Sep 17, 2018 · 50m · y-combinator
Joan Lasenby on Applications of Geometric Algebra in Engineering · Y Combinator
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gold bands on the timeline = statements, start to end. Hover to read, click to jump. CC turns on captions
In this interview, Cambridge researcher Joan Lasenby discusses the principles and practical applications of geometric algebra, highlighting how it unifies theoretical physics, revolutionizes computer vision and robotics, and overcomes traditional matrix limitations.
How this conversation actually went
Every chapter scored 0–10 on four independent dynamics. Hover any point for the reasoning behind the score. How this is scored →
The partners as informed peer 2.1 Guest teaching 6.8 Guest disagreement 0.8 The partners pushing back 0.4
speaking balance: gold is the partners, purple is the guest (3 minute bins)
Sharpest disagreement ▶ 22:27 Directly rejecting the reduced compute premise
Joan immediately and flatly corrects Craig's assumption that geometric algebra allows rendering with less compute, clarifying that it actually involves more underlying algebraic operations while providing superior developer abstraction.
Hardest push from the partners ▶ 38:48 Pressing on automated equation translation toolsCraig presses Joan on whether the community has built software ports to automatically convert conventional matrix equations into geometric algebra expressions.
Biggest teaching moment ▶ 9:30 Deconstructing the mathematical limits of the cross productJoan exposes the fundamental shortcoming of standard vector algebra by explaining how the cross product fails entirely in dimensions other than 3D because planes lack unique perpendicular vectors.
The partners hold their own ▶ 38:48 Connecting mathematical porting to software compilation paradigmsCraig applies practical software engineering intuitions about porting legacy codebases between environments like MATLAB and Python to gauge translation feasibility in geometric algebra.
the scores for every segment, with the reasoning behind each
| Chapter | Topic | The partners as informed peer | Guest teaching | Guest disagreement | The partners pushing back | Why |
|---|---|---|---|---|---|---|
| Defining Geometric Algebra from Historical Foundations to Modern Computation | 2 | 7 | 1 | 1 | Craig Cannon asks foundational questions about defining geometric algebra and its timeline relative to hardware. Joan Lasenby systematically breaks down Grassmann's outer product, Clifford algebra, and multivectors like bivectors and volumes. | |
| David Hestenes and Geometric Algebra as a Unifying Framework for Physics | 2 | 7 | 0 | 0 | Craig asks why geometric algebra matters today, and Joan outlines David Hestenes's realization that geometric algebra serves as a unifying language for spacetime physics, quantum mechanics, and linear algebra without matrices. | |
| Historical Obstacles to Adoption and Limitations of standard Cross Products | 2 | 8 | 1 | 0 | Joan explains the historical limitations of Gibbs' cross product to 3D and details how bivectors generalize quaternions and rotations across dimensions without numerical constraint degradation. | |
| Early Research Collaborations, Coordinate-Free Algebra, and Analytic Calculus | 2 | 7 | 2 | 1 | When Craig asks if geometric algebra reduces compute requirements, Joan directly clarifies that it doesn't compute faster, but rather enables coordinate-free formulations and analytic calculus on geometric objects directly. | |
| Conformal Geometric Algebra and Generalizations to Non-Euclidean Geometries | 2 | 8 | 0 | 0 | Joan introduces Conformal Geometric Algebra (5D representation), explaining how points, lines, planes, circles, and spheres become first-class algebraic objects and generalize seamlessly to hyperbolic and spherical geometries. | |
| Integrating Geometric Algebra with Modern Machine Learning Workflows | 2 | 7 | 1 | 0 | Joan openly acknowledges that geometric algebra does not replace deep learning for tasks like standard 2D segmentation, but excels at parameterizing and learning geometric entities in multi-camera dynamic motion. | |
| Practical Application Domains and Eliminating Matrix Translation Hacks | 2 | 7 | 1 | 1 | Joan contrasts geometric algebra with conventional computer vision matrix hacks where engineers blindly transpose matrices until code works, highlighting its clarity in thin-shell elasticity and mechanics. | |
| Educational Obstacles, Open-Source Tooling, and Code Porting Challenges | 3 | 6 | 1 | 1 | Craig inquires about automated porting tools between MATLAB, Python, and geometric algebra representations. Joan explains why direct syntactic porting is challenging due to the structural shift in how entities like spinors are modeled. | |
| The Viral Hacker News Paper and Overcoming Non-Commutative Mindsets | 2 | 6 | 1 | 0 | Joan reflects on her paper going viral on Hacker News and notes the mental hump students face when letting go of standard commutative algebra assumptions. | |
| Future Impact in Engineering Toolboxes and Personal Fitness Passions | 2 | 5 | 0 | 0 | Joan concludes by advocating for geometric algebra as an accessible tool for engineers and physicists, before sharing her personal passion for running and functional mobility. |
Statements from this episode (16)
Lasenby: Line-Based Vision Processing Is Classically Much Harder Than Point Clouds
“Lines are much more difficult classically in computer vision. Than points. A lot of reconstruction is done with points.”
