4 ms·

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 Wikipedi

by mcbits 10y ago

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 https://en.wikipedia.org/wiki/Spacetime_algebra

jacobolus 10y ago
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 http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf [gvision]: http://geocalc.clas.asu.edu/pdf/GrassmannsVision.pdf http://geocalc.clas.asu.edu/pdf/GrassmannsVision.pdf [chisolm]: https://arxiv.org/abs/1205.5935 https://arxiv.org/abs/1205.5935 [nfcm]: http://geocalc.clas.asu.edu/html/NFCM.html http://geocalc.clas.asu.edu/html/NFCM.html [planegeo]: http://web.archive.org/web/20011215062737/http://campus.uab.es/~PC00018/ http://web.archive.org/web/20011215062737/http://campus.uab.... [asu]: http://geocalc.clas.asu.edu/ http://geocalc.clas.asu.edu/ [cambridge]: http://geometry.mrao.cam.ac.uk/ http://geometry.mrao.cam.ac.uk/ [netadvance]: http://web.mit.edu/redingtn/www/netadv/Xgeomealge.html http://web.mit.edu/redingtn/www/netadv/Xgeomealge.html [lounesto]: https://users.aalto.fi/%7Eppuska/mirror/Lounesto/ https://users.aalto.fi/%7Eppuska/mirror/Lounesto/ [counterexamples]: https://users.aalto.fi/~ppuska/mirror/Lounesto/counterexamples.htm https://users.aalto.fi/~ppuska/mirror/Lounesto/counterexampl... [MacDonald]: http://faculty.luther.edu/~macdonal/ http://faculty.luther.edu/~macdonal/ [laga]: http://faculty.luther.edu/~macdonal/laga/index.html http://faculty.luther.edu/~macdonal/laga/index.html [vagc]: http://faculty.luther.edu/~macdonal/vagc/ http://faculty.luther.edu/~macdonal/vagc/ [cagc]: http://geocalc.clas.asu.edu/html/CA_to_GC.html http://geocalc.clas.asu.edu/html/CA_to_GC.html [invkinematics]: http://geocalc.clas.asu.edu/html/InvariantKinematics.html http://geocalc.clas.asu.edu/html/InvariantKinematics.html [gacs]: http://www.geometricalgebra.net http://www.geometricalgebra.net [unifalg]: http://geocalc.clas.asu.edu/html/UAFCG.html http://geocalc.clas.asu.edu/html/UAFCG.html [crystalsymmetry]: http://geocalc.clas.asu.edu/pdf/crystalsymmetry.pdf http://geocalc.clas.asu.edu/pdf/crystalsymmetry.pdf [crystalga]: http://geocalc.clas.asu.edu/pdf/CrystalGA.pdf http://geocalc.clas.asu.edu/pdf/CrystalGA.pdf [lgasg]: http://geocalc.clas.asu.edu/pdf/LGasSG.pdf http://geocalc.clas.asu.edu/pdf/LGasSG.pdf [gap]: http://geometry.mrao.cam.ac.uk/2007/01/geometric-algebra-for-physicists/ http://geometry.mrao.cam.ac.uk/2007/01/geometric-algebra-for... [gaet]: http://www.wiley.com/WileyCDA/WileyTitle/productCd-0470941634.html http://www.wiley.com/WileyCDA/WileyTitle/productCd-047094163... [Sobczyk]: http://www.garretstar.com http://www.garretstar.com [nfm]: http://www.springer.com/us/book/9780817683849 http://www.springer.com/us/book/9780817683849 [hva]: https://en.wikipedia.org/wiki/A_History_of_Vector_Analysis https://en.wikipedia.org/wiki/A_History_of_Vector_Analysis
- tomjakubowski 10y ago
  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!).
  - jacobolus 10y ago
    The nice thing about Markdown is that it still works pretty well anyway.
- jacobolus 10y ago
  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/Numerical_Relativity/bookGA.pdf http://www2.montgomerycollege.edu/departments/planet/planet/... Also some more links at http://www2.montgomerycollege.edu/departments/planet/planet/Numerical_Relativity/Cliff.html http://www2.montgomerycollege.edu/departments/planet/planet/...

