[newtonfractal]

Can someone speak to what the applications of geometric algebra are?

I see this topic show up often on HN, but no mathematicians I know in real life care about it, or even know about it. From googling geometric algebra and related terms, it just seems like more linear algebra to me - why the different name?

[Lichtso]

The way I see it mainly for these reasons:

  - You get one unified simple framework which replaces or incorporates vector, matrix, complex, quaternion, tensor and spin algebra (all of which would otherwise need their own incompatible notations). [1]
  - Objects and transformations of objects can be expressed by the same multivectors. In the computer graphics of today you would have let's say: Vectors for points and translations, quaternions for interpolatable rotations, matrices for chaining up transformations and you would have separate objects for rays, planes, etc. But in geometric algebra all of that can be expressed by one class: The multivector. [2] [3]
  - It generalizes the same way in all dimensions (which is not true for e.g. vector algebra and the cross product).
  - One can easily derive geometric calculus from geometric algebra, thus have derivatives and integrals (which is a lot harder to do when you have 6 different frameworks and notation systems)
  - Bonus: Because it is unified, it has less edge cases and you need less workarounds, possibly making it also more stable / robust.

It would be interesting if anyone could contribute if there are serious downsides except for not being widely used (mostly for historic reasons I guess).

[1]: https://en.wikipedia.org/wiki/Geometric_calculus#/media/File... [2]: http://projectivegeometricalgebra.org/projgeomalg.pdf [3]: https://bivector.net/3DPGA.pdf

[amitport]

Maybe it simpler that quaternions (debatable) or simpler than learning all of the concepts you've mentioned separately, but I wouldn't really call it a "simple framework".

[these_are_not_t]

I think that simple should be understood here as easy to use. As a crude analogy, the mechanical principles underlying a car with an automatic shift aren't simple, but the abstraction it provides makes driving much easier to learn.

A more mathematically grounded example could be the complex plane compared to the real line: when encountered for the first time, it is not simple in any way, but it offers a clean solution to many (originally real valued) problems (perhaps by the fact that it is algebraically closed?).

NB: the author of this library also made a very interesting presentation of geometric algebra (using this library as vehicle) for SIGGRAPH 2019: https://www.youtube.com/watch?v=tX4H_ctggYo

It has been posted a few times on HN before.

[Lichtso]

I agree that if you only need one specific part, than your needs are better met by using just one of the other frameworks. But in my experience that is rarely ever the case. Usually I have to work with at least 3 or 4 different ones. You don't even have to use all of the possibilities for a unified framework to be simpler already. So the word "simple" is meant in relation to the fragmented alternatives.

[yantrams]

This[1] can probably give you some insight into the applications side of things.

1 => https://blog.ycombinator.com/joan-lasenby-on-applications-of...

[peeterjoot]

Geometric algebra has some striking electromagnetism applications. We are able to eliminate the chaotic mess of signs, divergences and curls that we have in Maxwell's equations and write it as a single equation, that essentially has the structure D F = J, where D is a differential operator, F = E + I B is a combined electromagnetic field, and J contains the charge and current densities.

Geometric algebra is not unique as a mechanism to grouping Maxwell's equations into a more coherent structure. This can also be done with both the tensor formalism and differential forms. Each of these can encode Maxwell's equations into a single equation for the electric sources, plus an additional equation for the (zero) magnetic sources. So the fact that we can write all of Maxwell's equations as a single equation, when compared to the tensor formalism or differential forms, is only an incremental improvement.

However, some interesting opportunities are made available by putting Maxwell's equations into this single equation form. In particular, the system of equations is put into a form that we can invert using a Green's function for the differential operator. This gives us a way to solve all of Maxwell's equations in one fell swoop. It's well known that this is possible for the statics equations of electromagnetism, but we can also do this for the time dependent case, and obtain Jefimenko's solution of Maxwell's equation ( https://en.wikipedia.org/wiki/Jefimenko's_equations ) directly. That solution can be obtained without first having to resort to solving the second order potential equation, and differentiating those fields to find the electric and magnetic fields.

The geometric algebra form of the Jefimenko's solution is much simpler than the conventional form, because it is expressed in terms of a single electromagnetic field variable (F = E + I B). There are many other such examples, where things become simpler to express when the fields are grouped logically into a single entity.

A great reference that explores electrodynamics and many other geometric algebra physics topics is Doran and Lasenby's "Geometric Algebra for Physicists" ( https://www.cambridge.org/core/books/geometric-algebra-for-p... ).

Be warned that the reference above is a tough read. I have a for dummies exploration of some of the electromagnetism applications of geometric algebra in my book (free pdf available here: http://peeterjoot.com/writing/geometric-algebra-for-electric... ). I don't use the explicitly relativistic invariant form of Maxwell's equations used in Doran and Lasenby -- it is beautiful, but makes things somewhat more inaccessible. My book is fairly small, and is grouped into three pieces (geometric algebra basics, integration theory, and finally applications to electromagnetism.) I don't touch the mixed signature (projective, conformal, and relativistic) geometric algebras that ganja supports, as I tried to keep things as simple as possible.