Clifford Algebra: A visual introduction (2014)
slehar.wordpress.com
slehar.wordpress.com
I'm aware that Geomerics purportedly used it for doing global illumination calculations, but there was scant specifics (understandably) about the actual implementation. So we don't know in what capacity it was used.
I've heard it praised for physics applications, but I haven't seen anyone walk through, say, a simple rigid body simulator and show its improvement.
I've heard it spoken of as the future of ray tracing, but again I've not seen anyone willing to put an implementation (even a toy) up against a ray tracer that you could write in an hour with Gibb's vectors.
On the contrary, when I've tried to use it, the complexity of GA just makes it impractical. The problem with 4 or 5 dimensional GA for doing work in 3D, is that we end up dealing with 16 or 32 values per object (if you unify vectors, bivectors, etc), or else having five or six specialised forms of object plus their combinations (if you keep them separate), with potentially hundreds of specialisations of basic operation. This isn't just me, the GA libraries around have this bloat. I've just not found an application in which that massive increase in storage, computation and implementation complexity is worth it.
Which makes sense. In a regular vector based system, we might use vectors and quaternions (2 things) to model rigid bodies. We can do a lot with those. But it isn't clear what thing is difficult or inelegant to model with them, that a trivector/scalar would be the right tool for. So that extra flexibility to be able to store such exotic mathematical objects is a waste for computer and programmer.
I came to the conclusion that GA-advocacy at the moment is mostly a true-believer type thing. True believers will tell you over and over how important it is to modelling 3d space, but rarely show you a worked example of it being easier. I am not cynical enough to think it has no killer applications, but I've yet to find them. And I would dearly love to.
But with a mainstream bit of technology, you can go to any good university math department and sign up for a course on the subject. You can take your pick of 100 textbooks. You can read resources aimed at applying specific tools to any number of concrete practical uses. You can often assume that people with a math, science, or engineering degree are already familiar with the technology and understand how it works. You can go find open source implementations of tools built using that technology. Etc.
As a result, we don’t remember how hard all those tools were to learn, and how much work went into building up the conceptual ecosystem.
With a non-mainstream bit of technology, you get to choose from a small handful of resources, often aimed at an audience you aren’t properly a member of or aimed at uses you don’t care about and missing parts that you need. Often best approaches for solving particular problems haven’t been worked out yet. From one source to another, notation conventions and choices of model details can vary, making it hard to compare sources, notice errors, or communicate ideas. Concrete code implementations have seen little scrutiny and so are likely to be incomplete, inefficient, and buggy.
This is a problem with any difficult new bit of technology. Sometimes it takes decades or even centuries for a piece of mathematical technology to go from introduction to wide adoption. Sometimes they don’t make it or sit abandoned on the shelf for a while.
As far as I can tell mathematicians haven’t adopted Geometric Algebra for a simple reason: Mathematicians make their living proving new theorems, and GA is equally/less powerful (in a theoretical sense) than existing abstractions they have, so as a result isn’t particularly useful for their proofs. Its advantages are as a sort of “refactoring” of a bunch of existing technology, and it’s usually a career limiting move in any field to spend too much of your time refactoring, however advantageous it might be for humanity in the long run.
If you’re interested in physics applications, you should check out Hestenes’s book New Foundations for Classical Mechanics, which is pretty good. I’m particularly impressed with the completeness of the treatment of rotations and rigid body mechanics in chapters 5 and 7. General relativity is a field where Geometric Algebra already sees a fair amount of use in the academic literature. Hestenes has also written papers which sketch promising approaches to electrodynamics, quantum mechanics, etc., and the guys at Cambridge have been working on those a bit, but it’ll probably take a large group of others to really flesh it out.
> On the contrary, when I've tried to use it, the complexity of GA just makes it impractical. The problem with 4 or 5 dimensional GA for doing work in 3D, is that we end up dealing with 16 or 32 values per object (if you unify vectors, bivectors, etc), or else having five or six specialised forms of object plus their combinations (if you keep them separate), with potentially hundreds of specialisations of basic operation. This isn't just me, the GA libraries around have this bloat.
This is two tricky problems:
(1) In most cases Geometric Algebra solutions to problems don’t essentially require more computation than other approaches, but figuring out how to write efficient practical implementations (especially against our current hardware and software which is heavily optimized for solving matrix problems) is non-trivial and hasn’t really seen that much work yet.
