An Intro to Geometric Algebra | Whitworth Lecture Practice

Transcript

Oh, okay. Let's see. Okay, I think it's up. Well, let's go then. Alrighty.

Well, it looks like there are three people here. Hello. You get to see me in the in the raw view form. Okay. So, welcome to a short practice session that I will do for some a guest lecture that I will give at Witworth University tomorrow.

I just wanted to make sure that I have it nailed down before and yeah. Okay. So, it's just going to be uh what do you call it? I'm going to continue whether or not there are people still watching. Um but I will spend about an hour uh on one presentation and if I'm satisfied with it then I will just leave it at that and if I'm not then I might revisit a few parts and go over those parts again. But whether or not I do that I'm going to leave this video up on YouTube for people to view it.

So, even if you're um watching this later, it should still be helpful. Even if it's not uh the greatest presentation, I still think the information in the the lecture is good. So, without further ado, uh I should probably uh actually get a timer out. I forgot to do that. I meant to do that before.

So, go to my clock app. cannot wait for the day when I do not have a phone. Okay, so let's get into it. So, for those of you who don't know who I am, uh my name is Moab and I'm a Witworth University graduate. I graduated two years ago and I double majored in physics and mathematics.

And since then I have been uh I spent a little bit time of time in graduate school and then I got a job as a physics consultant, a private physics consultant and I've been doing research and teaching on my on my own time since then. And so I've become a big fan of geometric algebra and I think that it needs to be taught. And so here's a little bit of motivation for getting into it. It is uh It is able to be used easily in tons of different fields. For example, here are some recent publications um from a conference advances in geometric algebra in computer science and engineering.

This is a conference I attended last year and these are all publications in that and I actually know all of the people in this um except for Christian hockey. don't know him. Uh but Hamish Todd, the uh presenter for the or not the presenter, but the author for the second uh the second here, let me I'm going to get a laser pointer. The author for this second publication here is actually a friend of mine and he's a co-author um on my paper. Okay, rambling on a little bit, but the idea is that you can learn geometric algebra and be able to contribute and learn from all of these fields without having to learn new mathematics.

That's a little bit of a bold statement and you do have to learn a little bit of new math, but for the most part, you can go between them without um let's see, without much lag time. So, there are tons of benefits. They connect various areas of math like group theory, abstract algebra, linear algebra, topology, and differential geometry. And so it it's really good if you already know these things, then that just makes geometric algebra better. But if you don't, then geometric algebra makes it easier to learn those topics.

It's also what's called a universal formalism or a dimension agnostic formalism. And while it looks abstract, it's actually pretty much just all geometry. So a little bit more on this universal formalism. This equation here um is completely dimension agnostic and signature agnostic. What that means is that it doesn't really care about the space that it's in.

And because if you're working with spacetime then this becomes spacetime angular momentum or spacetime relativistic inertia. And if you're working in three-dimensional non-relativistic space, then this just becomes the normal linear momentum. And actually, I think it's not just linear momentum, but it also it also includes a rotational momentum as well. Uh, so I should probably fix that. And well, it's abstract, but again, it's rooted in geometry.

This expression is just telling you to associate a mass with a point m or with a point x and then look at the point x dot and then join them. Create a line between these two points. X dot represents the velocity. And so that makes sense that that's what the momentum would be. It's taking where the object is.

You know where it's going. Draw a line. That's the momentum. Now this presentation, this lecture will be in three main parts. There will be a little bit of a history and a presentation of two-dimensional geometric algebra.

Then a presentation of three-dimensional projective geometric algebra. And at that point, I'm going to focus much more on the concepts than the math because all of the math carries over from two dimensions because of this universal nature. And then section three, I'm going to be expanding or maybe not expanding, just introducing what I do for research and then providing uh resources for people who are more interested. So let's get into the history, shall we? Well, the history can't be talked about if we don't talk about William Kingdon Clifford. He was born in 1845 and he's the inventor of all of this.

