest-grade element, but this does not always square to -1. In the three main applications of geometric algebra (2D Euclidean space, 3D Euclidean space, and 4D Minkowski space), this does happen to be the case, but in general it is not true.

When I mentioned dividing vectors, I didn't mention the fact that when dividing, you have to "divide on the left" or "divide on the right" because multiplication is not commutative. I actually prefer just to divide by multiplying by the inverse, because then you don't have to worry about how you're dividing. I didn't mention this because I never mentioned inverses again after I introduced them.

Another issue with inverses is that while vectors have inverses, not all multivectors have inverses. Also, in other spaces, such as the Minkowski space used in relativity, even some vectors don't have inverses. Again, this was because I didn't really use inverses much after this.

This one is small, but we assumed several properties of the geometric product that we didn't prove or state, such as distributivity and associativity. To be precise, the set of multivectors along with addition and the geometric product forms an associative algebra. I defined the geometric product in a bit of a roundabout way and I couldn't find a good place to insert these facts. The kind of people this video is aiming at are those who don't know a lot of abstract algebra and so I assumed that they would assume associativity and distributivity.

Rotors actually have many different definitions, so don't always think of them as a complex exponential.

[sudgylacmoe]

I just realized a stupid mistake I made: at 8:20, this should actually be that the inner product is this length times ||u||. I don't know how I didn't notice this in all of my times watching this video while making it.


EDIT 2: Another correction: at 29:00, the equation should be i a · (b × c) = a ∧ b ∧ c, not a · (b × c) = i a ∧ b ∧ c.

[ConnorMcCormick]

Every "wait a minute" I have to pause to wipe tears from my eyes so I can see the screen

[mysteriousgrimreaper]

There’s something really beautiful about the fact that simply choosing to ignore “you can’t multiply vectors” leads to ALL OF PHYSICS

[gabitheancient]

geometric algebra feels like something that was missing from mathematics, it kinda explain things that were weird but proven, now everything makes sense

[adixo]

Imaginary Numbers being Pseudo-Scalars is better plot twist then some movies 😂

I work in applied electromagnetism and never learned so much in 45 min. You've made sense out of concepts that were always just out of reach. You've changed my life

[mjjohnson]

I don’t normally comment on YouTube, but as someone trying to understand higher level math concepts, this is one of the best videos for education I’ve ever seen, at least for the way my brain works; thank you so much!

[debbiegilmour]

As soon as you brought up "i", I thought "he hasn't..." but as the vectors were rotated through 90° I was like "he is!!!" then you dropped the big reveal and my face was all Ö

That was stunning and brave! Bravo!

[luismedrano]

I always had this problem with torque and angular momentum. I remember how after a Classical Mechanics class, I started searching for a more profound meaning to those concepts, since I was already in third year of Physics and still got no feeling of understanding the actual concept completely.

Now I watched this video, and all of a sudden my soul can finally rest. I mean, even Feynman, when trying to explain the conservation of angular momentum, talked about how the area was preserved. It makes total sense.

[mrpedrobraga]

This sounds like initiation in a cult, and i'm all for it.

[jamesperry]

Just realised that the number of components of a K-vector is described by the Kth row of Pascal’s triangle. Beautiful

[rescueafterhalfanddoubledo]

Middle school teacher: you can’t add scalar and vectors
Mathematicians: why the hell not

[sietsebuijsman]

I'm completely amazed. I found a Wikipedia article on Geometric Algebra a while ago, seemed interesting, but I didn't quite get it. The past few days I've been working with quaternions and now your video comes along. I almost cried when the "i" entered the stage, it's all so incredibly natural. Thanks for blowing my mind.

[Michallote]

I have noticed many people has had the same actual reaction that I had. I really cried, and I didn't understand why at first.

In my case I believe it is because I experienced something so potent that it almost dwarves everything I know in terms of physics. I can't help but feel for something so beautiful, so deep that my soul finally rests on knowing that those doubts I had and my inability to sometimes grasp concepts is perhaps justified. I am almost finishing mechanical engineering and I really felt there was still too much to many knowledge I lacked, not because of not knowing the existence of such topics but feeling I wasn't on top of many subjects to the degree my curiosity demands.

This was only mentioned briefly but the connection between tensors and complex numbers was something not even in my wildest dreams I could have come up and here it is JUST SO NATURAL. I always felt there was something missing from my education and I have looked desperately everywhere. I can't even begin to explain how relieved I am to finally arrive at this, not because it has solved all of my doubts, but because it gives me hope that I can perhaps eventually solve them.

It's a light on a path I thought was shoruded in darkness. I just want to scream, we should all have been introduced to this as kids. And I believe all of the physics education programme should be built around it. It's something you can immediately and intuitively relate to.

[kunalverma]

I swear I haven't had more surprises in 44 minutes in my entire life! This is one of the most underrated video on all of YouTube.

[stephendonovan]

Sudgy: *casually starts using tau
Me: wait a minute

[matthewfreudenrich]

I literally shouted when spinors came up out of no where. This is an absolutely amazing algebra and I am excited to see more of it.

[andrew]

I love that Maxwell's equation is basically the same as the tensor notion, which also condenses the 4 equations down to 1. It would be cool to see how geometric algebra relates to tensors

[smorcrux]

this feels like a video that would be unimaginably useful in the future, but now I just lost it around the 20 minute mark. It feels as if I will return to this in the future, and find it really helpful.

Edit: I remembered this video again now 5 months later, and I feel like I understand most of it! I've massively improved in maths in the past few months, and I'm even starting a bachelor's degree in just a few weeks.

[alberto]

I don't usually leave a positive comment on a video, but I'm making an exception. This is a masterpiece, the knowledge to produce this must be so utter. I haven't seen a book or video about the topic more enlightening. My most sincere congratulations, you really have a gift for this, consider making a career out of it.