(2) Existing programming languages really aren’t optimized syntactically for interacting with multidimensional objects in a natural way. Trying to cram a GA system into C++ or Java (e.g.) is just asking for a bloated and confusing syntactic nightmare. You see the same problem with trying to write code for doing matrix algebra in those languages, but at least we have a lot of experience with the domain and there are languages like Matlab or Julia or libraries like NumPy that make things much nicer. Nothing like those exist for Geometric Algebra, and it’s a non-trivial problem to solve.
The second issue, about performance, makes me slightly skeptical of the 'made with GA' claims of Geomeric's engine. I don't doubt there's a GA system in there doing something, but I wonder how much of the actual hard problems in global illumination it is handling, given that global illumination is primarily compute-bound.
As I said, I'd like to be proven wrong. The math does have nice features in the abstract (as you point out).
Here are a couple of hour-long lecture videos https://www.youtube.com/watch?v=cQp-kB9EOQs http://downloads.sms.cam.ac.uk/1160831/1160835.mp4 and the first 6 chapters of a book explaining the basics of the underlying theory are freely available online (the rest of the book is great too) http://www.chebfun.org/ATAP/
Basic concept is to approximate functions as high-degree polynomials, to within machine precision, and then do operations on those approximations instead of on the original functions, at each step chucking out any coefficients which are smaller than machine precision. This in many cases provides a dramatic speed boost over existing methods for taking integrals/derivatives, finding roots, solving differential equations, convolution, etc. etc., while staying extremely accurate, much better than discrete solutions.
It’s not universally applicable to all functions (for obvious reasons) or all problems, but the team responsible has really done an incredible amount of solid work on the project.
Apparently it’s 10+ years old now, but nobody I’ve asked has ever heard of it before. Which is kind of surprising, because it seems amazing to me so far.
Unfortunately stuck as a Matlab library for the moment (there are partial ports to NumPy and Julia but they are very feature poor compared to the Matlab version).
A worthwhile OT. Maybe you could submit it as a link?
I want to write some examples (as long blog posts or similar) in a few concrete science fields showing how to use these tools in practice and submit those. I feel like the chebfun guys haven’t really figured out some good introductory hooks yet, or done much marketing/outreach: most of their output is papers about approximation theory which takes some amount of effort to figure out and only get seen by academics working in the same subfield, and the practical examples on their site are pretty small pieces (and don’t have comparisons vs. alternative approaches) so it’s hard to see the big picture. Those lecture videos are pretty good, but an hour long lecture is a somewhat big commitment and hard to skim.
BTW, I’m serious about the email thing: shoot me a note (address in my profile) and I’d love to hear about your attempts to implement Geometric Algebra in code. One of my goals for the next few years is to at some point sit down and write a nice web-based introduction to GA filled with interactive diagrams.
Not sure how useful it is in day-to-day reality, but at the very least it might make other GA implementations seem simple by comparison.
[EDIT] Here's the link for the chip: http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=6...
- Cross product only works in 3d
- Why is it that the cross product seems mysteriously related to determinants?
- Multivariate calculus as typically taught is very ugly
- Complex numbers for 2d, quaternions for 3d, then what?
- Rotations/reflections using linear algebra
- Cauchy's integral formula, Stokes' theorem, Helmholz decomposition, etc.
- Etcetera
GA just makes all of that much nicer and brings it together into a coherent framework.
Here's a simple problem that illustrates why GA is nice: given 3 points A,B,C in 2-d space with integer coordinates, determine whether B is on the line segment AC. Now do it in 3-d, and then in n-d.
"Hestenes ... is the prime mover behind the contemporary resurgence of interest in geometric algebras and in other offshoots of Clifford algebras as ways of formalizing theoretical physics."
