#94 - Joan Lasenby

Hey, how's it going? This is Greg Cannon, and you're listening to Y Combinator's podcast. Today's episode is with Joan Lazenby. Joan is a university reader in the Signal Processing and Communications Group of the Cambridge University Engineering Department.

She's also a college lecturer and director of studies in engineering at Trinity College. In this episode, we talk about Joan's research into 3D reconstruction for multiple cameras and her interest in geometric algebra. If you'd like to learn more about Joan's work, I'll link it up in the description. Alright, here we go.

So Joan, as we walk through geometric algebra, I think the best place to start might be through a more tangible example. So you're doing a project with drones here at Cambridge. Can you explain that first? Yes, so we're doing a project with drones.

This is joint with the Architecture Department. And what we'd like to do is use drones to look at the built environment, generally, and the built environment has made up of lots of lines. So what we would like to do is to do a lot of our processing, our vision processing, with lines. And lines are much more difficult, classically, in computer vision than points.

A lot of reconstruction is done with points. You've got a point cloud. You can get structure from motion. Some motion capture, for instance.

When you see someone in the suit with all the ping pong balls, they're connecting points. They're all points, exactly. But then even not with motion capture, just with cameras that are moving, you can get points, you can match points. So we would like to do this with lines.

Lines are difficult, and our mathematical framework that we will use for this is geometric algebra. Okay, so let's define it. Let's start there. Me, shall I define it?

Okay. And we'll see if it ties in with what you've heard about it. So do you want some history? Yeah, let's start.

So, Grasmann was a mathematician, and Grasmann had something called an outer product. So, for example, I can take two vectors, I can put a wedge between them, a wedge product, or an outer product, and I get A wedge B. So this quantity is now a different thing, and I'll explain what that is. So Grasmann had this outer product, and Clifford, William Clifford, who was actually at Trinity, before I moved to London.

So he came along, and he extended this outer product. He effectively had an inner product plus an outer product. So for example, if I have two vectors, A dot B, where the dot is an inner product, giving me a scalar, and A wedge B, which gives me this other thing, is a Clifford product. So, seems like a strange thing to do, that he had an algebra for this Clifford product, which is called Clifford algebra.

And it's been in the mathematics literature and research programme forever, since Clifford died in the 1870s. He died at the age of 34, I think, but of TB. So he did a lot, but clearly could have done much more. So, but as a kind of applied tool, it was really used.

And it was David Hesseners who came in the 1960s and said, gosh, look at this, Clifford called it geometric algebra, I'm going to call it geometric algebra, and I'm going to do all these wonderful things for this. So effectively, that's the background. The basis is, imagine I have scalars, so just numbers, vectors, so things with a magnitude and a direction. In 3D, we can think of this.

Five vectors, they're planes. So things that have two vectors that mix a plane. So a plane would have a position and a magnitude and a handedness. So suppose I have three points that make up a plane, I can sort of go from A to B to C, or from A to C to B.

So if I take A wedge B to form my plane, and then B wedge A will give me minus the plane. So you have to then start to think of these planes as geometric objects which have a sign as well. So if I take vectors A wedge B wedge C, it gives me a volume. Again, it will be an oriented volume.

If I live in four dimensions, A wedge B wedge C wedge D give me a four volume. So at some point, I get to the highest element in space. In 3D, that will be a volume. I can't go any bigger.

And that has a special place in my algebra. So imagine I have in 3D, scalars plus vectors plus bivectors plus trivectors. So points, lines, planes, volumes. I have an algebra which takes these things as objects, and I can add them.

I can multiply them. I can differentiate the structure. So it's a kind of abstract concept, but it is amazingly powerful. So let's see the kind of rough idea of...

Right. How Clifford algebra moved to geometric algebra. And this is all predating computers. So this isn't being rendered anywhere yet.

