I recently had a bit of an epiphany around GA the other day (and Dual numbers, and Screw Theory with Twists and Wrenches): I realized that I had it all backwards. I’d seen GA as an alternative formulation for projective geometry and for Maxwell’s equations, Dual Numbers for auto-differentiation, and Screw Theory for rigid-body dynamics; the question that had always bothered me was “how in the hell did they manage to take these engineering problems and derive these crazy linear algebra theories to solve them?”<p>The realization was that it went the other way around! Clifford Algebras are an interesting linear algebra formulation. As it turns out, by appropriately defining the e^2=0, e^2=1, and e^2=-1 axes of the general Clifford Algebra framework, a bunch of these engineering problems <i>map</i> into the algebra and then adopt a super compact notation for solving them.
In addition to this, there is the book, <i>Projective Geometric Algebra</i>:<p><a href="http://projectivegeometricalgebra.org/" rel="nofollow">http://projectivegeometricalgebra.org/</a><p>and for a more grounded approach, the book series from Make:<p>- Geometry: <a href="https://www.goodreads.com/book/show/58059196-make" rel="nofollow">https://www.goodreads.com/book/show/58059196-make</a><p>- Trigonometry: <a href="https://www.goodreads.com/book/show/123127774-make" rel="nofollow">https://www.goodreads.com/book/show/123127774-make</a><p>- Calculus: <a href="https://www.goodreads.com/book/show/61739368-make" rel="nofollow">https://www.goodreads.com/book/show/61739368-make</a>
I am sorry if I dont understand. What is this link about? Courses, video tutorials? articles? I understand it has something to do with geometric algebra and that is about it. Can someone shed some light with all the visualizations on the home page?
An article that argues against GA:
<a href="https://alexkritchevsky.com/2024/02/28/geometric-algebra.html" rel="nofollow">https://alexkritchevsky.com/2024/02/28/geometric-algebra.htm...</a><p>TLDR:<p>- GA tends to attract a lot of crackpot. In fact most mathematicians avoid the name Geoemtric algebra and call it Clifford algebra to disassociate with them.<p>- Most of the usefulness of GA comes from just exterior algebra and exterior product/wedge product is more important than geometric product.<p>- It shows up in spinor field in physics but this does not concern most people that do not work in quantum physics.<p>My rudimentary view towards GA:<p>- It is useful in game physics since rotors can represent n-dimensional rotation in 2^{n-1} numbers instead of n^2 numbers as 2^{n-1} < n^2 when n <= 6. You can use PGA if you want to add translation as well. It is also better in interpolation.<p>- Outside of this you should just probably just learn exterior algebra instead.
As an aside, I'm getting lots of stuttering when scrolling the website on a variable-refresh rate monitor + an M1 laptop. This is a static-content website – a profiling tells me consent SVG animation and their ganja.gs [1] math library accounts for this. Not a good sign for the engineering applications of this theory...<p><a href="https://github.com/enkimute/ganja.js">https://github.com/enkimute/ganja.js</a>
Bivectors and higher show up a lot in the math of 4D+ geometry, too. There's a fascinating SIGRAPH 2020 talk [0] (with a paper [1]) by the maker of 4D Toys [2] and Miegakure (a 4D game in the works, [3]) explaining 4D geometry collision physics (with a good bivector intro in addition to the collision math). It's a great read and/or watch that smoothly covers everything from building basic intuitions to the gritty technical math of extending standard physics models to N-D spaces.<p>[0]: Bivector section timestamp: <a href="https://www.youtube.com/watch?v=JpxZQxXxMWY&t=479s" rel="nofollow">https://www.youtube.com/watch?v=JpxZQxXxMWY&t=479s</a><p>[1]: <a href="https://marctenbosch.com/ndphysics/" rel="nofollow">https://marctenbosch.com/ndphysics/</a><p>[1]: PDF: <a href="https://marctenbosch.com/ndphysics/NDrigidbody.pdf" rel="nofollow">https://marctenbosch.com/ndphysics/NDrigidbody.pdf</a><p>[2]: <a href="https://4dtoys.com/" rel="nofollow">https://4dtoys.com/</a><p>[3]: <a href="https://miegakure.com/" rel="nofollow">https://miegakure.com/</a>
Did Cixin Liu (Three Body Problem author) took Geometric Algebra as inspiration for some elements in his trilogy? There's reminiscent terminology in the story, like weapons names (dual vector foil).