Geometric Algebra · HackerPapers discussions

Geometric Algebra — mentioned by kragen

Wed, 07 Aug 2024 14:59:08 GMT

thank you very much for the context! wrt ewd1300, https://www.cs.utexas.edu/~EWD/ewd13xx/EWD1300.PDF is a scan in the original handwriting. it's been a number of years since i read it, and i'd forgotten most of its contents, though evidently i've incorporated some of them subconsciously. and i'm not sure i finished it last time i've been using ∨ and ∧ for infix max and min, respectively, which is a generalization of their boolean-algebra use and appropriate to their use in the theory of lattices (the partial-order kind of lattice, of which the usual total order is a special case).…

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Geometric Algebra — mentioned by Qem

Mon, 28 Mar 2022 13:56:26 GMT

I second this thinking. You may be interested in reading about Geometric Algebra[1] and Infinitesimal Calculus[2], as two other alternative frameworks for the symbolic tooling used in the physical sciences. Together with symbolic manipulation and calculation software, I also think alternative, simpler symbolic frameworks may help solve what I call the "receding shoulders climbing problem". There's a phrase attributed to Newton "If I have seen further it is by standing on the shoulders of Giants". The problem today is that the giants became so big, one can spend many years studying and still…

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Geometric Algebra — mentioned by jacobolus

Wed, 03 Nov 2021 17:58:55 GMT

The author went to MIT, worked for NASA, and now lives in San Francisco https://twitter.com/mferraro89 It is unexceptional (indeed, expected) to get through an American undergraduate science or engineering degree without ever taking an abstract algebra course (much less the 2+ apparently expected of German pure math students). But in any event, the top post here by Garlef is barking up the wrong tree. Division algebras, field extensions, and galois theory (per se) are not the tools to use for studying arbitrary-dimensional geometry. What you want is Clifford algebra (which Clifford himself…

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Geometric Algebra — mentioned by jacobolus

Mon, 07 Dec 2020 09:32:00 GMT

> I've yet to see [...] Most implementations I have seen do much more than quaternions. How hard did you look? Here’s an example you can play with in a browser, https://enkimute.github.io/ganja.js/ But what is implemented in computers is not 1:1 with the algebraic tools that can be profitably employed to solve geometry problems symbolically. GA is a rich language full of useful tools which can describe/model many kinds of problems. Coming from a math background, let me recommend you start with https://arxiv.org/abs/1205.5935

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Geometric Algebra — mentioned by bollu

Fri, 27 Nov 2020 12:44:03 GMT

For anyone looking for a sane way to study this stuff, what finally made sense to me was (1) study discrete differential geometry from Keenan Crane's notes: https://www.cs.cmu.edu/~kmcrane/Projects/DDG/ ; (2) Read these notes on geometric algebra that formally lay down what the hell a GA space is and what operations one can perform on them: https://arxiv.org/pdf/1205.5935.pdf (Geometric Algebra: Eric Chisolm)

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Geometric Algebra — HN submission

Fri, 27 Nov 2020 11:43:08 GMT

Geometric Algebra (2012)

147 points; 57 discussion replies on HN. Paper and source history. A submission is not an endorsement.

Geometric Algebra — mentioned by LolWolf

Tue, 08 Sep 2020 00:52:45 GMT

It’s kind of the same, actually. You can treat the individual entries as “components” in a 2^d-vector with special multiplicative structure, which essentially transforms in specific ways. These axioms of GA lay that out more clearly: https://arxiv.org/pdf/1205.5935.pdf (specifically axiom 6)

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Geometric Algebra — mentioned by jacobolus

Thu, 11 Apr 2019 15:52:35 GMT

What is your background/perspective? For introductory motivation, check out https://www.shapeoperator.com/2016/12/12/sunset-geometry/ And perhaps follow up with the GA for Computer Science book, or if you like (or want to learn about) Newtonian mechanics, try Hestenes’s book New Foundations for Classical Mechanics. For people with a mathier background, I would recommend https://arxiv.org/abs/1205.5935

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Geometric Algebra — mentioned by dahart

Sat, 03 Nov 2018 16:33:01 GMT

Geometric algebra has a definition that is more specific than any use of geometry to teach Elementary Algebra on real numbers. https://en.m.wikipedia.org/wiki/Geometric_algebra https://en.m.wikipedia.org/wiki/Clifford_algebra For teaching geometric algebra, there are curricula and textbooks available... http://geometry.mrao.cam.ac.uk/home/introduction-to-ga/ https://arxiv.org/abs/1205.5935v1 http://www.faculty.luther.edu/~macdonal/GA&GC.pdf http://assets.cambridge.org/052148/0221/sample/0521480221WS.... http://geometricalgebra.org/downloads/ga4cs_toc.pdf https://www.amazon.com/dp/0792355148/

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Geometric Algebra — mentioned by jacobolus

Sat, 03 Nov 2018 00:35:02 GMT

You might enjoy https://arxiv.org/abs/1205.5935

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Geometric Algebra — mentioned by jacobolus

Sun, 13 Nov 2016 07:18:25 GMT

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…

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Geometric Algebra — mentioned by jacobolus

Sun, 13 Nov 2016 03:17:37 GMT

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…

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Geometric Algebra — mentioned by jacobolus

Thu, 08 Sep 2016 00:50:18 GMT

You might find this clearer than other sources: http://arxiv.org/abs/1205.5935 Also check out http://geocalc.clas.asu.edu/pdf/LGasSG.pdf

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Geometric Algebra — mentioned by jacobolus

Tue, 05 Jul 2016 21:49:26 GMT

Kalid: you should really check out geometric algebra (a.k.a. Clifford algebra). It will give you a much deeper understanding of what i is and what the exponential function is, and how they generalize to higher dimensions and more complicated models, and it will help stitch together the weird inconsistent little fragments of understanding provided by imaginary numbers, quaternions, matrix algebra, Lie theory, differential forms, etc. into a more unified/cohesive model. This is the model all high school and college students will be taught in 100 years, or perhaps even in 50 years, and it will…

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