You mean in hardware? Yeah. So we have a community, not a massive community, but there are lots of people who are very interested in potentially getting instructions for chipsets, etc. And PGA is it is.

And of course, we have quite a lot of programs that people have been building, so that the community can test these things out. And so what rekindled it in the 60s to make it happen now? So David Chestner was doing his PhD, and he was a physicist and looking at space time. So basically his PhD turned into a book called Space Time Algebra from which basically in the 80s people got hold up and started to get interested in.

Even though David had been working on it throughout since the 60s, what started David on the Clifford algebra? I am actually not quite sure. I'll ask him. Yeah.

And then so... So he very quickly realized that this algebra simplified a lot of space time physics. Right. So that's kind of what I wanted to get at that.

What was the realization that made it like, why is this important now? In effect, space time physics, quantum physics, relativity is extremely complicated. And is, you know, it's the area where you do have to have a lot of background knowledge and a lot of background mathematical knowledge. You do need to be proficient in, for example, for general relativity, you need differential geometry.

You need a lot of these mathematical systems. You need a lot of tensor analysis. So, I mean, David could see that with this algebra, he could work entirely within an algebra of geometric objects, transformations between these objects. Everything stayed in the algebra.

Transformations, like linear transformations functions, were geometrically intuitive, were mappings of objects, objects, they weren't just tensors. We didn't have to go to another space, you know, sort of dual space as you do in differential geometry. So things became easy. And he started to see that you could interpret a lot of things like, do matrices.

Well, actually, they're not matrices at all. They were, you know, elements of the algebra, and immediately they became easy to deal with. So, his big motivation was that here was a unifying language for mathematics and physics, basically. So, if you know this language, you can not only do rigid body dynamics in engineering, classical mechanics, you can do linear algebra without matrices and without tensors.

And you can do complex things in quantum space-time physics with the same mathematical system. And this is not theorized, this is proven. It's all good. It's just a clarification.

Maybe this might work someday. The part of the problem is that you're next to what you might be. Well, if it's so wonderful, why does everyone use it? So, A, Clifford died when he was 34, and it was at the point where Gibbs, heavy side, came in and produced vector algebra and the cross product.

And so, have you ever thought about cross product? So, I have two vectors, and I take A cross B. It gives me the vector, which is perpendicular to the plane. Now, that's all very nice, but it only works in three days.

It doesn't work in any other dimension. Because in a plane, in a non-product of the plane, there's no concept of a perpendicular to a plane. But of course, we will grow up with vector calculus, linear algebra, matrices, and then we have our research areas. And it's very difficult to actually listen to somebody if they come along and say, I've got this super-duper new mathematical system that you ought to take notice of.

Someone comes up to me and says, look at this. I will say, well, I've got my own research, and it takes me all my time to do what I'm doing. And so, what plane did you see that you wanted to follow this path? So, you're the truth.

Sure, that was a goal. So, the truth is, Michael's been, met David Testus, Anthony, and Lays and B. He became fairly obsessed with geometric algebra. He is a cosmologist.

He could see that there was this thing which told him what the co-she-remen equations are, for people who know it. That the power matrices and the direct matrices in space and quantum physics were just really interpretable. He became obsessed with it. Totally obsessed.

So, Anthony had a PhD student called Chris Doran. Chris and Anthony, they wrote a book on geometric algebra for physicists. And it was very difficult for me to actually talk to my husband at that point without actually finding out something about it. Because he thought it was one of the most exciting things he'd ever seen.

And you were at the time just pursuing math at PhD? What were you? I was pregnant at the time. So, I was just trying to have a baby.

So, I did have some time actually. Because I was, you know, not that much time, of course, when you got a small child. But, yeah, I was a gone-back as an engineer. I was a postdoc in engineering.

I was doing imaging. I was imaging flames. That was a two-year postdoc position. But I actually then began to realise that this algebra would be really useful in parts of engineering, particularly things which involve rotations.