As such, it is respectful for me to give him the halo. He discovered geometric algebra in 1878 and unified a bunch of different areas of math, the quaternians, inner products, outer products, the complex analysis, and then proceeded to die the year after that at 33 years old. And so he wasn't able to really promote his work much, which is a massive bummer because it is super cool. And it wasn't really rediscovered until the 1960s and by de David David Hestinis and he was really cool about it. However, he was very divisive about this to other physicists.

He'd be like, "Why aren't you using geometric algebra? This is better." And that's a little bit uh of a bummer because it turned a lot of people off geometric algebra. However, um it is now experiencing a resurgence which is amazing. That is something I'm very happy about, happy to see. I started getting into geometric algebra in what 2020 and to see that there are so many people getting into it now, it makes me feel uh very happy. I know it start the the momentum started before 2020, but it feels like it's grown even more since then.

One more thing, if you're interested in learning more about William Kingdom Clifford, he wrote this little snippet of text on the space theory of matter after he read Reemon's uh work on differential geometry. And he predicted that all motion and forces are results of curvature in space and curvature in time. And so he predicted Einstein's theory of general relativity 40 years before it actually happened. And so I think not not enough of the geometrical I um I don't know how to say this. Not enough is attributed to William Kingdom Clifford Clifford because he died so young.

Okay, let's get into two dimensional geometric algebra now. So the question that we all need to be asking is how did Clifford unify these branches of math? And he did that through the geometric product which for the vector form is written as vector a * vector b equals a dob plus a wedge b. a dob is the inner product of vectors. A wedge b is the outer product. But what does this mean? It's easier to think about this geometrically.

If I just take two vectors and then I collide them into each other, then I'm going to get the component that's parallel and I'm going to get the component that's perpendicular and that's exactly what the geometric product does. And it results in a real number a scaler for the uh part parts that are per or parallel and then result results in a new object called a B vector uh which encodes the information of the perpendicular components of the vectors. And it's this simple product that enables all of geometric algebra to unfold. So we're going to talk about two dimensions. We write it as G2, the geometric algebra of two-dimensional space.

And it's defined as the real numbers over the basis vectors X and Y. Geometrically, they form the basis of two-dimensional space as we should expect. And so now we're going to compare these basis vectors with this geometric product and see how they behave. If we take x and smash it into itself, then we only get a real number because it's parallel to itself. And so it only returns a real number.

And because it's a unit vector, it just gives you positive one. And the same thing happens with y. Y is parallel to itself. When you smash it into itself, you just get a real number. You get no by vectors.

The real interesting stuff comes when we start considering when these uh perpendicular vectors start smashing into each other. If we consider x and y then when we smash x and y into each other we don't have any parallel component. So we only get a bi vector. So we get the by vector called x wedge y. But what happens when we do y * x? We just get y wedge x.

Well, what's the difference? The difference lies in orientation. So for y wedge x, we take x and we slide it along y. And that gives us a right-handed orientation. And we do the exact opposite for x wedge y. And so now we get two opposite um orientations.

And that that should tell us something namely that these are oriented negative relative to each other. So they this in mathematics is called anti-commutativity. When an object when you flip its order is negative of itself then this is anti-commutative. Okay. So now we found that by vectors are anti-commutative objects.

Then the geometric product can be reexpressed um in terms of components of vectors. I'm sure that everyone is interested in that especially when I actually present this in person. They're all computer science students. So uh they're all going to be interested in the components. A can be expressed as a linear combination of the x and y directions.

And same thing with b. The inner product then is going to be the sum of the x components times each other and then the y components times each other. Or you can express it as the lengths of both vectors times cosine of the theta. That's the angle between a and b. And lastly for a wedge b we have ax b y minus a y bx.

This is the two-dimensional determinant for those familiar with that. or something more familiar to them might be that a * b the length of a length of b * sin theta um where theta is the angle between them and then we end up with this xy I hope you agree with my writing x wedge y here and just simply xy because they're fully equivalent since the vectors are perpendicular to each other. Okay, so now we're going to focus on xy a little bit. specifically we're going to square it. Now we know that it anti-commutes and so we can now insert this into the squaring and when we do that we end up with a -1.

