6 ms·

Exploring Physics with Geometric Algebra, Book I [pdf] (2016)

130 points by adamnemecek 9y ago

adamnemecek 9y ago
I've been talking on HN basically non-stop about geometric algebra (aka Clifford, Grassman, Exterior algebra) which I've been exploring as of late. I'm not a math person but GA is possibly the most exciting math field to me. It's a replacement for linear algebra but unlike LA it actually makes fucking sense (e.g. cross product generalizes beyond 3 and 7 dimensions, but there's a lot more). Basically it unifies the geometric concepts of a line and a circle (in GA, a line is just a circle with points on the circle at infinite distances). As a result, you can do very useful computations very easily. E.g. a point times a point gives you a line connecting the points. It kind of makes sense, because generally, you care more about objects around you as opposed to all objects in one direction to infinity. It's also a generalization of the complex plane beyond 2D. I finally kinda understand complex numbers. So much shit is easier in this formalism, esp. anything related to reasoning about space. This includes but isn't limited to computer vision, crystallography, graphics, physics (all the quantum shit makes sense in GA, Maxwell's equations too, in GA, Maxwell's equations are reduced from 4 to 1!!!, https://slehar.wordpress.com/2014/03/18/clifford-algebra-a-visual-introduction/ https://slehar.wordpress.com/2014/03/18/clifford-algebra-a-v...). For example there's something called CSVM (Clifford SVM) which learns any given manifold, unlike an SVM which learns a hyperplane. It also replaces quite a bit of differential geometry[0], as you you can answer questions from differential geometry without resorting to calculus. I feel like some sort of burden has been lifted from me, because I feel like so much higher mathematics is a lot more accessible in this formalism. But your mileage might vary. [0] pls note that some time ago I didn't give a fuck about differential geometry but somehow I do now. Check out this project https://github.com/reloZid/algeosharp https://github.com/reloZid/algeosharp it’s c# but cross platform.
- ianai 9y ago
  What have been your applications of it?
  - adamnemecek 9y ago
    I’m looking into learning some electromagnetic/photonic engineering. But like I’m astounded by the number of applications. I generally don’t learn math for it’s sake and GA is applicable to virtually everything.
  - eggy 9y ago
    I started down the GA path in 2005 with the purchase soon after of Geometric Algebra for Physicists, and other than learning the simplicity of expressing things in this algebra, I have not been able to dive deep into its uses. I have checked out Versor[1] for code stuff, and there was the promise of potential hardware implementations of the algorithms necessary to be able to make gains in the human program GA code to the pipelines and bits on CPUs, GPUs and other hardware. I think the day is coming for an explosion of GA in everything, since it has been simmering behind the scenes for many years thanks to many, Dave Hestenes[2] in particular. I love his book New Foundations for Classical Mechanics. I program in APL and J[3], and hope that the array programming will segue into a winning GA library/module and way of approaching coding GA to make it more of a player in my work rather than my hobby. EDIT: Versor.js is also available for a spin! [4] [1] http://versor.mat.ucsb.edu/ [2] http://geocalc.clas.asu.edu/ [3] jsoftware.com [4] https://github.com/weshoke/versor.js/
    - vanderZwan 9y ago
      In JavaScript there is also: https://enkimute.github.io/ganja.js/ https://enkimute.github.io/ganja.js/
- shanusmagnus 9y ago
  I've been interested in this for some of the reasons you mention, but never took the leap because I worried that even if I understood a bunch of useful stuff in the GA formulation, I wouldn't be able to back-port my knowledge to the more traditional path and understand papers / algorithms built on that knowledge. Be interesting to hear your perspective on that.
  - adamnemecek 9y ago
    Haha, this isn't my concern that much anymore (it kinda used to be). Idk, this sentiment just kinda disappeared? I guess I realized that I can be really productive in this formalism. You are right, porting things is annoying but it's one of the things that each gram is super valuable so you might not need to do it as much.
- madhadron 9y ago
