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The speaker in the video runs a community for people interested in geometric algebra.

Check out the demo https://observablehq.com/@enkimute/animated-orbits

Job the discord https://discord.gg/vGY6pPk

GA seems to be getting more popular, but I still see vector algebra used in areas such as robotics where GA is clearly superior. In that application the small performance difference between GA and vector algebra is irrelevant, but the advantages of GA are huge.

In game engines the vector algebra method is slightly faster, so the elegant programming model is often sacrificed in the name of performance.

That, I can understand.

Why physicists use at least 4 separate formulations of the EM equations I can't understand, especially considering the vector version is the worst yet the most popular.

How can it be faster considering that they are equivalent?

> How can it be faster considering that they are equivalent?

Computing `ln(e^x)` and `x` are equivalent, but the latter is faster.

Well, to be fair, these are equal, not just equivalent.

Computers don't care about mathematical equivalence. For instance, dividing by a number is generally slower than multiplying with the inverse of that number. Also, memory access patterns can make a difference of night and day.

I'm not sure though where exactly the idea that GA is slower comes from. It's all down to the application and implementation. Perhaps it's because GA generalizes so well to higher dimensions that many libraries are overly generalized and thus slower than vector algebra libraries that need special cases anyway.

The 3D case is often handled using a conformal projective geometric algebra, which is 5D and requires 2^5 = 32 elements for a general multivector. This is twice that of a 4x4 matrix, and and eight times the size of a 4-element homogeneous vector as typically used in most 3D engines.

Of course, there are all sorts of optimisations. Most GA libraries are actually based on code-generation and support various "subsets" of the full GA to efficiently represent things like vectors only. From what I've seen, even a well-tuned library has approximately a 25% overhead for hot paths and 50-100% is typical.

This is a bit like representing a simple rotation like the hand of a clock, with either just one angle, or a two-element unit vector pointing in the direction of the rotation. The one-element angle has a bunch of special cases, like have to be checked to see if it goes past 360 degrees and then reset back to the 0..360 range. Representing this with two numbers just requires multiplication with a matrix, which involves no conditionals or modulo arithmetic.

This is analogous to the vector vs geometric algebra. Typically, GA has no special cases and "just works", but it does so by "uncompressing" the compact vector representation. It's going to be slower.

Reducing Maxwell's equations from the 4-equation Heaviside form to a single equation doesn't seem like necessarily a good thing. Each of the four has a clear physical meaning in the vector calculus formulation, which may be lost when moving to a single equation.

Yeah, dF = 0 is not very enlightening...

http://home.lu.lv/~sd20008/papers/essays/Maxwell's%20equatio...

It's all about inertia. Everybody either already knows vector algebra or will have to learn it anyway, to understand the vast majority of existing literature.

Geometric Algebra may be superior, but it's "one more thing" to learn/teach. It might catch on, but that might take another century of preaching.

Find a good laugh in 40 minutes of the YouTube about people in the old days do not have -ve no. and 0. Also, the surprise way to do algebra by geometry is fun. Studying pre-socrates at the moment and there is a speculation whether the "unbounded" guy is also a geometric guy.

Do you know if they've done work on formulating the Vlasov equation (coupled evolution of EM fields and phase-space density of populations of charged particles) in this way?

I know that Glasser and Qin at Princeton are working on geometrical view of plasma physik. They aim to get a view of particle-in-cell methods (basically a way to discretize the Vlasov equation using sparse sampling) using a geometric view point expressed using discrete exterior calculus. But I am not sure how much of that is ready or published.

I did research on PIC in grad school.

In the process I gained ... a healthy skepticism of the Qin group's papers. The man himself is sharp, though. Interested to see what they come up with.

Just for reference re that "Siggraph 2019 presentation":

https://en.wikipedia.org/wiki/Geometric_algebra

"The geometric product was first briefly mentioned by Hermann Grassmann" (1809 – 1877) "In 1878, William Kingdon Clifford greatly expanded on Grassmann's work to form what are now usually called Clifford algebras in his honor (although Clifford himself chose to call them "geometric algebras"). For several decades, geometric algebras went somewhat ignored, greatly eclipsed by the vector calculus." "The term "geometric algebra" was repopularized in the 1960s by Hestenes."

The background image on:

https://bivector.net/

is William Kingdon Clifford.

>It's almost a joke. To me it's reminiscent of looking at beginner programmers. You see them do crazy things like calculate a date "next month" by taking apart the pretty-printed date string, parsing to find the month, realising that sometimes the day part is one digit and sometimes it's two, then having to worry about m-d-y or d-m-y formats, building little tables of "days per month" and leap years... and so on.

This illustrates the entire field of programming as we know it. It actually happens among all programmers but for less obvious things. Especially with design patterns. I've seen programmers use so much structure for the sake of a design pattern on something that could otherwise be 10x simpler.

That's awesome