The case against geometric algebra

https://alexkritchevsky.com/2024/02/28/geometric-algebra.html

#HackerNews #geometricalgebra #mathematics #techdebate #criticalthinking #algebra

#GAME2026 is happening and streaming #geometricAlgebra

https://bivector.net/game2026.html

Lloyd's relaxation is a cousin of k-means that’s still used for quantization, dithering, and stippling. With the pieces from yesterday, it’s only about 10 lines of code…

https://en.wikipedia.org/wiki/Lloyd's_algorithm
#geometer #geometricalgebra

I’m interested in #GeometricAlgebra (and #ExteriorAlgebra, #CliffordAlgebra, #ExteriorProduct and #WedgeProduct). I’m trying to work up an intuition for a few things. Thoughts on this welcome!

* It’s very intuitive that adding vectors means something and is useful. Join pencils end to end and now you have the pencil of their path. I have much less intuition that adding #bivectors is useful.

* In 3D, the wedge product of two vectors is an oriented area (bivector). And the wedge of an area and a vector is a volume. This makes sense. But the wedge of two areas in 3D is zero. Always (right?!). Is it even a well typed operation?

* If I have an area (say, some solar panels) and a direction (say incident sunlight), I think I can wedge them to get collected light (as a pseudo scalar). How do I know to wedge here instead of dot product? What’s the intuition for that so that the question becomes absurd?

* Is there a most simplest toy problem for playing with these to work up intuition? I think maybe solar panels (with area and orientation) and incident light (with intensity and direction) is reasonable? Because it just about makes sense to add oriented panels. And possibly even directed incident light?

Thanks!

If I had been taught "geometric algebra" (also known as the Clifford algebra) in high school (and university), my life would be different. Especially applied to physics.
I discovered it many years later, too late.

One example:
https://marctenbosch.com/quaternions/

Fun fact: Maxwell equations are just *one* equation using geometric algebra.

#algebra #physics #geometricalgebra

https://www.youtube.com/watch?v=eY6GuTFfYpQ

An Interpretation of Relativistic Spin Entanglement Using Geometric Algebra

#geometricalgebra #physics

It's Friday and I've gotten hyper-fixeated by #geometricalgebra

Specifically, I'm trying to figure out what directions an ant on the surface of a unit sphere would have to travel to intercept another one moving with a constant know speed.

There is a really nice relationship between tangent vectors and rotors, but solving for the time to intercept involves, among other things, taking the log of a bivector, which I didn't even think was possible until today