Understanding Geometric Algebra

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1. jacobolusNov 2016
   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

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2. eggyJul 2018
   I am working my way through Understanding Geometric Algebra by Kenichi Kanatani up to the Grassmann Algebra chapter, so I can then pick up John Browne's Grassmann Algebra Volume 1 Foundations.

   Kenichi's book works from basic vector calculus, Hamilton, Grassmann and Clifford. Grassmann was not even a mathematician by trade, but a school teacher and a linguist. He translated the Rig Veda. His big mathematical work did not get the attention when written as it did posthumously. I know things with cool names like differntial forms and fiber bundle space [1]. I'm becoming quite the Grassmann fanboy!

   I am thinking if I can learn these algebras (Grassmann, Clifford, etc...) I will be able to deal with more abstract thinking in geometry and handle it more concisely and logically. Almost like when I use J programming language for math! [2]

   I think Quarternions don't hold after 3 dimensions, so what about Octonions?

     [1]  https://galileospendulum.org/2012/04/26/everything-is-geometrical-hermann-grassmanns-algebra/
     [2] jsoftware.com

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