And maybe I should talk about rotations. Because in engineering and physics, rotation, the way geometric algebra deals with rotations is totally key. So, have you heard of quaternions? I've heard of them, yeah.

So, quaternions are, well, go back. People rotate with rotation matrices. So, you have, say in three dimensions, you have a three by three matrix. You act on a bet and it rotates it.

Now, a rotation has just three degrees of freedom. We have nine. Nine components and a three by three matrix. And so, they're all constrained.

So, rotation matrices are not numerically nice to deal with because you have to keep them on the manifold. You have to make sure, if you change, if you update, rotation matrices, you have to make sure it obeys constraints. So, it's one of the things we use. Now, in graphics, et cetera, and satellite motion, it's long been known that the best things to use is quaternions.

Other methods are Euler angles, like rotation about the x-axis, the y-axis and z-axis. But quaternions have been particularly nice because they are minimally parameterised. They have three components. They are smooth.

They don't suffer from singularity problems. And Hamilton created quaternions as an extension of complex numbers. So, complex numbers. Everybody, people know that complex number i.

i effectively rotates in the plane, and multiplication by i. So, Hamilton spent many, many years in his later life trying to extend complex numbers to three dimensions. And he came up with quaternions. So, quaternions have these elements i, j, k, which also were to minus one.

So, it's like three imaginaries. So, no, to start with, that's pretty awful. But everybody knew. They were great to the libraries.

People have been using quaternions since the early days of satellites. So, if you actually look at code, et cetera, people will use this for rotations. Now, very early on, if it's in David's book, you see that if I square one of these bivectors in three days, it's a second three day for now. If I square it, if it's a unit back there, I get minus one.

So, I have a real object that squares to minus one. Okay. Which is kind of telling me that i, complex the unit imaginary and this j, k are probably unit planes. It turns out that the quaternions are just rotations.

The i, j, k give you rotations about the unit planes in three dimensions. The x, y, the y, z and the x, z plane. So, immediately you see that you can have complete generalizations of things that do rotations in any dimension. And so, at that point, you're doing this postdoc position and you realize that you could apply it.

Yes. Not to flames, but to, to, mainly computer vision. Okay. And what was the stage, like the state of computer vision at this time?

What year was this? 1993, 1993. Okay. So, not much happening.

Yeah, but a lot of the, you know, it was, it was really starting to move forward. There was no machine learning in computer vision, but it was all geometry. It's basically all geometry. So, people had used, projectable, you know, people had been using the ideas of projective geometry in computer vision for a long time, which is a four dimensional.

Okay. Space. So, it was matrices. So, I rotate and translate things.

And I have lots of cameras. I want to find from my images the rotations and translations between my cameras. Once I've done that, I can triangulate and I can do 3D reconstruction. Okay.

There was also at that stage a lot of Bayesian statistics were coming into computer vision. So, tracking things in images and finding most probable, the most probable tracks in crowds. So, computer vision was really starting to take off. Okay.

And so, you, you finished up your two year, you're creating these flames, these graphics essentially. What do you jump into to actually give it a go? So, I was extremely lucky because by that point I had two small children. I don't know if you heard the Royal Society in the Royal Society, the body in London.

Okay. So, you probably have because it's, it's has journals and lots of, you know, historical people with big names in history were fellows of the Royal Society. So, the Royal Society has something called University Research Others. And I applied to do, it was probably quite a step for them because I probably, I applied to do applications of geometric algebra in engineering.

And they gave it to me. So, I then had a five year effectively postdoc. I could choose what teaching I did in the engineering department were very good. You know, I didn't, I didn't need to do a lot of teaching or admin, but I basically had kind of five years to try and get this off the ground.

Kind of figured it out. And what was an example of an early project? So, so another project was actually with, I think at the time, the internet wasn't really like it quite like it was today. But there was a mailing list and there was somebody who called Eduardo by Rocha Rachano, who's now in Mexico, but he put this email out on some listing, anybody working with Compute to Vision and Geometric algebra.