  A few nits: * GA is not a replacement for linear algebra. It's a vector space equipped with a certain product. All of linear algebra still applies to it. There's just more structure. What it does replace is Gibbs-Heaviside vector analysis. * Unifying a line and a circle is a classical part of projective geometry. It can be written in Clifford algebra straightforwardly, but the identification is centuries old at this point. * Quaternions and octonions are the generalization of complex numbers beyond the plane. A lot of stuff just happens to reduce to the same thing in 2D. * Maxwell's equations are reduced to one in any relativistic formulation. The hard part is rewiring your brain to work in spacetime instead of space plus time. * It doesn't replace differential geometry. It's a convenient notation for parts of it, the way that bra-ket notation in quantum mechanics doesn't replace the underlying structure, it's just a notation adapted to it. But, yes, it's cool stuff.
  - zardo 9y ago
    >* Quaternions and octonions are the generalization of complex numbers beyond the plane. A lot of stuff just happens to reduce to the same thing in 2D. Would an octonion by any other name, not smell as sweet? The even graded sub-algebras in GA are isomorphic to complex numbers, quaternions, etc... For me at least, coming at the same structure from a completely different direction is the best bang for the buck in increasing understanding.
    - vanderZwan 9y ago
      Do you happen to be a tauist? ;)
      - zardo 9y ago
        I was reading Shakespeare yesterday. I guess it leaked. 4*pi is of course, the most reasonable circle constant.
        vanderZwan 9y ago
        Ah yes, the "when in doubt, annoy everyone equally" approach. I approve.
- andars 9y ago
  It seems like you are confusing vector algebra with linear algebra. The cross product is not a part of the theory of vector spaces and linear transformations. Geometric algebra is pretty cool, but it is a potential replacement for vector algebra, not linear algebra. It is defined on a vector space, which means that the theory of linear algebra (concepts like linear transformation, bases, etc.) applies. As a sidenote, reducing Maxwell's equations in the 4-equation Heaviside form to a single equation is not necessarily a boon. Each of the four has a clear physical meaning in the vector calculus formulation, which may be lost when moving to a single equation.
- eigenstuff 9y ago
  Oh, man, I'm getting Hanson's "Visualizing Quaternions" for Christmas and I can't wait, so I'm definitely gonna print off that visualizing Clifford algebra article and go through it with my sketchbook this evening to hold me in the meantime. Thank you!
deleted 9y ago
[deleted]
elteto 9y ago
I loved the axiomatic approach followed in the introduction! It builds intuition from the very beginning. At some point I could not follow anymore as the notation and the concepts started getting more complicated and I lack the math background to keep up. One of my favorite undergrad classes was linear algebra. I wonder if there are any linear algebra books out there that follow this approach of "build the world, one axiom at a time". My sweet spot would be something aimed at someone who is not a mathematician but has some math background (like an engineering unndergrad) and can follow along with a bit of effort and study. Any recommendations?
- adamnemecek 9y ago
  > I loved the axiomatic approach followed in the introduction! It builds intuition from the very beginning. At some point I could not follow anymore as the notation and the concepts started getting more complicated and I lack the math background to keep up. Patience. It relies on mathematics that are probably foreign to you, so it makes sense.
- JadeNB 9y ago
  > I wonder if there are any linear algebra books out there that follow this approach of "build the world, one axiom at a time". Basically any math book—so most textbooks don't qualify as math books—will do things this way. SUMS (http://www.springer.com/series/3423 http://www.springer.com/series/3423) and most MAA publications (https://www.maa.org/press/books/book-series https://www.maa.org/press/books/book-series) are good places to look. I didn't see one in a glance at the MAA series, but SUMS has Further Linear Algebra (http://www.springer.com/us/book/9781852334253 http://www.springer.com/us/book/9781852334253). I haven't used it personally, but it looks like it has a chance of being the sort of thing you want.
- esmi 9y ago
  I've found this LA textbook follows the "build the world" approach. http://joshua.smcvt.edu/linearalgebra/ http://joshua.smcvt.edu/linearalgebra/
agumonkey 9y ago
what a beautiful pdf
- Osmium 9y ago