So this is an object that now squares to -1. Well that's just like an imaginary unit. And so if you're thinking that this is involved with rotation somehow, then your intuition is dead on. Now let's consider two vectors times each other, we can write it as their uh magnitudes times cosine of their angle and then the magnitudes of s of their angle and then we have this unit by vector which behaves like an imaginary unit. So the first thing to do we pull out their magnitudes.

So we just get these cosiness and ss and then we can just re label this a and b to d. d stands for dilation factor because d is just going to take a vector and make it bigger or smaller. So it dilates the vector and then we can just move directly into oiler form where we write this as an exponential because cossine theta plus an imaginary unit time sin theta gives an exponential and because this x and y behaves just like the imaginary unit we get this exponential and this we can form an object from and it's called r a rotor because it rotates objects and if you have a and b with separation theta here. Then if you take some arbitrary vector C and you apply this rotation, then you're going to rotate from C by this angle to a new vector. And so no matter where you are in the space, this rotor will rotate by the separation of the original vectors.

And if you're interested in seeing an actual calculation, then feel free to pause this and just look at this. Um this is indeed these are indeed the transformation equations for two dimensions. Okay. Now there's a little something I didn't say and that it's this is not a universal way to rotate by mapping from the vector to the rotor times the vector. And what I mean is that in higher dimensions you actually run into some problems.

Now, because this is introductory, I'm not going to explain exactly how this works, but I'm just going to give you the solution. If you're more interested in this, then there are uh easier easier ways for you to find out than this lecture. But typically, what we do is we redefine the rotor to rotate by half the angle and we just put a a minus sign just for convention. And notice that I flipped the this to be y and x just so that you can see that why the minus sign is there. So we have a minus sign there.

And now we apply the rotation on both sides of the vector. Now I know this is definitely not satisfying because you're like why is there a dagger? Well dagger is a new operation. It flips the product order. So if I have something time something time something then that flips the order of their products. So this flips the order of the products x or yx.

So we get xy. But we know the xy is the negative of yx. So a minus sign now cancels and we have yx again. And all this means is that the dagger just gives you the inverse of the rotation of the rotor. And so for those of you familiar with quatronians, this is actually the definition of the quatron transform.

And in three def or three dimensions, it's completely equivalent. And so this is another aspect of the universal formalism of geometric algebra. This equation for rotation is exactly the same in any dimension and in any signature which is super cool. Okay, so now we're going to do a quick recap. This is the algebra of two dimensions.

It's G2. And when we take the basis vectors and multiply them, we get one or a B vector. and the B vector squares to negative one. And then when we want to rotate something, we create a rotor and then we map from one vector to the rotor applied on both sides of the vector where this R dagger is going to be the inverse of the original rotor. And remember geometrically this corresponds to flipping the um the imaginary unit sign in the rotor.

Okay. So, oh, this is also um is the the full multi vector basis as it's called is 1 x y and x y. Um here I see some comments. Uh I'm going to pause my timer here. And then let's see what are is the RVR dagger form still true in one and two dimensions.

Uh yeah, it's it holds in all dimensions. Um it's a it's a universal form. Oh, and also thank thank you for the uh the nice compliments. But okay, uh moving now from that, I'm going to resume my timer. Um now we're going to talk a little bit about the geometry of the of this.

So there's a traditional view where you interpret the uh unit one as just kind of like a point. Um it just doesn't it's the point, but it's not really considered. And then the vectors are just vectors. They're little sticks oriented in space. And then the bve vectors are the sticks glued together and they give you a whirling motion.

But we're going to work with something that I believe is while fully equivalent more powerful. And I call it the mirror view. And thank you to Hamish Todd for the name because he's the one I believe who came up with it or at least he's the one who told me. So mirror view the unit one represents the space itself. Not in exttrinsic to anything or intrinsic to anything.