Lasenby: Engineers are developing geometric algebra instruction sets for FPGAs and chipsets
“We have a community, not a massive community, but there are lots of people who are very interested in, in you know, potentially getting instructions for chip sets, et cetera. FPGA is, it is. So, and of course we have quite a lot of Programs that people have be…”
Lasenby: Geometric algebra unifies classical mechanics, linear algebra, and quantum physics
“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 sy…”
Lasenby: The vector cross product only works in three dimensions
“Now, that's all very nice, but it only works in three D. It doesn't work in any other dimension. Because in a plane, I've got no perpendicular. You're stuck. In four dimensions, there's no concept of a perpendicular to a plane.”
Lasenby: Rotation matrices are numerically inconvenient due to manifold constraints
“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 the constraints.”
Lasenby: Quaternions prevent singularity issues and parameterize rotations smoothly
“Quaternions have been particularly nice because they are minimally parameterized. They have three components. They are smooth. They don't suffer from singularity problems.”
Lasenby: Geometric algebra enables easy calculus on coordinate-free objects
“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. So, because I have this algebra of objects. I can…”
Lasenby: Geometric algebra requires more underlying compute but simplifies code
“Underlying, you've got an algebra of A much bigger algebra than three-dimensional space where I've got three vectors. So actually, computationally, there's more going on. There's more going on. Yeah, but at a higher level, you know, I can get code to do all th…”
Lasenby: Conformal geometric algebra turns geometric shapes directly into algebraic objects
“Well, what this gets you is that points, lines, planes, circles, and spheres become objects in the algebra. They're objects. You give me a, it's a C. This big C is a circle. It's a trivector in my five dimensional space. And rotors, Which are these, this class…”
Lasenby: Conformal geometric algebra handles non-Euclidean geometries by altering invariants
“Now, if you have a different underlying geometry, so if you have hyperbolic or spherical geometry. Then in this algebra, you have to change, you, in conformal algebra, Euclidean geometry is the thing that keeps the point at infinity invariant. Then if I keep o…”
Lasenby: Geometric algebra adds no new capabilities, only intuitive clarity
“Geometric algebra won't really give you anything that you can't do conventionally. What it might enable you to do is to see how, how to do that thing.”
Lasenby: Geometric algebra eliminates trial-and-error matrix hacks in computer vision
“If people have worked with computer vision, they will know that often things don't work. So instead of a rotation matrix R, they try R transpose. And instead of a translation vector T, they try R transpose T. And they mess around until it works because it's ki…”
Lasenby: Computation is no longer a bottleneck for geometric algebra
“I don't think it's not computational anymore, because we're building up more and more tools.”
Lasenby: Geometric algebra provides huge simplification in electromagnetism
“Electromagnetism is also, I should have mentioned this, is a field whereby you really get huge simplification.”
Lasenby: Geometric algebra requires unlearning standard commutative mathematics
“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. So it makes perfect sense once you're into it, but there's a little, you know, there's a little h…”
Lasenby: Geometric algebra empowers practitioners with geometric intuition over advanced math
“A lot of people have a lot of geometric insight But maybe not the mathematical sophistication. And I think this will certainly sort of give them a big advantage because it seems to me to be the way the world works. This, if it's a unifying language, it's what …”