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Clifford Algebra: A visual introduction (2014) (slehar.wordpress.com)
111 points by mcbits on Nov 12, 2016 | hide | past | web | favorite | 24 comments

I like the idea of Clifford Algebras a lot, though this article is written like the whole thing is crackpottery and has a lot of weird irrelevant rants.

It feels extremely natural to incorporate bi- and tri-, etc, vectors into physics -- far more natural than the awkward Pauli matrices and pseudovectors and pseudoscalars that it's normally explained in terms of. I'm totally convinced that it's correct to, for instance, interpret the magnetic field as a bivector-valued field.

If you want something rigorous but which stays focused on application and implications for computing, this is perfect: http://geometricalgebra.org/

There's a more modern implementation of those concepts in a C++ library here: https://github.com/wolftype/versor

+1 on the book recommendation, I own a copy and it was very eye-opening.

I'm very interested in Colapinto's work. His dissertation, Articulating Space, has a lot of neat ideas and applications in it (http://versor.mat.ucsb.edu/ArticulatingSpace.pdf). I especially appreciate his explicit emphasis on exploration encouraging others to build on past work and discover what these formulations enable.

I'm curious in what way versor is 'more modern'. My understanding is that its main benefit over Gaigen is that there's no code-generation step, and provides more general library for various kinds of I/O. I have been tempted to use versor, but for reasons explained below I'm leaning toward Gaigen for optimized implementation.

What is attractive about Gaigen (2) is that it will generate code for Java, C++, and C# from the same specification. I'm working on a relatively high-level implementation in Clojure, and the possibility of compiling down to something which will run efficiently 'everywhere' is appealing. In particular, there's huge benefit of being able to remain within a highly interactive, dynamic environment for prototyping and experimentation. If I can work out the kinks of interfacing with Java there, then it should be relatively feasible to target C# (for Unity) and C++ for Unreal Engine (I'm especially focused on VR).

Also, the Gaigen authors literally wrote the book, so I suspect the least impedance mismatch in trying to work from their generated code. That said, versor is also obviously heavily influenced, so I wouldn't anticipate problems there.

At one point, earlier in my exploration, I had concluded that versor would definitely provide the most bang-for-buck. I've shifted, but I think that may be more a matter of my goals and requirements. I'm interested in any direct experience you have with versor or other GA software.

> I'm curious in what way versor is 'more modern'.

Well, it uses modern C++ idioms that didn't exist whet Gaigen 2 was created (e.g. C++11). Mostly, these eliminate the performance-related need for a code generator (and incidentally, serves as great example of how much further C++11 moved the language in practical terms).

But more importantly to me, I think versor is a better-designed library/API and it implements more of the things that are needed to effectively use geometric algebra in practice. Most of these "additional things" are documented in Articulating Space, which you've already linked to. If you use versor, you can make direct use of a working, tested implementation (instead of trying to rewrite them to work with Gaigen 2's API).

Versor is also actively maintained, which I consider a feature.

That said…Gaigen 2 is perfectly serviceable and performant, so if it meets your needs, I don't think you should feel like you've made a bad choice. It works, it's fast, and as you pointed out, it generates output for languages other than C++. Prior to discovering versor, that's what I used, too.

That was useful. If and when I need pure C++, or want a fast path to 'additional things', it sounds like versor is the way to go.

There is a lot to sort through in coming up to speed with this paradigm, and each new source helps build the practical picture. Thank you for contributing to that.

I think the rants and informal/outsider tone of the article are what I liked about it. It came up while trying to get some background intuition for the Wikipedia article on "spacetime algebra"[0], which follows the Wikipedia math article tradition of assuming the reader already knows 99.9% of the subject.

[0] https://en.wikipedia.org/wiki/Spacetime_algebra

Here’s a comment I made elsewhere recently:

* * *

You might start with teasers like Hestenes’s papers [“Reforming the Mathematical Language of Physics”][oersted], [“Grassmann’s Vision”][gvision], etc.

For a summary from a mathy perspective, try Chisolm’s [book-like thingy on arxiv][chisolm]

If you want a whole book of concrete problems to solve at an advanced undergraduate physics student level, try Hestenes’s [New Foundations for Classical Mechanics][nfcm]. This is the best source I’ve seen anywhere about understanding complex rotations in mechanics problems.

If you want to solve some plane geometry problems, this thing looks fairly accessible [Treatise of plane geometry through the geometric algebra][planegeo] (I haven’t looked too closely).