It's just the space. There's no like if I have an object then with reference to itself it can't have any orientation and so the space itself has no orientation that's represented by a scalar. Then for the vectors they now represent lines in two dimensions because the lines all have the same orientation that is given by the vector. And now the bi vector is the intersection of these two lines and rotates in the direction given by the orientation of their bi vectors. You'll notice that if you were to put this to the left of the red arrow, you'd end up with the same construction for here.

And so really the mirror view has the traditional view inside itself, but now you're also considering things as these bigger objects. And so this is going to be very important as we move on to three dimensions because we're going to be working with the mirror view. Okay. Now, we're going to get into 3D projective geometric algebra. But if we're not going to focus too much on the math because all of the math for this carries over from two dimensions.

And so if you want to do all of it for any of it for yourself, you can just copy what I did for two dimensions and do it for three dimensions. just knowing that there are more degrees of freedom, but it's exactly the same. What I really want you to focus on is the uh intuition and the geometry getting getting a feel for how this works. So, we're going to first go into it by introducing the full multi vector basis for 3D 3D geometric algebra. This this part is not projective.

We're going to take the 3D algebra and then explain that and then add in the projective dimension. So for full uh for the full multi vector basis we have three vectors. We have the x y and z direction. For b vectors we have the xy by vector, the y z by vector and the z x vector. And of course there's now a new object called a trvector that is the product of all three of these basis vectors.

What do these look like? Well, we're going to be using the mirrorbased view. So, one gives space itself. And so, when we now look at the vectors, they're going to give planes because the plane can now extend through all of space and have the orientation given by the vector. So, for the x vector, we have a plane perpendicular to the traditional vector. We have the same thing for y and for zed.

So, then what are the bi vectors? it's their intersection. So when you intersect X and Y, you get this axis right here with an orientation going around it. And for Z X, you get an axis again with an orientation going around it. And then for YZ, you get the same thing, another rotation axis with a clockwise rotation going around it. And then finally for the tri vector, that is called the origin and it has an orientation given by the right-h hand rule.

Okay. Now, if you're interested in just the notation for what this algebra is, it's G3 and it's the real reals extended over X, Y, and zed. Okay. Now, we introduce the projective aspect of it. But before I get into that, it looks like there's another question.

So, let me pause my presentation timer. Let's see. Um, yeah, that's a good point. Yes, how the scalar part can have negative and positive. Um, they're kind of uh that that's something that I've wondered about myself whether or not it's some kind of reflection within itself or not.

I guess you could consider it yeah as a sort of mirroring which um so that's a good that's a good point. Um I should probably include that is in it but yeah. Oh no, the I forgot the equation was behind my head. Yeah. Um well, it won't be for the actual pre presentation.

Uh okay, we'll just move on. Uh I'll have to remember that and explain whatever is there uh if that happens again. Okay, moving on again to this. So now we're moving into projective geometric algebra. We introduce a new vector.

I'm labeling it as infinity because it is going to be a vector infinitely far away from all the other vectors. We'll see how this is interpreted geometrically shortly. It does result however in all of these new multiv vectors. It results in three new by vectors and in four or three new tri vectors and in one new quadro vector. Okay, the plane at infinity is the vector infinity.

And so it's a plane that wherever you look is infinitely far away from you and perpendicular to you. The closest I could represent this on screen is a a sphere with um orientation given by the direction of the sphere vectors coming out of it. However, it's important to note that the vector is not curved. It has no curvature. This is just a way of representing it because I can't have something infinitely far away from something on a screen.

But now you're going to you're apol I'm going to apologize for these really poor drawings. But these are the intersection lines now between the x vector and the plane at infinity and then the z vector and the plane at infinity and then the uh the y vector and the plane at infinity. They give these lines at infinity. um right here. So whereas we had these axes that go through the origin before, now we have these axes that are sort of infinitely far away.

