Geometric Algebra
12 comments
·February 26, 2025tonyarkles
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?”
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 map into the algebra and then adopt a super compact notation for solving them.
leumassuehtam
I like to think of an algebra the same as a language. You want to define and constrain your language such that you can describe elements of your universe is clear and consice.
For example, if I am trying to describe a art piece in the museum, I need concepts and words that connect what I am seeing to reality. In the same way in electromagnetism, proper time is part of the reality and it is an experimental fact, so it must be included in the algebra. Another example is a functional programming language where you identify high-level functions as the building concepts of many computer science problems.
Defining an algebra tailored for the problem you're working with actually highly constrain the space, and this is what makes many problems trivial.
vivzkestrel
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?
TheCleric
I don’t know what it is but that site jumps around constantly on my phone. Can’t read it because none of the text sits still.
spartanatreyu
Just block the "join us on Discord" banner with your adblocker.
Super easy on android with Firefox and uBlock Origin.
1. Load page
2. Tap firefox's menu icon
3. Tap Extensions
4. Tap uBlock Origin
5. Tap "Enter element picker mode"
6. Tap the annoying moving text
7. In the element picker, select the correct layer (you don't want to delete the background image behind the moving text, you want to delete the moving text so tap through the layers until you get the right one. For this website you want "###disc" rather than the default "||bivector.net/images/banner.jpg").
8. Tap Create
You now have a custom filter for that website that removes that annoying element from appearing.
You now also have the knowledge to remove anything from any website on your phone.
Jtsummers
It's their bouncing "Join us on Discord!" text. At its smaller font size it fits on one line, and then it increases to the larger size and spans two lines. This causes everything to shift up and down. It's a poorly thought out idea.
colinthompson
If you rotate your phone and view in landscape mode the problem goes away, on iPhone at least. Not ideal but at least it’s something.
simojo
The "Join us on Discord!" banner changes size periodically and bumps the content down each time. Zooming out keeps it from happening, if your eyes can take it ;)
ludwigschubert
Are you also on a mini? ;-) Just because I really like bivector.net, despite its questionable web design choices: zooming out one level should make it so the animated “join our discord” no longer causes reflows. This should be fixed, but as a workaround until then for those of us on slimmer viewports…
koolala
This link doesn't really explain anything directly.
Qem
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).
In addition to this, there is the book, <i>Projective Geometric Algebra</i>:
http://projectivegeometricalgebra.org/
and for a more grounded approach, the book series from Make:
- Geometry: https://www.goodreads.com/book/show/58059196-make
- Trigonometry: https://www.goodreads.com/book/show/123127774-make
- Calculus: https://www.goodreads.com/book/show/61739368-make