Two websites are [geocalc.clas.asu.edu][asu] and [geometry.mrao.cam.ac.uk][cambridge]. Also see the [link page at The Net Advance of Physics][netadvance], and this [mirror of Lounesto’s site][lounesto] (he passed away a while back). Lounesto liked to publish [collections of counterexamples][counterexamples].

There are a number of journals, conferences, etc. Try a google search for “Clifford Algebra”.

If you want an introductory undergraduate textbook which tries to teach both geometric algebra and traditional matrix algebra, you could look at [MacDonald][]’s [Linear and Geometric Algebra][laga]; I don’t think Hestenes is really on board with this approach, but there’s not too much else pitched at a similar audience. MacDonald also has a book [Vector and Geometric Calculus][vagc] which I suspect will be substantially easier to work through than Hestenes and Sobczyk’s [Clifford Algebra to Geometric Calculus][cagc]. (I haven’t looked at either of MacDonald’s books)

If you’re interested in geometric modeling for robotics, computer graphics, computer vision, or similar, check out [these papers][invkinematics] and then look at the book [Geometric Algebra for Computer Science][gacs]. The “conformal geometric algebra” model proposed there is pretty neat. Also see [these papers][unifalg] about related topics.

If you’re interested in crystallography, check out the papers [“Point Groups and Space Groups in Geometric Algebra”][crystalsymmetry] and [“The Crystallographic Space Groups in Geometric Algebra”][crystalga].

If you’re interested in Lie theory or representation theory, check out this paper [“Lie Groups as Spin Groups”][lgasg].

If you’re a physicist / physics student / electrical engineer / etc., try the book [Geometric Algebra for Physicists][gap] or perhaps [Understanding Geometric Algebra for Electromagnetic Theory][gaet]

For a slightly different perspective, take a look at [Sobczyk][]’s [New Foundations in Mathematics: The Geometric Concept of Number][nfm]. Sobczyk has some interesting papers about representing geometric algebras using (real or complex-valued) matrices, which might be helpful if you want to write fast numerical code on current computers, which have been optimized to do matrix math.

If you’re interested in the history, I recommend Crowe’s 1967 [History of Vector Analysis][hva].

I believe you can find the collected mathematical papers of Clifford online if you do a google search, or buy a used copy of a nice version published by Chelsea in the 1960s; the recent AMS reprint is awful quality.

Grassmann’s two mid-19th century Ausdehnungslehre books have been relatively recently translated into English, as has Peano’s late-19th century book Geometic Calculus, all three by Kannenberg.

[oersted]: http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf

[gvision]: http://geocalc.clas.asu.edu/pdf/GrassmannsVision.pdf

[chisolm]: https://arxiv.org/abs/1205.5935

[nfcm]: http://geocalc.clas.asu.edu/html/NFCM.html

[planegeo]: http://web.archive.org/web/20011215062737/http://campus.uab....

[asu]: http://geocalc.clas.asu.edu/

[cambridge]: http://geometry.mrao.cam.ac.uk/

[netadvance]: http://web.mit.edu/redingtn/www/netadv/Xgeomealge.html

[lounesto]: https://users.aalto.fi/%7Eppuska/mirror/Lounesto/

[counterexamples]: https://users.aalto.fi/~ppuska/mirror/Lounesto/counterexampl...

[MacDonald]: http://faculty.luther.edu/~macdonal/

[laga]: http://faculty.luther.edu/~macdonal/laga/index.html

[vagc]: http://faculty.luther.edu/~macdonal/vagc/

[cagc]: http://geocalc.clas.asu.edu/html/CA_to_GC.html

[invkinematics]: http://geocalc.clas.asu.edu/html/InvariantKinematics.html

[gacs]: http://www.geometricalgebra.net

[unifalg]: http://geocalc.clas.asu.edu/html/UAFCG.html

[crystalsymmetry]: http://geocalc.clas.asu.edu/pdf/crystalsymmetry.pdf

[crystalga]: http://geocalc.clas.asu.edu/pdf/CrystalGA.pdf

[lgasg]: http://geocalc.clas.asu.edu/pdf/LGasSG.pdf

[gap]: http://geometry.mrao.cam.ac.uk/2007/01/geometric-algebra-for...

[gaet]: http://www.wiley.com/WileyCDA/WileyTitle/productCd-047094163...

[Sobczyk]: http://www.garretstar.com

[nfm]: http://www.springer.com/us/book/9780817683849

[hva]: https://en.wikipedia.org/wiki/A_History_of_Vector_Analysis

One source I forgot to add (edit window for my comment has passed or I’d add it in context above):

Alan Bromborsky, An Introduction to Geometric Algebra and Calculus http://www2.montgomerycollege.edu/departments/planet/planet/...