And likewise continuing this, we have points as well. Now let's talk a little bit more about the interpretation of lines infinitely far away. How do we deal with this? What we're going to do, um I keep forget. Okay, I keep forgetting to put this there. Move on for that.

What we do is we go to the origin and we look outward. What's the origin? It is the XY and Z point. And now we look out to infinity and we see two perpendicular lines. We have the line infinity X and the line infinity zed. Note, however, that these lines aren't going in the z direction or x direction.

They are perpendicular to it. So the x direction if this is the x uh infinity line then the x direction is going that way at least um I don't know how uh how is this okay it's opposite for me and the camera okay so for for you guys it's going that way so that's important to note and there's also their intersection lies infinitely far away but the important thing is that we can now take a line that is near us at the origin and goes out to infinity So now this line goes outward to infinity. And if we rotate it around this axis, we can treat this line as an axis and rotate it. So we rotate it around an axis at infinity. Well, this just looks like a rotation.

What is it actually? To do that, to figure that out, we need to look above it. And so now, as we look above, the lines straighten out because lines that converge at infinity are parallel. What this tells us is that rotations about axes at infinity are just translations. And so the cool thing about this means that we can now go from this geometric algebra of 3D space. We can include one zeros squaring vector and that now gives us a space that has both rotations and translations.

And so what this means is that for objects in G3, everything must be oriented and passed through the origin. However, in the projective version, you can have objects anywhere. They can be located anywhere because if you have translations, you can create an object at the origin, then translate it out of the origin. And what's the real difference between something existing outside of the origin and being created at the origin and then being moved to that po that spot. And so that's the power of projective geometric algebra.

It's by including the concept of infinity to get rotations and translations. We're going to focus a little bit on how rotations actually or are dealt with in this algebra. So recall in two dimensions we had the rotor defined with a negative sign 1/2 the angle theta that we want to rotate in this plane and then we map from vector A to the rotor applied to both sides of the vector where this is the reverse and the reverse is the inverse for the rotor. And in 3D we have exactly the same thing except we have now a generic bvector B. And this B vector B has an arbitrary orientation.

It is a linear combination of all basis by vectors in the three-dimensional algebra. And the rotation is is done in exactly the same way as it is in two dimensions. Let's look at this geometrically because I know the algebra is a little bit, you know, it's not the not the best way to learn sometimes. So if we have all of these different basis axes that tell us how to rotate, then B is just some arbitrary axis. We rotate by some amount to get a different axis.

And now we're rotating around this axis. That's what this rotor does. It rotates around this random or not random but this arbitrary axis. Okay. Now we're going to focus on translations.

And you might have noticed that. Let's let's go back just for a second. Pay special attention to the form of these equations. Did they change? No, they did not. I reabeled R to T and I got rid of the minus sign because of convention.

But these are exactly the same equations. So applying translations is exactly the same method as applying rotations except now we're considering these lines that are at infinity. This is the line at infinity in the z that pulls you in the z direction or pushes you in the z direction. This is the line at infinity that pushes you in the x direction. This is the line at infinity that pushes you in the y direction.

And so an arbitrary direction B is just an arbitrary line at infinity that pushes you in that direction. So you can rotate in any direction and you can translate in any direction using this formalism. Okay. So now rotations and translations can be done at the same time using what's called a motor where you literally just compose a rotation and a translation in this equation. this generally or this gives a screw motion like what is shown on the screen but more generally than that you can describe very complicated motions.

So this is I'm not a very good coder but this is something I did about a year ago um and it's something just some basic physics. So now I have an object that is rotating with a very complex angular momentum and it is accelerating downward. There is a force of gravity in this simulation. I've slowed down the time so you can see it, but this is done using just these this concept of rotating and translating at the same time using the exact same math. And so it's kind of amazing that you can use these relatively simple geometric concepts to do pretty complicated things.

And if you want to do this yourself, then this is a link and a QR code to the document that was prepared by other people that allows you to do this yourself. It will look a little scary if it's your first time going into this kind of thing on your own. But it, trust me, it's really not that scary. It's it's just very different from how things are normally taught. But it's actually, when you get down to it, much more simple.

