Comment by adamnemecek | Hacker News Reader

Comment by adamnemecek | Hacker News Reader

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-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 it’s c# but cross platform.

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.

>* 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.

Do you happen to be a tauist? ;)

I was reading Shakespeare yesterday. I guess it leaked.

4*pi is of course, the most reasonable circle constant.

Ah yes, the "when in doubt, annoy everyone equally" approach. I approve.

Of course, you haven't lived until you did as the astronomers do, and did exams stating that you may consider Pi = 4 and Pi^2 = 10

> All of linear algebra still applies to it.

Why would I use it tho? Sure, if I upgrade from a tricycle to a warp drive space ship, I can still use the tricycle by why would I when I need to haul shit across space.

> Unifying a line and a circle is a classical part of projective geometry.

Sure, but in Euclidean geometry, you don't really do this. Like it's essential go GA.

> Quaternions and octonions are the generalization of complex numbers beyond the plane.

Yes, I meant that it generalizes the concept of complex plane, not the plane itself.

> It doesn't replace differential geometry.

That is correct, however I still believe that you can rephrase certain DG questions as GA questions so I said that it replaces some of it.

> * 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.

Perhaps GP meant to say "GA subsumes linear algebra". Yes, LA is still valid, but a lot more intuitive within the context of GA if you ask me.

If I recall correctly Maxwell's equations are usually reduced to 2 in relativistic formuliations: dF = 0 and d⭑F = J.

GA purports to be useful in combining F and ⭑F into a single object (or rather having a derivative operator D = (d + d⭑)) so that DF = J captures everything. But I don't find that that insightful.

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.

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!

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.

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.

What have been your applications of it?

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.

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/

In JavaScript there is also:

https://enkimute.github.io/ganja.js/

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