[HN Gopher] Introduction to Clifford Algebra (2006)
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Introduction to Clifford Algebra (2006)
Author : beefman
Score : 125 points
Date : 2020-02-14 16:31 UTC (6 hours ago)
(HTM) web link (www.av8n.com)
(TXT) w3m dump (www.av8n.com)
| adamnemecek wrote:
| If this interests you, you should check out the bivector
| community https://bivector.net/.
|
| Join the discord https://discord.gg/vGY6pPk.
|
| Check out a demo https://observablehq.com/@enkimute/animated-
| orbits
|
| Also at the end of February, there is geometric algebra event in
| Belgium. https://bivector.net/game2020.html All the big names in
| the field will be there.
| DreamScatter wrote:
| Check my clifford algebra implementation
|
| https://grassmann.crucialflow.com
|
| The Grassmann.jl package provides tools for doing computations
| based on multi-linear algebra, differential geometry, and spin
| groups using the extended tensor algebra known as Leibniz-
| Grassmann-Clifford-Hestenes geometric algebra. Combinatorial
| products included are [?], [?], [?], *, [?], ', ~, d, [?] (which
| are the exterior, regressive, inner, and geometric products;
| along with the Hodge star, adjoint, reversal, differential and
| boundary operators). The kernelized operations are built up from
| composite sparse tensor products and Hodge duality, with high
| dimensional support for up to 62 indices using staged caching and
| precompilation. Code generation enables concise yet highly
| extensible definitions. The DirectSum.jl multivector parametric
| type polymorphism is based on tangent bundle vector spaces and
| conformal projective geometry to make the dispatch highly
| extensible for many applications. Additionally, the universal
| interoperability between different sub-algebras is enabled by
| AbstractTensors.jl, on which the type system is built.
| Squithrilve wrote:
| Does anyone know any other good resources (esp. books) on this
| subject? (Clifford/geometric algebra)
| SAI_Peregrinus wrote:
| In addition to gibsonf1's recommendations (not books), the
| following books are good. The first 2 are more pure
| introductions to the math, the third is applying it to physics,
| the fourth to CS.
|
| Linear and Geometric Algebra, by Alan Macdonald:
| http://www.faculty.luther.edu/~macdonal/laga/
|
| Vector and Geometric Calculus, by Alan Macdonald:
| http://www.faculty.luther.edu/~macdonal/vagc/index.html
|
| Application to physics: New Foundations for Classical Mechanics
| by David Hestenes:
| https://books.google.com/books/about/New_Foundations_for_Cla...
|
| Geometric Algebra For Computer Science by Dorst, Fontijne, and
| Mann: http://www.geometricalgebra.net/index.html
| gibsonf1 wrote:
| I highly recommend https://bivector.net/
|
| In the docs section, the Geometric Algebra Primer by Jaap Suter
| is excellent. http://www.jaapsuter.com/geometric-algebra.pdf
| lazycrazyowl wrote:
| Clifford Algebras: An Introduction
|
| Author(s): D. J. H. Garling Series: London Mathematical Society
| Student Texts 78 Publisher: Cambridge University Press, Year:
| 2011 ISBN: 1107096383
| aesthesia wrote:
| I've read a number of introductions to Clifford algebras, and I'm
| always left with the question of what the geometric product is
| supposed to mean. The wedge product and dot product are easy to
| understand and have obvious interpretations. But other than being
| a gadget from which you can extract these other products, I don't
| see what the geometric product is for, or why it should be the
| primary object of consideration.
| edflsafoiewq wrote:
| Despite the name the motivation for the geometric product is
| principally algebraic, ie. it's useful for doing algebraic
| manipulation. It does not, AFAIK, possess any geometric meaning
| outside of special cases.
|
| (It's "geometric" in the sense it doesn't depend on a choice of
| basis I guess.)
| vmchale wrote:
| Neat, thank you!