And now I'm going to make it seem like I'm an idiot by showing you what the equations are because when you look at these you think, okay, there there's no way that that's simple. But trust me, it really is. These are only all the equations you need to do that animation that I had before. You just need those four things and you can update those to create it. And the cool thing about these equations is that they work in any dimension, any signature.

You can do this in Menovsky space, in purely hyperbolic space, in parabolic space or in uklitian space. These work for all of it. And what it is conceptually, this gives you something called the force or depending upon your um space, the fork, which is the force and the torque. And that's just the rate of change of your inertia. That's what this is.

And you might recognize this from the first equation I showed in this presentation where you literally take the position of a mass m and you create a line between it and its change in time. And so if you have the inertia right here and you just look at the change in inertia in time, you get the force or fork. And then this will tell you how the bi vector is changing in time and the bi vector will tell you how the motor is changing in time. So if you go through all of these then you're able to uh model basically any situation in any dimension or signature of space. Now this is not my area of expertise so I'm not going to stay on this much longer than I have to but it is very very cool.

Now we'll talk about my research a little bit. So I work primarily in geometric and algebraic physics which is a subcomponent of mathematical physics. I study a lot of the algebra of physical space which is the three-dimensional algebra. Also the light cone algebra, the space-time algebra and the durac algebra and they're very related to these other algebbras, the complex quaternians, the dual quatronians and the hyperbolic quturnians. But right now my research is focused on the geometry of ignor little groups and the geometry in particle physics.

So like what's what's the geometry for scattering amplitudes and all of these different transformations. So I also have a focus in pedagogy. So distinguishing interpretation from theory that's something that I really love to do. Uh I haven't really focused much on my YouTube channel in terms of this but it's something that I engage in actively on my discord channel and also when I'm talking with other other students or other physicists. So I also like to uh clarify spinners because that's something that's an area of uh study that is very very confusing and so I try to my best to make the concepts easier for other people to understand which is why I have um two videos on spinners on my channel which are very different ones on algebraic vial spinners and then ones on powi spinners but I feel like both of those videos I'm not to pat myself on my back, but I feel like both of those videos explain what those concepts are better than any other subject can.

And maybe maybe you don't agree. And if so, then I think that would make for a very good uh debate and maybe I could improve upon my methods. But another part of pedagogy is obviously the stuff I do in this channel. It's thinking of algebra in terms of geometry and vice versa, geometry in terms of algebra. So focusing a bit more on my geometric and algebraic physics uh background.

I have a recent publication the Vner little group for photons is a projective subal algebra and this uh is a paper wherein I tried to solve and in my opinion successfully solve a problem that was given by Vner. Um he discovered that some transformations of that the transformation of light its transformation group was isomeorphic to the plane of two dimensions. So it had both translations and rotations. And he didn't know how to construct this physically. He knew that if you had a sphere and then you put a line on or a plane on top of the sphere, you could do it.

But he didn't exactly know what does that. Well, I demonstrated that you could do it using geometric algebra and using the spac-time algebra specifically and I actually found that you can also do it in two the oneplus 2dimensional manovsky space and actually in general you can do it in any dimensional manovsky space and in whenever you do it you'll always generate a subalgebra of the entire algebra and so that's something I'm particularly happy about um and if you're interested in reading that I have the QR code for it up there. Um, and then some other resources. If you're interested in Discord servers where you can learn this stuff, I first and foremost recommend bvector.net. That is where all of the best people are to help you learn and they have all these resources.

So, go there. If you're more interested in discussing my videos, then I would say uh go to Wizardheim because that's where I am more often than not. And but if you're really really just wanting to focus on geometric algebra, then go to bvector.net and then check me out too. But uh again, it's not if you really want to learn, go to bi vector. And then of course my channel.

