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.
+1 on the book recommendation, I own a copy and it was very eye-opening.
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.
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.
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.
* * *
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.
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 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').
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.
And, for the record, anything aspiring to be a grand unified theory of mathematics has to go far beyond algebra.
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...
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.
but I'm a little bit skeptical of this given that properties of things like "electric motors" can be observed in the real world.
What? No. No. That's not how Math works.