| dang wrote:
| Related from 2016: https://news.ycombinator.com/item?id=12938727
|
| 2015: https://news.ycombinator.com/item?id=9746051
|
| Related a bit more generally:
|
| 2017 https://news.ycombinator.com/item?id=15932739
|
| https://news.ycombinator.com/item?id=14947065
|
| 2016 https://news.ycombinator.com/item?id=13239632
| vtomole wrote:
| Clifford algebra is a big part of quantum computation. The
| Clifford gates (https://en.wikipedia.org/wiki/Clifford_gates)
| along with Magic state distillation
| (https://en.wikipedia.org/wiki/Magic_state_distillation) can be
| used to perform fault-tolerant quantum computation.
|
| Edit: Clifford groups are not the same as Clifford algebras. I
| was wrong!
| knzhou wrote:
| That's the Clifford _group_ , though. Is it actually related to
| the Clifford algebra, beyond being named after the same guy?
| vtomole wrote:
| A group is an algebraic structure. Please reference https://e
| n.wikipedia.org/wiki/Clifford_algebra#Clifford_grou...
| knzhou wrote:
| Yes, I'm aware of that. But _an algebra_ is a very
| different thing from "an algebraic structure".
| lisper wrote:
| I'm not sure "very different" is a fair characterization.
| The two are closely related:
|
| https://en.wikipedia.org/wiki/Algebraic_structure
|
| An algebraic structure on a set A (called the underlying
| set, carrier set or domain) is a collection of operations
| on A of finite arity, together with a finite set of
| identities, called axioms of the structure that these
| operations must satisfy. In the context of universal
| algebra, the set A with this structure is called an
| algebra,[1] while, in other contexts, it is (somewhat
| ambiguously) called an algebraic structure, the term
| algebra being reserved for specific algebraic structures
| that are vector spaces over a field or modules over a
| commutative ring.
|
| Examples of algebraic structures include groups, rings,
| fields, and lattices.
| joppy wrote:
| The common use of "an algebra" in mathematics is a ring
| with a bit of extra structure.
| klodolph wrote:
| I'm going to agree with knzhou here. Unfortunately, the
| terminology in mathematics can be misleading.
|
| It is important to note context, and note the part where
| the article you quoted uses the works "ambiguously",
| because the word "algebra" has more than one meaning.
|
| In this case, a group is not an algebra (because we are
| talking in the context of algebras over a field or ring,
| not universal algebras).
|
| It is unfortunate that the words are defined this way,
| but you have to deal with it. A "universal algebra" is a
| very different concept from an "algebra" (over a ring or
| field) even though one is an example of the other.
|
| It's like saying that "book" is a very different concept
| from "The Great Gatsby".
| monoideism wrote:
| Huh? I thought a a group, ring, etc. was precisely an
| example of a "universal algebra". You seem to contradict
| yourself, at times agreeing with this statement ("even
| though one is an example of the other"), at times not ("a
| group is not an algebra").
|
| Edit: Wolfram Mathworld agrees with me: "Universal
| algebra studies common properties of all algebraic
| structures, including groups, rings, fields, lattices,
| etc." http://mathworld.wolfram.com/UniversalAlgebra.html
| vtomole wrote:
| oops yeah my apologies!
| Koshkin wrote:
| No, it's not related.
| vtomole wrote:
| You are right. My confusion. Sorry.
| dktoao wrote:
| Kinda makes me want to go back to college (I'm an EE) just to re-
| learn all the stuff I remember being so mind bending with this
| elegant new framework. Also, just so I can be THAT guy who always
| argues with the professor. Anyone know of any PhD openings that
| could use a maverick like me? :) (/s kinda)
| DreamScatter wrote:
| going to college won't really help you, I quit college so I can
| abandon traditional math to completely devote myself to
| geometric algebra based math, here is my algebra implementation
| for example:
|
| https://grassmann.crucialflow.com
|
| it isn't taught at universities, it is self taught.. at the
| university level you are going to be artificially held back
| more than you would by studying it independently
| Random_ernest wrote:
| I was in this very situation, thinking I want more than "just"
| EE. Applied for a PhD position in a branch where mostly
| mathematicians work, could not be happier with the decision.