So, I actually contacted him. And those early days, we did quite a lot of translating all the classic, projective geometry, computer vision, which was quite mathematical at the time, integer geometric algebra. Gotcha. Okay.

And I mean, this may sound kind of basic, but one of the really nice things about putting, putting your problem, et cetera, integer geometric algebra is you have an origin, an original, so some inertial frame. Everything is with respect to that. I rotate, translate that. I don't have matrices, so I don't have to worry about coordinate systems.

I don't stack up coordinate system upon coordinate system and then worry about what on earth that translation. What coordinate system is that in? And with Vision, where you're measuring things in an image. So what coordinate frame is that in?

It is quite confusing. And I know this, in fact, because students are very confused about it all. When you've got a rotation, where's it with respect to in these complicated systems? This just makes life very, very easy.

And you almost can't go wrong. It was a goal of this five year period to then apply it to some like product use or just do the basic research and see how it goes. At that time, it was just really seeing what we could get out of it. So I did some work with a company called SpaceSpace, who's their motion capture company in the States, not often Berkeley.

And looking at algorithms to calibrate cameras, because one of the things I haven't mentioned is that there was another aspect to this system, which is a system that was used to be used in the system. Which is extremely useful. So I've said that I have these geometric objects. My rotations are objects.

So I can write down coordinate free expressions. But not only can I write them down, I can differentiate with respect to them easily. Because I have this algebra of objects, I can do calculus on them. And that's quite hard to do conventionally, because you've got people can do it, but you know, you're differentiating to a matrix or a tensor or a vector and all this.

So it's a much harder. You can do it component wise. But if I want to get close form solutions, doing the analytic stage of the calculus, is extremely useful. And so the notion was that you could do it with less compute, like you could render these things using geometric algebra faster.

Never. Okay. So you can do pretty fast. It's much easier to program up.

It's intuitive. You can think of what you want to do. And I can program it up at this high level. Underlying, you've got an algebra of a much bigger algebra than three dimensional spaces.

So actually, computation is more going on. But at a higher level, I can get code to do all this for me. At a higher level, I can certainly rotate this object to this object. Okay.

And then in terms of the state of computer vision from then until now, what has progressed to make this drone project possible? Okay. So this is an interesting question. So David, Hessen, and Garrett Sockic wrote a book.

This is a sort of real reference book of Clifford Algebra to geometric calculus. It's got everything in there. It's not a book you read. It's a book that you go to.

It's like a reference. Yeah. I'm sure they meant it for you to read. And that was 1984 or so.

In 1999, David gave a talk at a conference. He'd done a paper with Hongbo Lee about something which was in the final few pages of this book. Of course, not many of us have ever got to the final few pages of this book, which you call conformal geometric algebra. So this is truly stunning for graphics and for originally projects, projects which use vision.

So conformal geometric algebra is a five dimensional space. So imagine you take Euclidean space, the space we live in, you effectively add on two more vectors. One is a point of infinity. Projective geometry effectively adds on one more vector.

So this is this five dimensional space. So you get this five space and you say, well, okay, what this is going to get me. Well, what this gets you is that points, lines, planes, circles and spheres become objects in the algebra. So objects, you give me a C, this big C is a circle, it's a trifecta in my five dimensional space.

And rotors, which are these, this class of objects that rotate, encompass rotations, translations, dilation, and rotations, translation, dilation. So you've suddenly got I can, for example, set up, you give me three points, I immediately get the circle that goes passes through, you give me four points, I give you the sphere, it's an object. I use my rotor, I rotate it, I intersect them. It's a beautiful language for graphics.

So, but because, for 1999, lots of people, a third number people. So, Leo Dost and Steve, a man in Daniel Fontaine in Steve in Canada, Leo Italian Amsterdam, rotable geometric algebra of computer scientists. And a lot of that was based on this conformal, just what you could do, just how easy it was to do things. So, because lines are just objects, because I know how to sort of compare one line to another, because I can intersect lines of planes, keeping this nice algebra.