  Agreed! I think I've seen it before as a theme, but I'm not sure what it's called. Wish there was a CSS version available somewhere. (Edit: thought it looks like it's not using small-caps variants for numerals, which is a shame)
  - wrs 9y ago
    It's a variation on a LaTeX style called classicthesis (http://www.miede.de/#classicthesis http://www.miede.de/#classicthesis) based on Robert Bringhurst's Elements of Typographic Style. Unfortunately this particular version has messed up some things (notably line length), so use the original!
  - deleted 9y ago
    [deleted]
- cyberpunk0 9y ago
  Beautiful as it may be idk where I'm expected to find the time to read 500+ pages of it
Mugwort 9y ago
The main problem I have with geometric algebra is that it's an alternative framework to express exactly the same things. Some things are expressed or derived easier in G.A. e.g. the Kerr solution of General Relativity, electron Zitterbewegung, etc. The problems with G.A. as I see it... 1) It's a specialized tool that does not generalize well to simple everyday physics problems. Don't get me wrong, I like G.A. and learned it from Hestenes' papers almost 20 years ago. I just don't see the point in trotting out G.A. to solve certain problems or illustrate some theory any more than I would abandon Gibbs vector calculus and insist on doing all calculations in 4-vector notation using field tensors and differential forms. I wouldn't even want to use it to teach differential geometry. Once someone knows these subjects I'd say go for it and learn G.A. It's actually very easy once you know the more conventional stuff. 2.) I don't know how things are now but in the past G.A. had the feeling of a cult theory. Unlike most cults, this one had substance. The problem learning the material first with G.A. is that you won't be able to understand other papers, books or solved problems of others. You'll be dependent on using G.A. and unable to make sense of Goldstein or Spivak. 3.) I'm all for alternative ways of doing things. If you have time to burn there's a "Bird Tracks" method of doing calculations in Lie algebras which is very different than the traditional Cartan matrix, Dynkin diagrams. I advise you to avoid learning these topics the first time around in a weird mathematical language. After you know you're way around these topics go for it.
- adamnemecek 9y ago
  1.) I think that therein lies the reason it's not more popular. You can achieve the same things in other formalisms but GA generally makes the most sense to me. Simplification to the point of obviation is a big win for me. 2.) Diversify. I picked it up because I felt like I can mold this better for my problems than LA. And I'm more about publishing code than publishing papers. So I care more about making it "my own" than explaining things to people. 3.) Coincidentally Lie algebras are also easier in GA. Look at this https://upload.wikimedia.org/wikipedia/commons/thumb/1/14/E8Petrie.svg/180px-E8Petrie.svg.png https://upload.wikimedia.org/wikipedia/commons/thumb/1/14/E8... and don't tell me that the symmetries seem more circular than linear.
- vanderZwan 9y ago
Koshkin 9y ago
To me, the text feels more like a set of notes that are meant to be read by the author. Even the "introduction" must be completely incomprehensible to a reader who is uninitiated in whatever topic the author is trying to present. Geometric Algebra is a beautiful subject (best explained in Hestenes' Oersted lecture). On the other hand, in practice, as the proposed "geometry of physics" it has been long replaced by the machinery of differential forms, with a vast amount of literature devoted to it; a short introduction to this you can find, for example, in an article by Ted Frankel: http://www.math.ucsd.edu/~tfrankel/the_geometry_of_physics.pdf http://www.math.ucsd.edu/~tfrankel/the_geometry_of_physics.p.... (The Geometric Algebra being discussed here is not to be confused with the subject of the marvelous book by Artin that bears the same title.)
- vanderZwan 9y ago
  The author is pretty much in agreement with you. From the preface: > These notes are more journal than book. You’ll find lots of duplication, since I reworked some topics from scratch a number of times. In many places I was attempting to learn both the basic physics concepts as well as playing with how to express many of those concepts using GA formalisms. The page count proves that I did a very poor job of weeding out all the duplication. These notes are (dis)organized into the following chapters
evo_9 9y ago
Would be cool if it were interactive like this site: http://immersivemath.com/ila/index.html http://immersivemath.com/ila/index.html