Um it's weird that I'm putting it in this uh live stream because when I'm presenting in person, it's not going to be weird at all. Okay. Wow. Um so that's kind of it. I felt like it was kind of a rough presentation.

That was my first time presenting it. But let's uh yeah, let's go with that. I'm going to look through the comments now. I've seen a bunch and we can go through questions and if anyone has questions, I'll happily answer them and then I can also go to earlier parts of the presentation and clarify stuff too. Okay.

So, okay. So yeah, the 4D vectors of the form X, Y, Z, and W, where W represents the infinity basis. Yeah, I'm not exactly sure what the construction is for that, but uh I've talked a little bit with my adviser at Wit or my old adviser at Witworth and yeah, he's mentioned stuff like that. I think it's uh pretty cool that basically the same concepts are used but in different fields of mathematics. I think it's pretty easy to visualize in geometric algebra.

So I think that's one of the big bonuses in the fact that it's dimension agnostic. Yeah. Um let's see. Do I Oh, I remember seeing Hamish's Twitch stream about generic geometric algebra. Is this a new way to visualize CL2 compared to traditional VGA? Um no.

So it's not a new way to visualize at least with the with the mirrorbased view. um the mirrorbased view that I talked about earlier. Let's go there. Yes. Let's continue all the way back to Yes.

this part right here. So, the mirrorbased view isn't necessarily new. Uh I believe that uh Deart or was it was a Carton? I can't remember who which one exactly but one of those mathematicians actually used the mirrorbased view where they thought of things as as hyper hyperplanes in their respective spaces like a hyper plane in two dimensions is a line a hyper plane in three dimensions is a plane a hyper plane in four dimensions is a volume you know so on so the mirrorbased view isn't necessarily new but it's radical in the sense that it's not widely adopted but you don't need to have degenerate um you don't need to have degenerate algebbras to be able to use this view. You can use it outside of degenerate algebbras. So I hope that answers the question.

Um yeah uh if anybody has any more questions I'll answer them. But if uh if not, then I think I'm going to end the stream because I'm I'm sat it wasn't a perfect stream, but I'm satisfied with how it went and I think I'll do even I think I'll do well enough tomorrow. I just need to make sure that I go a little bit longer than this because it's an hour long. It's an hourong presentation and I only took half an hour to do it. So maybe I should slow down my speaking.

But okay, let's see. I'm gonna wait like 30 more seconds to see if there are any questions and then I'll get off. Oh, hello to Nadia. Um, I saw that. It's weird uh kind of talking to people in chat.

I've never done this before. It's my first time streaming. Yeah, thank you. Thank you, Maxwell, for your uh insight into these uh into the uh traditional version or the matrix version. That's that's cool to think about.

I deal a lot with the matrix versions of stuff in physics. So, uh it's it it's thinking it's making me think of like an analog but in computer science instead of physics. Okay. All right. So, I'm going to go ahead and end it.

And I think that this was a pretty cool stream. Uh, I think I might start streaming more often now um to talk about different things in physics. Just ask questions on my Discord server if you want me to talk about something over a stream and then I will because uh Oh, hang on. Someone did ask a question. magics.

If you want to find the intersection of two lines or the distance between, what algebra would be best for three dimensions? So then I would say it's the it's the geometric projective geometric algebra of threedimensional space. So it's let me get there. It's this algebra right here. Oh, my head is blocking the Let me let me move my Let me move my monitor or my head. Okay.

Yes. So, it's this algebra right here. This would be the best for finding the intersection of two lines. So, lines are going to be bio vectors in this case. And basically you you'll take two lines and then you'll take the wedge product the outer product between them and that will give you the intersection.

Um after finding the distance uh there are some you just manipulate the algebra around to do that. I don't know the specifics just because while I while I do know more about geometry than the average physicist my specialization isn't in the construction uh or in in all of the nitty-gritty details of geometry. But if you want to get into it then go ahead. I mean, the literature is out there waiting to be read. So, yeah, and everyone in bctor.net will probably be very helpful with that.

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