Description from ABE books:
Item Description: Gordon Breach Publishing Group, Switzerland, 2015. Hardback. Book Condition: New. 2nd Revised edition. 235 x 155 mm. Language: English Brand New Book. This small book started a profound revolution in the development of mathematical physics, one which has reached many working physicists already, and which stands poised to bring about far-reaching change in the future. At its heart is the use of Clifford algebra to unify otherwise disparate mathematical languages, particularly those of spinors, quaternions, tensors and differential forms. It provides a unified approach covering all these areas and thus leads to a very efficient ‘toolkit’ for use in physical problems including quantum mechanics, classical mechanics, electromagnetism and relativity (both special and general) – only one mathematical system needs to be learned and understood, and one can use it at levels which extend right through to current research topics in each of these areas. These same techniques, in the form of the ‘Geometric Algebra’, can be applied in many areas of engineering, robotics and computer science, with no changes necessary – it is the same underlying mathematics, and enables physicists to understand topics in engineering, and engineers to understand topics in physics (including aspects in frontier areas), in a way which no other single mathematical system could hope to make possible. There is another aspect to Geometric Algebra, which is less tangible, and goes beyond questions of mathematical power and range. This is the remarkable insight it gives to physical problems, and the way it constantly suggests new features of the physics itself, not just the mathematics. Examples of this are peppered throughout ‘Space-Time Algebra’, despite its short length, and some of them are effectively still research topics for the future. From the Foreward by Anthony Lasenby.
There are a lot of pretty pictures in this article but I found it to be almost completely opaque. This is a subject that deserves a book, a class, a teacher.
> This entire analysis of Clifford Algebra was based on my own foundational assumption that mathematics is not a human invention, but more of a discovery of the essential principles of computation in the brain.
And now I'm suspicious. I prefer an objective mathematical treatment and not one with some self-described "radical" agenda.
The "big wins" are that the notation is simpler and easier to reason about, and everything is coordinate-free. As an example, there is a model of 3 dimensional euclidean space that can be represented using a 5-dimensional GA (with 4,1 metric signature). Within this model, you have lines, planes, circles, spheres, etc; but you also have rotations, reflections, translations; and all of these things are just standard elements of the mathematics with no special cases. So you can multiply a translation and a rotation and you get back the composition of those two things. Or you can multiply a sphere by a translation and you get back a sphere that has been translated by the specified amount. Or if you want to smoothly interpolate a rigid body motion (translation+rotation) between two points, you can take the logarithm of that motion. And again, this can all be done symbolically without resorting to coordinates - except at the very end, perhaps, if you want to map it onto an actual coordinate system :)
In terms of people applying it to more "advanced" subjects like Minkowski space or differential geometry - I'm not so sure. I haven't spent too much time looking into that.
For example, is there a way to use geometric algebra to better handle tensor manipulation (decomposition etc.)? That all gets pretty messy once you leave matrices and vectors.
But IMHO this is different.
You combine the current j and charge density rho into a single four-vector current J; you combine the electric field E and magnetic field B into a single bivector F. (In spacetime, not just space, so the space of bivectors has dimension (4 choose 2) = 6 = 3+3, which is the right amount for representing E and B.) Then the Clifford-algebra analogue of the "del" operator from conventional vector calculus does div-like things and curl-like things in the right combination to encode all of the Maxwell equations.
Of course, you can do more or less the same thing without calling it Clifford algebra (or geometric algebra) and you can find it done in physics textbooks. Usually this means a whole lot of tensor formulae; see e.g. https://en.wikipedia.org/wiki/Covariant_formulation_of_class.... Geometric algebra gives a nice framework for writing it all down, though.
In two dimensions, that's impossible, because there's no other direction for that new vector to point. In more than three dimensions, that's impossible, because there's no single direction for that new vector to point.
It's a hacky way to express the more general concept of orthogonality and handedness: The magnitude of the product we want is proportional to how orthogonal the two vectors are, and the sign is due to a handedness convention we've adopted.
Well, if two vectors are orthogonal to any degree at all, they span a two-dimensional surface, so why not define a product which takes two one-dimensional vectors and makes a signed two-dimensional surface out of them, with an area proportional to their lengths and how orthogonal they are and a sign due to a handedness convention? That's the wedge product in Clifford algebra.
The fun comes when you realize that these signed areas are quaternions and can be used to rotate vectors in arbitrary-dimensional spaces simply by picking vectors which span the desired plane of rotation and choosing apt multiplication rules.
The real fun comes when you realize that choosing your metric cleverly gives you one plane where rotations are hyperbolic instead of Euclidean (circular), so you can model accelerations as rotations in the space-time plane. Which is Special Relativity. Which is neat.
This lays it all out nicely with pictures and everything: http://www.av8n.com/physics/clifford-intro.htm
And here is SR done as hyperbolic rotations in a space-time plane: http://www.av8n.com/physics/spacetime-welcome.htm
The relationship between Quaternions and the 3-dimensional cross-product corresponds to the relationship between the octonions and this 7-dimensional cross-product. Normed Division Algebras are a fascinating and worthwhile subject to look into.