130 points by adamnemecek 9y ago

I've been talking on HN basically non-stop about geometric algebra (aka Clifford, Grassman, Exterior algebra) which I've been exploring as of late. I'm not a math person but GA is possibly the most exciting math field to me. It's a replacement for linear algebra but unlike LA it actually makes fucking sense (e.g. cross product generalizes beyond 3 and 7 dimensions, but there's a lot more). Basically it unifies the geometric concepts of a line and a circle (in GA, a line is just a circle with points on the circle at infinite distances). As a result, you can do very useful computations very easily. E.g. a point times a point gives you a line connecting the points. It kind of makes sense, because generally, you care more about objects around you as opposed to all objects in one direction to infinity. It's also a generalization of the complex plane beyond 2D. I finally kinda understand complex numbers. So much shit is easier in this formalism, esp. anything related to reasoning about space. This includes but isn't limited to computer vision, crystallography, graphics, physics (all the quantum shit makes sense in GA, Maxwell's equations too, in GA, Maxwell's equations are reduced from 4 to 1!!!, https://slehar.wordpress.com/2014/03/18/clifford-algebra-a-visual-introduction/ https://slehar.wordpress.com/2014/03/18/clifford-algebra-a-v...). For example there's something called CSVM (Clifford SVM) which learns any given manifold, unlike an SVM which learns a hyperplane. It also replaces quite a bit of differential geometry[0], as you you can answer questions from differential geometry without resorting to calculus. I feel like some sort of burden has been lifted from me, because I feel like so much higher mathematics is a lot more accessible in this formalism. But your mileage might vary. [0] pls note that some time ago I didn't give a fuck about differential geometry but somehow I do now. Check out this project https://github.com/reloZid/algeosharp https://github.com/reloZid/algeosharp it’s c# but cross platform.