It becomes quite, almost easy to see what I have to do to implement a variety of algorithms. So, it's quite hard to intersect two spheres. People could do it, but in the past... Well, it's easy to imagine how it works, but yeah.

Yeah, but you've got equations, but here we just do have operators that do it, we're between objects. And one of the beautiful things is, we live in a Euclidean, what extensive purpose is there, what we see as Euclidean. Now, if you have a different underlying geometry, so if you have an hyperbolic or spherical geometry, then in this algebra, you have to change, in conformal algebra. Euclidean geometry is the thing that keeps the pointed infinity invariant.

Then if I keep other things invariant, I get these other geometries, and I can just use my standard apparatus and do exact same things, rotate my objects in hyperbolic space and rotate my objects as a spherical space, move them around. So, it is a beautiful language of geometry. And I haven't even touched it on. Physics was a lot of decidospace, which a lot of cosmologists work in, is a different geometry.

Okay, gotcha. Which you can use this for. Right, which is why it was appealing to your husband 30 years ago. Exactly.

So he started out by being amazed at how things like Pauli and the Dirac matrices spin as well, all just trivial in this algebra. Then he started to realize that these complicated transformations, which are all written in tense notation. Or actually, if you put them in geometronic algebra, they are mappings between real things like bivectus to bivectus, or bivectus to vectors, things like this. And as soon as you see it in this way, it enables you to interpret things and then maybe move on.

And so, for example, so Anthony and Steve Gold, Chris Lauren and David Tessin are interested in a series of gravity in flat space, which produces all. So, there's the irregular gravity you can understand. If you want to stop the algebra, basically. Okay, gotcha.

And so, as you were saying before, the reason why people aren't picking up geometric algebra is that you become kind of in a certain track, and you know what you know. But right now, what are people using for modern computer vision to do comparable work? So, computer vision is massively advanced. Of course, yeah.

So, today, people are really now moving from geometry to machine learning. So, you're using a little bit of geometry, but you are learning to segment things, recognize things by giving lots and lots of images. But you know, you still have lots of geometric problems. So, we still have to extract things from images if we've got moving cameras and things like that.

But we have, I suppose, it's a case of geometric algebra won't really give you anything that you can't do conventionally. But what it might enable you to do is to see how to do that thing. So, for example, if I ask you how close this one line to another, I have a way of doing that in my algebra. If I could sit down and write it in conventional.

Of course, yeah. But could I have actually thought of that conventionally? Probably not, because I'm not clever enough. So, okay.

So, you know, I need a set of tools, which makes sense to me, geometrically and physically. And I can then think of other people, think about how to extend that. Okay. And so, in your day-to-day research, how are you then applying machine learning?

Because many of your PhD students are working on exactly that. Okay. So, it's almost impossible to avoid machine learning. Yeah.

You can try, but you can't avoid it. So, I have two sort of strands. So, some students are applying conventional machine learning techniques, or conventional, you know, they change all the time. Neural networks, recurrent neural networks, LSTMs, two classical data, like medical time series data.

And I don't know how you can use geometrical algebra for that. Okay. So, and, you know, doing image segmentation, et cetera. So, classical image segmentation, well, you know, I'm not sure how you can get geometrical algebra into that.

But as soon as it comes to anything involving, like a moving camera, moving drone, multiple moving cameras, having streams of images, you want to match things, you want to triangulate, et cetera. Then, it is almost the only way I know how to do it. Okay. So, it's that aspect of it.

And then, can we extend it? So, can we then, you know, we know how to parameterize in this algebra on my lines, my planes, et cetera. Can I learn them? So, can I learn these geometrical objects?

So, are we working on analyzing both sort of moving images, which we're going to extract lines, lines, and also motion? So, motion. Can I actually parameterize my problem in terms of my geometric objects and learn them? Okay.