|
| Do it. Scratch that itch while you still can.
| hackernewsname wrote:
| What steps did you take to go from industry back into
| academia?
| msla wrote:
| Here's the page one up in the directory structure:
|
| https://www.av8n.com/physics/
|
| It's got a lot of very interesting math and physics information.
| m4r35n357 wrote:
| Looks really impressive. For those interested in this sort of
| thing, another huge and varied collection of
| mathematical/physics articles is located at
| https://www.mathpages.com/
| OldGuyInTheClub wrote:
| He's the guy that built the shark for 'Jaws' while an
| undergraduate. Very very capable, to say the least. Saw him in
| the halls during my postdoc but never had the occasion to talk
| with him.
| msla wrote:
| > He's the guy that built the shark for 'Jaws' while an
| undergraduate.
|
| Fascinating. That thing famously never worked well, and the
| movie was better for it.
| OldGuyInTheClub wrote:
| Denker describes the development as a case study in
| "Experimental Techniques in Condensed Matter Physics at Low
| Temperatures." His chapter is on electromagnetic shielding
| and grounding - important for animated sharks and
| microKelvin measurements. Spielberg is indirectly
| referenced as the director getting impatient with delays
| caused by all sorts of hidden electrical problems.
|
| The book is a compendium of tips and techniques from
| graduate students in Cornell's famous low temperature
| physics lab. Although published in 1988, it is sufficiently
| general to be valuable today.
|
| https://books.google.com/books?id=8tJMDwAAQBAJ&pg=PP16&lpg=
| P...
| playing_colours wrote:
| Is interest in GA just a local fashion or there are objective
| reasons in recent revival of interest?
| Koshkin wrote:
| I think it's both. Still, in the manner it's happening, the
| surging abundance of tutorials on Geometric Algebra somehow
| feels worrisome. This looks all too similar to the ever growing
| number of guides on what are monads and how to use them in
| programming. For most people - kind of makes sense, sort of
| interesting, sometimes inspiring, practically useless...
| virgil_disgr4ce wrote:
| Why does it worry you?
| Koshkin wrote:
| This is almost like, for instance, why are there so many
| popular accounts on quantum mechanics (and new ones keep
| popping up every so often). This makes me think they are
| all wrong somehow (and to a significant degree they indeed
| are - which is quite understandable in this case, as QM is
| a tricky subject); looks like that's what the author of the
| next one should think, too.
| monoideism wrote:
| If you're using a functional programming languages, monads
| are actually quite useful. It's not just a theoretical thing.
|
| I can see how you might not want to use them in JavaScript
| (although many folks do), but it's quite natural to use them
| in Scala, Haskell, or OCaml.
| m4r35n357 wrote:
| I never got the hang of direction of cross products, still
| don't know which hand rules to use for eg motors & generators
| (well I think I knew once but as never at ease).
|
| This looks like what I wished I had learned instead!
| virgil_disgr4ce wrote:
| I also have noticed the sudden and relatively intense interest
| in this subject: there have been at LEAST 3-4 front-page-
| ranking links on HN in just the past week (that I've seen at
| least). An interesting spontaneous zeitgeist in the comp-sci &
| related communities.
| dktoao wrote:
| Read a little bit of the introduction. If you are familiar with
| using complex numbers and vector cross products, you will see
| the advantage pretty quickly
| oddthink wrote:
| Does anyone have a good summary of how this relates to the
| differential geometry world with its n-forms and n-vectors? For
| example, I'm used to thinking of the wedge product as operating
| over n-forms and requiring a metric (or volume element)
| transformation to work over vectors. Similarly, I don't see any
| discussion of behaviors under coordinate transformations.
| DreamScatter wrote:
| My website is based on differential geometric algebra:
|
| https://grassmann.crucialflow.com/dev/algebra
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(page generated 2020-02-14 23:00 UTC)