ianai 9y ago
What have been your applications of it?
- adamnemecek 9y ago
  I’m looking into learning some electromagnetic/photonic engineering. But like I’m astounded by the number of applications. I generally don’t learn math for it’s sake and GA is applicable to virtually everything.
- eggy 9y ago
  I started down the GA path in 2005 with the purchase soon after of Geometric Algebra for Physicists, and other than learning the simplicity of expressing things in this algebra, I have not been able to dive deep into its uses. I have checked out Versor[1] for code stuff, and there was the promise of potential hardware implementations of the algorithms necessary to be able to make gains in the human program GA code to the pipelines and bits on CPUs, GPUs and other hardware. I think the day is coming for an explosion of GA in everything, since it has been simmering behind the scenes for many years thanks to many, Dave Hestenes[2] in particular. I love his book New Foundations for Classical Mechanics. I program in APL and J[3], and hope that the array programming will segue into a winning GA library/module and way of approaching coding GA to make it more of a player in my work rather than my hobby. EDIT: Versor.js is also available for a spin! [4] [1] http://versor.mat.ucsb.edu/ [2] http://geocalc.clas.asu.edu/ [3] jsoftware.com [4] https://github.com/weshoke/versor.js/
  - vanderZwan 9y ago
    In JavaScript there is also: https://enkimute.github.io/ganja.js/ https://enkimute.github.io/ganja.js/
shanusmagnus 9y ago
I've been interested in this for some of the reasons you mention, but never took the leap because I worried that even if I understood a bunch of useful stuff in the GA formulation, I wouldn't be able to back-port my knowledge to the more traditional path and understand papers / algorithms built on that knowledge. Be interesting to hear your perspective on that.
- adamnemecek 9y ago
  Haha, this isn't my concern that much anymore (it kinda used to be). Idk, this sentiment just kinda disappeared? I guess I realized that I can be really productive in this formalism. You are right, porting things is annoying but it's one of the things that each gram is super valuable so you might not need to do it as much.
madhadron 9y ago
A few nits: * GA is not a replacement for linear algebra. It's a vector space equipped with a certain product. All of linear algebra still applies to it. There's just more structure. What it does replace is Gibbs-Heaviside vector analysis. * Unifying a line and a circle is a classical part of projective geometry. It can be written in Clifford algebra straightforwardly, but the identification is centuries old at this point. * Quaternions and octonions are the generalization of complex numbers beyond the plane. A lot of stuff just happens to reduce to the same thing in 2D. * Maxwell's equations are reduced to one in any relativistic formulation. The hard part is rewiring your brain to work in spacetime instead of space plus time. * It doesn't replace differential geometry. It's a convenient notation for parts of it, the way that bra-ket notation in quantum mechanics doesn't replace the underlying structure, it's just a notation adapted to it. But, yes, it's cool stuff.
- zardo 9y ago
  >* Quaternions and octonions are the generalization of complex numbers beyond the plane. A lot of stuff just happens to reduce to the same thing in 2D. Would an octonion by any other name, not smell as sweet? The even graded sub-algebras in GA are isomorphic to complex numbers, quaternions, etc... For me at least, coming at the same structure from a completely different direction is the best bang for the buck in increasing understanding.
  - vanderZwan 9y ago
    Do you happen to be a tauist? ;)
    - zardo 9y ago
      I was reading Shakespeare yesterday. I guess it leaked. 4*pi is of course, the most reasonable circle constant.
      - vanderZwan 9y ago
        Ah yes, the "when in doubt, annoy everyone equally" approach. I approve.
andars 9y ago
It seems like you are confusing vector algebra with linear algebra. The cross product is not a part of the theory of vector spaces and linear transformations. Geometric algebra is pretty cool, but it is a potential replacement for vector algebra, not linear algebra. It is defined on a vector space, which means that the theory of linear algebra (concepts like linear transformation, bases, etc.) applies. As a sidenote, reducing Maxwell's equations in the 4-equation Heaviside form to a single equation is not necessarily a boon. Each of the four has a clear physical meaning in the vector calculus formulation, which may be lost when moving to a single equation.
eigenstuff 9y ago
Oh, man, I'm getting Hanson's "Visualizing Quaternions" for Christmas and I can't wait, so I'm definitely gonna print off that visualizing Clifford algebra article and go through it with my sketchbook this evening to hold me in the meantime. Thank you!

deleted 9y ago

[deleted]

elteto 9y ago

I loved the axiomatic approach followed in the introduction! It builds intuition from the very beginning. At some point I could not follow anymore as the notation and the concepts started getting more complicated and I lack the math background to keep up. One of my favorite undergrad classes was linear algebra. I wonder if there are any linear algebra books out there that follow this approach of "build the world, one axiom at a time". My sweet spot would be something aimed at someone who is not a mathematician but has some math background (like an engineering unndergrad) and can follow along with a bit of effort and study. Any recommendations?

adamnemecek 9y ago
> I loved the axiomatic approach followed in the introduction! It builds intuition from the very beginning. At some point I could not follow anymore as the notation and the concepts started getting more complicated and I lack the math background to keep up. Patience. It relies on mathematics that are probably foreign to you, so it makes sense.
JadeNB 9y ago
> I wonder if there are any linear algebra books out there that follow this approach of "build the world, one axiom at a time". Basically any math book—so most textbooks don't qualify as math books—will do things this way. SUMS (http://www.springer.com/series/3423 http://www.springer.com/series/3423) and most MAA publications (https://www.maa.org/press/books/book-series https://www.maa.org/press/books/book-series) are good places to look. I didn't see one in a glance at the MAA series, but SUMS has Further Linear Algebra (http://www.springer.com/us/book/9781852334253 http://www.springer.com/us/book/9781852334253). I haven't used it personally, but it looks like it has a chance of being the sort of thing you want.
esmi 9y ago
I've found this LA textbook follows the "build the world" approach. http://joshua.smcvt.edu/linearalgebra/ http://joshua.smcvt.edu/linearalgebra/

agumonkey 9y ago

what a beautiful pdf

Osmium 9y ago
Agreed! I think I've seen it before as a theme, but I'm not sure what it's called. Wish there was a CSS version available somewhere. (Edit: thought it looks like it's not using small-caps variants for numerals, which is a shame)
- wrs 9y ago
  It's a variation on a LaTeX style called classicthesis (http://www.miede.de/#classicthesis http://www.miede.de/#classicthesis) based on Robert Bringhurst's Elements of Typographic Style. Unfortunately this particular version has messed up some things (notably line length), so use the original!
- deleted 9y ago
  [deleted]
cyberpunk0 9y ago
Beautiful as it may be idk where I'm expected to find the time to read 500+ pages of it