And so, for instance, like, you know, someone doing computer vision with a self-driving car, like, are they applying the same techniques that you're applying to get these lines? Or what would they use? I think, no. So, so, you know, there's huge amounts of research.

Yeah, yeah, of course. And who knows what they're doing. But primarily, it's people are using, if it's not like, if it's not LIDAR. Right, yeah.

Single camera, multiple camera, lots of data, Bayesian methods for segmentation. Following lines is easy. You know, it's not that's not matching them or trying to reconstruct them. You just kind of following them.

It's really recognizing if it's a person, if it's a road, if it's a tree, and you've got multiple sensors. So, you know exactly where your GPS is really accurate these days. So, no, is it? Yeah.

I think I don't know. So, I think the question I'm kind of getting around to is, like, where are the other applications? Right? So, like, in your instance, like, you're rotating in a camera, you want to map something, makes a lot of sense, like, and then you can move it around, but because you have the lines you can recreate the shape.

What are the other use cases? Well, you know, I do think that it's real use is this, the fact that it unifies. It's a unifying language. So, if I know this, I can just work things out more easily.

Instead of trying, I mean, if people have worked with computer vision, they will know that often things that work. So, instead of a rotation matrix R, they try R transpose. Instead of a translation vector T, they try R transpose T. And they mess around until it works because it's kind of confusing.

Okay. You've got no such problem here. It's very straightforward. Now, okay, that's not a good reason to use it if you're proficient enough with classical techniques.

But I can go into fields like thin shell elasticity. No, and there is a student who is here at moment doing this. That is quite in full field. But if we put it in terms of, like, I have these rotors, I have a surface, I can stretch them, I can translate them.

Then it's just in the language of computer vision. It's not a differential geometry. It's not in anything else. It's a totally understandable process.

Right. Okay. So you could see a world where an artificial intelligence understands this and then can apply it anywhere. And then you can then understand it.

Yeah. Yeah. I think that's where strength lies. And in, and in your physics.

I think, I think, you know, because it unifies lots of different quantum mechanics and relativity, and it's not unifying quantum relativity yet. But you can see that it is a system where you might be able to think of different ways for it. Right. Okay.

So it's actually a better tool. Better tool. In a better tool. Okay.

People are extremely clever. So they will find ways of doing things that are stunningly clever. But are complicated. Right.

And so where do you see limits right now? I think it's not computational anymore because we're building up more and more tools. So I can give you a website and, you know, you can try it out and you have to install anything that is online, you know, type in notebook so you can have a play with it. So we are, you know, we are a community which is certainly moving forward.

There still isn't, not a lot of us. So it's not taught. It's not taught. So, you know, teach my four years in each processing.

But I don't teach them geometry algebra. Okay. And is there an interest there to learn or they're just like, I don't know. When you don't know about it, you can't really be interested in it.

Okay. So it's a, it's a, and students have no hand-gops. So they learn it and I think, great. You know, it's another tool.

Yeah. Yeah. And use it. Okay.

So, um, it really, it enables you. I mean, I have a lot of confidence that if you give me a paper on, uh, no, cinch shell elasticity, it's going to be tough because it's tensed everywhere and I've got frames and frames and frames and dual frames and things, but I can eventually understand it. So these different fields, if I was a colleague, Alan McGrobe, he was using a sort of form of it in structures. So there are fields.

Electromagnetism is also, I should have, I should have mentioned this, is a field whereby you really get huge simplification. So I am fairly confident I can model, um, electromagnetic fields and do maybe some new engineering things using this if I had time. Okay. Well, I was, we were talking about programming before we started recording at lunch and about going from that lab to Python, right?

Have people tried to create a port? In other words, like, oh, you have this traditional equation, we can port it over and then you can understand it and see the value. Has that happened yet? Um, no.