by mcbits 10y ago

jacobolus 10y ago

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 http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf [gvision]: http://geocalc.clas.asu.edu/pdf/GrassmannsVision.pdf http://geocalc.clas.asu.edu/pdf/GrassmannsVision.pdf [chisolm]: https://arxiv.org/abs/1205.5935 https://arxiv.org/abs/1205.5935 [nfcm]: http://geocalc.clas.asu.edu/html/NFCM.html http://geocalc.clas.asu.edu/html/NFCM.html [planegeo]: http://web.archive.org/web/20011215062737/http://campus.uab.es/~PC00018/ http://web.archive.org/web/20011215062737/http://campus.uab.... [asu]: http://geocalc.clas.asu.edu/ http://geocalc.clas.asu.edu/ [cambridge]: http://geometry.mrao.cam.ac.uk/ http://geometry.mrao.cam.ac.uk/ [netadvance]: http://web.mit.edu/redingtn/www/netadv/Xgeomealge.html http://web.mit.edu/redingtn/www/netadv/Xgeomealge.html [lounesto]: https://users.aalto.fi/%7Eppuska/mirror/Lounesto/ https://users.aalto.fi/%7Eppuska/mirror/Lounesto/ [counterexamples]: https://users.aalto.fi/~ppuska/mirror/Lounesto/counterexamples.htm https://users.aalto.fi/~ppuska/mirror/Lounesto/counterexampl... [MacDonald]: http://faculty.luther.edu/~macdonal/ http://faculty.luther.edu/~macdonal/ [laga]: http://faculty.luther.edu/~macdonal/laga/index.html http://faculty.luther.edu/~macdonal/laga/index.html [vagc]: http://faculty.luther.edu/~macdonal/vagc/ http://faculty.luther.edu/~macdonal/vagc/ [cagc]: http://geocalc.clas.asu.edu/html/CA_to_GC.html http://geocalc.clas.asu.edu/html/CA_to_GC.html [invkinematics]: http://geocalc.clas.asu.edu/html/InvariantKinematics.html http://geocalc.clas.asu.edu/html/InvariantKinematics.html [gacs]: http://www.geometricalgebra.net http://www.geometricalgebra.net [unifalg]: http://geocalc.clas.asu.edu/html/UAFCG.html http://geocalc.clas.asu.edu/html/UAFCG.html [crystalsymmetry]: http://geocalc.clas.asu.edu/pdf/crystalsymmetry.pdf http://geocalc.clas.asu.edu/pdf/crystalsymmetry.pdf [crystalga]: http://geocalc.clas.asu.edu/pdf/CrystalGA.pdf http://geocalc.clas.asu.edu/pdf/CrystalGA.pdf [lgasg]: http://geocalc.clas.asu.edu/pdf/LGasSG.pdf http://geocalc.clas.asu.edu/pdf/LGasSG.pdf [gap]: http://geometry.mrao.cam.ac.uk/2007/01/geometric-algebra-for-physicists/ http://geometry.mrao.cam.ac.uk/2007/01/geometric-algebra-for... [gaet]: http://www.wiley.com/WileyCDA/WileyTitle/productCd-0470941634.html http://www.wiley.com/WileyCDA/WileyTitle/productCd-047094163... [Sobczyk]: http://www.garretstar.com http://www.garretstar.com [nfm]: http://www.springer.com/us/book/9780817683849 http://www.springer.com/us/book/9780817683849 [hva]: https://en.wikipedia.org/wiki/A_History_of_Vector_Analysis https://en.wikipedia.org/wiki/A_History_of_Vector_Analysis

tomjakubowski 10y ago
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!).
- jacobolus 10y ago
  The nice thing about Markdown is that it still works pretty well anyway.
jacobolus 10y ago
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/Numerical_Relativity/bookGA.pdf http://www2.montgomerycollege.edu/departments/planet/planet/... Also some more links at http://www2.montgomerycollege.edu/departments/planet/planet/Numerical_Relativity/Cliff.html http://www2.montgomerycollege.edu/departments/planet/planet/...