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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 ma

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!

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!