Mugwort 9y ago

The main problem I have with geometric algebra is that it's an alternative framework to express exactly the same things. Some things are expressed or derived easier in G.A. e.g. the Kerr solution of General Relativity, electron Zitterbewegung, etc. The problems with G.A. as I see it... 1) It's a specialized tool that does not generalize well to simple everyday physics problems. Don't get me wrong, I like G.A. and learned it from Hestenes' papers almost 20 years ago. I just don't see the point in trotting out G.A. to solve certain problems or illustrate some theory any more than I would abandon Gibbs vector calculus and insist on doing all calculations in 4-vector notation using field tensors and differential forms. I wouldn't even want to use it to teach differential geometry. Once someone knows these subjects I'd say go for it and learn G.A. It's actually very easy once you know the more conventional stuff. 2.) I don't know how things are now but in the past G.A. had the feeling of a cult theory. Unlike most cults, this one had substance. The problem learning the material first with G.A. is that you won't be able to understand other papers, books or solved problems of others. You'll be dependent on using G.A. and unable to make sense of Goldstein or Spivak. 3.) I'm all for alternative ways of doing things. If you have time to burn there's a "Bird Tracks" method of doing calculations in Lie algebras which is very different than the traditional Cartan matrix, Dynkin diagrams. I advise you to avoid learning these topics the first time around in a weird mathematical language. After you know you're way around these topics go for it.

adamnemecek 9y ago
1.) I think that therein lies the reason it's not more popular. You can achieve the same things in other formalisms but GA generally makes the most sense to me. Simplification to the point of obviation is a big win for me. 2.) Diversify. I picked it up because I felt like I can mold this better for my problems than LA. And I'm more about publishing code than publishing papers. So I care more about making it "my own" than explaining things to people. 3.) Coincidentally Lie algebras are also easier in GA. Look at this https://upload.wikimedia.org/wikipedia/commons/thumb/1/14/E8Petrie.svg/180px-E8Petrie.svg.png https://upload.wikimedia.org/wikipedia/commons/thumb/1/14/E8... and don't tell me that the symmetries seem more circular than linear.
vanderZwan 9y ago

Koshkin 9y ago

To me, the text feels more like a set of notes that are meant to be read by the author. Even the "introduction" must be completely incomprehensible to a reader who is uninitiated in whatever topic the author is trying to present. Geometric Algebra is a beautiful subject (best explained in Hestenes' Oersted lecture). On the other hand, in practice, as the proposed "geometry of physics" it has been long replaced by the machinery of differential forms, with a vast amount of literature devoted to it; a short introduction to this you can find, for example, in an article by Ted Frankel: http://www.math.ucsd.edu/~tfrankel/the_geometry_of_physics.pdf http://www.math.ucsd.edu/~tfrankel/the_geometry_of_physics.p.... (The Geometric Algebra being discussed here is not to be confused with the subject of the marvelous book by Artin that bears the same title.)

vanderZwan 9y ago
The author is pretty much in agreement with you. From the preface: > These notes are more journal than book. You’ll find lots of duplication, since I reworked some topics from scratch a number of times. In many places I was attempting to learn both the basic physics concepts as well as playing with how to express many of those concepts using GA formalisms. The page count proves that I did a very poor job of weeding out all the duplication. These notes are (dis)organized into the following chapters

evo_9 9y ago

Would be cool if it were interactive like this site: http://immersivemath.com/ila/index.html http://immersivemath.com/ila/index.html