Also some more links at http://www2.montgomerycollege.edu/departments/planet/planet/...

This comment, chock full of fantastic citations, makes me wish that HN supported at least a small subset of Markdown syntax (even just Markdown links would be great!).

The nice thing about Markdown is that it still works pretty well anyway.

I'm assuming you're referring to:

> 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.

I don't see what the issue with that is. It's an interesting hypothesis, and he's not alone in thinking about it. It's already a common stance that effective/valuable/interesting mathematical structures are direct consequences/reflections of the universe's own structure, and are seen more as discoveries than inventions. The author's proposal is just refining the question: have we come to this conclusion because it's an intrinsic property of the universe, or does it always seem present because we can only contemplate the universe through the human brain, which does its own structuring?

As an example of respected thinkers thinking along these lines, Sir Arthur Eddington's The Philosophy of Physical Science works pretty well (Though, to be fair, it's not as if he's never been called 'out there').

The nonstandard definitions and notation, the lack of proper typesetting, random clip-arty images, the precociousness of "Grand Unified" whatever, and the general pretense that this isn't covered in most intro algebra graduate courses, all of that doesn't help his case. :(

I guess if it were purporting to be a text book rather than a blog post, these would be issues. As it stands, he's just someone on the internet sharing his thoughts.

I'm not sure what you mean by 'precociousness' there, since I imagine you're not attempting to compliment the author—but he's hardly alone in considering Geometric Algebra to be a kind of foundation/'unified theory' for large swaths of mathematics.

The other critiques seem mostly along the lines of, "he doesn't dress like us", which, while maybe a little off-putting to the in-crowd, shouldn't earn him an automatic dismissal or derogation.

Of course people can write and wear whatever they like. I am simply explaining why it smells like crackpottery, since it seems you thought it was only due to that one sentence.

And, for the record, anything aspiring to be a grand unified theory of mathematics has to go far beyond algebra.

Here, start with these: (1) http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf (2) http://geocalc.clas.asu.edu/pdf/GrassmannsVision.pdf (3) http://geometry.mrao.cam.ac.uk/wp-content/uploads/2015/02/Im... (4) https://arxiv.org/pdf/1205.5935v1.pdf (5) http://www2.montgomerycollege.edu/departments/planet/planet/...

This book is also great if you are interested whatsoever in computer modeling of geometry, e.g. for robotics, computer vision, computer graphics, VR, etc. http://www.geometricalgebra.net The “conformal geometric algebra” model developed there is a really amazing recent idea. If you don’t want to buy a book see the papers at http://geocalc.clas.asu.edu/html/UAFCG.html or this Ph.D. thesis http://www2.eng.cam.ac.uk/~rjw57/pdf/r_wareham_pdh_thesis.pd...

That's fair. In my initial comment I was using the quote as an exemplar of an idea running throughout the article--but my reply was directed more to the three comments implying the author is a crank than to yours in particular. As for the unified theory bit, I agree (you can see this in my previous comment) that Geometric Algebra wouldn't cover all of mathematics.

There's a whole bunch of weirdness spread throughout the post, like the section starting "Now we come to the concept of inversion, 1/x, a concept you learned so long ago that you probably can’t even remember learning it, but in all the time since then, nobody ever pointed out to you that it is a conceptual impossibility!" and continuing through the author getting het up about the one-point compactification of the plane.

There's something about clifford algebras that brings out the fanatic in people, the first time I heard about them was Pertti Lounesto posting all sorts of rants in sci.math back in the day.

I went a little off the deep end a bunch of years ago learning geometric algebra. For a similar, but slightly different and cool take, check this out [0]. It's a take on multivectors, dyadics, and differential forms, which really shine in E&M. It's a shame this isn't taught to undergrads with linear algebra and vector calc.

[0] http://onlinelibrary.wiley.com/book/10.1002/0471723096

The author writes: "Clifford Algebra reveals, for example, that the apparent chirality in electromagnetism, i.e. the right-hand rule for electric generators, and the left-hand rule for electric motors, turns out to be actually an artifact of the math used to describe the world, not a property of the world itself. It turns out that electromagnetism has no chirality, as revealed by Clifford Algebra"

but I'm a little bit skeptical of this given that properties of things like "electric motors" can be observed in the real world.

"The truth, or validity of Clifford Algebra is confirmed by Occam’s Razor, it provides a simpler model of mathematical objects than does vector algebra [...]"

What? No. No. That's not how Math works.

I think this stuff is all well known, and goes under the modern heading of "multilinear algebra".

Besides its application to graphics, I wonder if this could be applied to type algebra.

Sounded interesting until the crank shit at the end.

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