Okay. Okay. Not really. Not really.

I mean, there are, I mean, most of the code is, um, enabling you to do things in a kind of transatlantic way, like I can wedge together two vectors or multiple vectors together. I can conformal, I can form my sphere visualizing and I can do numerical, I can do lots of numerical computations with it fast. So we can now do it quickly. But porting things is a kind of difficult one.

Yeah, like you're saying, well, you know, here's my equations in terms of powerly spinners. Or does it look like in Juma to Cartier? Well, kind of looks the same except your spinners aren't spinners. They're, you know, uh, so it's a difficult, that's a very hard, um, question to answer.

Right. Cause it's not so simple. It's like, oh, this is how you call this function. It's not the same thing at all.

Right. Okay. So this is sort of a weird random tangent. But before when we met, you were telling me that a couple of years ago, someone posted on Hacker News, one of your papers.

Yeah. What was it? What was it about? How did all this happen?

So it was an invited paper, um, the Millennium Edition 2000 of philosophical transactions in rural society. So we wrote a paper, which was telling, just saying how GMA was a unifying language and look at all these great things we can do in computer science, engineering and physics. Yeah. And this is the way the world works and this is, you know, you should, you should do it.

So it, you know, some citations and it sat there until bizarrely, um, it was posted. Yeah. And it was a friend of mine who reads these things, texted me and said, well, your paper is number one read on this Hacker News. And I said, well, what's that?

Yeah. I don't know. So hard to know how it, how it emerged and hard to know how many, I mean, where the some people who looked at that and now are in our geometric algebra community, who, you know, started getting interested and came along. Um, throughout the world now, there are groups in the 1990, there were not many.

Yeah. There were little groups who were really keen on it. And now there are groups, you know, almost everywhere. They're not big.

Sure. But, um, interesting. Did you, did you read the comments when I went up there? I did.

Yeah. And then we arranged from, wow, this looks really cool to this looks crazy, you know, uh, and impossible to understand. We, you know, I thought, no, that doesn't, you know, that wasn't, you know, that wasn't, you know, that wasn't the, we're not trying to communicate that. No, we didn't want to say that to gosh, this looks as though I can do everything, you know, so it went, they went very detailed.

It's more people beaming in and saying, um, do be short things. Although I don't know whether I read them all. That's an issue of the internet. You have to comment quickly.

Yeah. It falls off the front page and then you're coming. Yeah. Right.

Exactly. Right. But, um, what, what do the folks who are, are there naysayers? Is that a community?

Sorry, what? Are there naysayers of geometric algebra? Sorry. It's going to be a thing.

Um, um, say so. I mean, there are people who think, yeah, that looks interesting, but really I can do it anyway. Yeah. And I have, I am, I am the world's expert in, you know, probably matrices or whatever.

Why do I need to put my matrix in terms of a vector? Um, and so, you know, people think in different ways. Um, to them, that is the way of doing it. That's always been the way of doing it.

Um, so I don't think there are people who say, oh, this is complete rubbish. There are just people who say, oh, yeah, but, you know, what's it going to get me? Okay. Um, and I don't think people, it's not the kind of thing where you can say it's wrong, because it's not wrong.

Right. It's just a tool, which, you know, depends whether you want to invest the time because you think about multiplying vectors together and things like this. Yeah. It's, it's anti-commutative.

It's not a commutative algebra. So immediately you throw away everything you've learned as a kid and through school and through university. It makes perfect sense once you're into it. There's a little, you know, there's a little hump that's a learning curve.

Yeah. If there are some things away. Yeah. And it's much easier for younger people to do that.

Sure. That makes sense. Because I have no real prejudices. Right.

Okay. They say, yeah, it's another thing. I was like, you know, I'm not going to go. Yeah.

Another algorithm. I'll program it up now. Okay. And so yeah, I don't, I think probably there are people who do think, well, why should I bother that?

