Geometric Algebra

algebra (e.g., https://arxiv.org/pdf/1205.5935) so i've been thinking about changing. dijkstra's choice of ↑ for max (supremum) and ↓ for min (infimum) are tempting a second unfortunate collision is that recent versions of apl use ∧ and ∨ for infix lcm and gcd, which is also a generalization of their boolean-algebra use and appropriate to their use in the theory of lattices; it's just using a different partial order i think dijkstra's choice of ';' for relational composition is rather unfortunate, though it's probably a better choice than my choice of simple juxtaposition in binate (http://canonical.org/~kragen/binary-relations) which uses ';' for the relational product, which i think is an equally unfortunate choice. • is the conventional operator for relational composition, and it doesn't introduce such fatal ambiguities when typeset, but it poses human-interface problems with a normal keyboard (how do you know how to type it?) and may be confused with some other kind of infix dot operator when handwritten, such as the infix function-application operator dijkstra explains here a reasonable alternative is the ∘ conventionally used for functional composition, which is the same thing as relational composition when the relations in question happen to be functions, and so generally introduces no semantic ambiguity, as well as less typographical ambiguity in handwriting. it has the same human-interface problem as • with respect to quantification, in http://canonical.org/~kragen/sw/dev3/paperalgo i've taken the approach ka-ping yee introduced in python, of treating quantified operators like Σ, Π, ∀, ∃, etc., as ordinary functions whose argument is a set or sequence; then we can notate the sequence with the moral equivalent of a list-comprehension. (the notation i use for such comprehensions in paperalgo unfortunately will not fit in this margin, but the web page displays it well.) dijkstra's choice here of ⟨i: i < 10: i²⟩ privileges dummy variables ranging over consecutive integers, but as he implicitly points out, you can use any logical proposition, such as ⟨i: i ∈ S: i²⟩, and while my choice privileges iterations over concrete sequences, you can similarly iterate over any set i'm more interested in expressing algorithms, while dijkstra was more interested in proving properties of algorithms, and some of our differences in notational choices reflect this, such as this last one

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

can be used on arbitrary manifolds, in non-metrical contexts, etc. Basic geometric algebra should be taught to advanced high school students and all undergraduates studying any technical subject. Math students looking for a math-style introduction to geometric algebra should try Chisolm (2012) https://arxiv.org/abs/1205.5935

becoming the practical tool of choice in many geometric computing problems, and among niche groups of physicists. ”Reforming the Mathematical Language of Physics”, http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf “Grassmann’s Vision” http://geocalc.clas.asu.edu/pdf/GrassmannsVision.pdf “Imaginary Numbers are not Real” http://geometry.mrao.cam.ac.uk/wp-content/uploads/2015/02/Im... “Geometric Algebra” http://arxiv.org/pdf/1205.5935v1.pdf (this one is a good place to go if you get stuck in another source). (Or books might be better sources for going in depth. Search for New Foundations for Classical Mechanics, Geometric Algebra for Computer Science, Geometric Algebra for Physicists) You’d also probably enjoy Hestenes’s work on modeling in physics teaching, e.g. http://modeling.asu.edu/R&E/ModelingThryPhysics.pdf http://worrydream.com/refs/Hestenes%20-%20Modeling%20games%2... http://modeling.asu.edu/R&E/Notes_on_Modeling_Theory.pdf http://modeling.asu.edu/R&E/Hestenes-ModelingTheory2007.pdf