Um, but I don't think they would say, no, it's the wrong thing to do. You know, one shouldn't do that. One should use matrices. Okay.

So, you know, educated opinion, where do you see this really taking hold in the next, I don't know, like thinking about it in a practical sense. So many people who are listening to my C-podcast are like entrepreneurs or engineers are studying. Where do you see the people applying things like this? Um, so clearly in the, in the fields of fundamental theoretical advances, so, you know, physics, or the aspects of physics that we're not quite sure about, it's a big thing.

Where I see it will have some effect. It enables you to think, and it enables you to think in different ways. In engineering, I think there are lots of fields and I don't know whether the, is there a killer application where if I could do it, everyone would say, whoa, this is the way to go. Right.

I don't know. Um, as I say, I, I would, what I would like to see is that people, people had it in their toolbox. Yeah. Because, um, you know, there are lots of very, very clever people around who can cope with very hard, sophisticated physics and maps.

There are a lot of people who are maybe not quite so clever, and they need the right tools in order to do these complex things. And I see it as, actually, this provides these people with a real, you know, a lot of people have a lot of geometric insight, but maybe not the mathematical sophistication. I think this will certainly sort of give them a big advantage, um, because it seems to me to be the way the world works. This, if it's a unifying language, it's what we should be writing our equations in.

Okay. Um, so I'm not sure that's answered your question. No, no, that's a great answer. Yeah.

I believe that. Um, if you weren't working on this, do you have thoughts on where you might apply your, your energy? Well, it's interesting because I have always had, um, I'm a runner. So I've been a runner since I was a kid.

I, I am absolutely convinced that even if you don't run, you know, as you get older, you need to move and you need to keep your body moving independently. You need to make the muscles move independently. You need to keep healthy. And I, I probably would be doing something which tried to get everybody out and moving and that, that was not what you were expecting.

No, it's great. Yeah. But, you know, I look at people and as I get older and life could be so much better for them if they kept mobile and they, they looked after their body. That's quite a obsession of mine.

But anyway, yeah. That was great. That was so awesome. Yeah.

I guess then my last question is if someone wants to learn more about this and actually start trying it out, where should they go? Well, so lots of books available and I, I, people probably be annoyed with me if they don't mention their books. So, the David Hessen says three books, space time algebra, probably not the place to start, new foundations of classical mechanics and geometric algebra, Clifford algebra to geometric completers. Now they, they great books, Michael's been Anthony and Chris Doran have a book, Geometric Algebra Physicists, Leodor, Stephen Mann and Daniel Fontaine have a book which is Geometric Algebra for Computer Scientists.

It's a word about a, by a conchano has books which are more folks on robotics. So there are lots of books out there, lots of review articles, lots of conference proceedings, lots of code now. So people, you can kind of get code for Matlab, C for Python. We've been using now a package by an American guy called Alex Olsenovitch who wrote a Clifford package and we've been integrating it into, I know, a nice one of my students who go, who you met, has created a web version.

So the big problem with a lot of these things, you have to download the package, you've got the package, you've got to get the number, you've got to get all these other things. You've got to get it all working. You've got to get it all working. So if you're not about your windows, it's like impossible, not impossible.

But your average person would probably say, you know what, no. But we were looking to try and get a web version so people can actually go on, try it out with some really files and a bit of graphics so you can see it. So there's loads of stuff out there at the moment. Where can I find that if they want to?

Is it on your side? It is Hugo and Alex have been working on this and I can give you the web address. We can put it in the blog post. Yeah, yeah, they're just releasing it to the world.

Okay. Yeah, perfect. So you can have people tested out and say, email back and say, this doesn't work. Yeah.

All right. Thank you so much for your time. Okay. Be no pleasure.

All right. Thanks for listening. So as always, you can find the transcript and video at blog.Y Combinator.com. And if you have a second, it would be awesome to give us a rating and review wherever you find your podcast.

See you next time. Bye.