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Geometric Algebra
- jeerycharls 2y ago[dead]
- TheCleric 2y agoI 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.
- Jtsummers 2y agoIt'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.
- ludwigschubert 2y agoAre 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…
- simojo 2y agoThe "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 ;)
- spartanatreyu 2y agoJust 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.
- colinthompson 2y agoIf 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.
- enkimute 2y agofixed it. mea culpa.
- koolala 2y agoThis link doesn't really explain anything directly.
- phrotoma 2y agoReally needs an about page. WTF am I looking at?
- Qem 2y agoDid 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).
- wizviper 2y agoA dual vector is just a linear-algebraic concept, no need for GA.
- at_compile_time 2y agoIt's also a bivector in 3D GA, no need for linear algebra.
- WillAdams 2y agoIn addition to this, there is the book, <i>Projective Geometric Algebra</i>: http://projectivegeometricalgebra.org/ http://projectivegeometricalgebra.org/ and for a more grounded approach, the book series from Make: - Geometry: https://www.goodreads.com/book/show/58059196-make https://www.goodreads.com/book/show/58059196-make - Trigonometry: https://www.goodreads.com/book/show/123127774-make https://www.goodreads.com/book/show/123127774-make - Calculus: https://www.goodreads.com/book/show/61739368-make https://www.goodreads.com/book/show/61739368-make
- vivzkestrel 2y agoI 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?
- Kaijo 2y agoThe third video in the "Featured Introductions" section near the top of the page, called "A Swift Introduction to Geometric Algebra", is a very good starting point.
- agnishom 2y agoIt is a collection of resources about "Geometric Algebra". It is basically a vector space equipped with a anticommutative product. Turns out that this formalism is very useful in elegantly representing a number of operations (like rotation) in the domain of physics or computer graphics.
- tonyarkles 2y agoI 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 2y agoI 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.
- Sharlin 2y agoAs it usually goes. See also Wigner's classic essay "The Unreasonable Effectiveness of Mathematics in the Natural Sciences". https://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences https://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness...
- matthberg 2y agoBivectors 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. [0]: Bivector section timestamp: https://www.youtube.com/watch?v=JpxZQxXxMWY&t=479s https://www.youtube.com/watch?v=JpxZQxXxMWY&t=479s [1]: https://marctenbosch.com/ndphysics/ https://marctenbosch.com/ndphysics/ [1]: PDF: https://marctenbosch.com/ndphysics/NDrigidbody.pdf https://marctenbosch.com/ndphysics/NDrigidbody.pdf [2]: https://4dtoys.com/ https://4dtoys.com/ [3]: https://miegakure.com/ https://miegakure.com/
- bigbacaloa 2y ago[dead]
- barrenko 2y agoWould this be useful for e.g. modelling terrains, or is used in something like GIS?
- howling 2y agoAn article that argues against GA: https://alexkritchevsky.com/2024/02/28/geometric-algebra.html https://alexkritchevsky.com/2024/02/28/geometric-algebra.htm... TLDR: - 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. - Most of the usefulness of GA comes from just exterior algebra and exterior product/wedge product is more important than geometric product. - It shows up in spinor field in physics but this does not concern most people that do not work in quantum physics. My rudimentary view towards GA: - 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. - Outside of this you should just probably just learn exterior algebra instead.
- rhelz 2y ago// tends to attract a lot of crackpot. // Come on. You know what else attracted a lot of crackpots? The internet. If you are criticizing math, criticize the math, not the people. // just learn exterior algebra instead of// YMMV, but I like to know where the mathematical concepts came from. GA gives a nice origin story, see below: // Most of the usefulness of GA comes from just exterior algebra // Dot products come from the geometric product. If e1 & e2 are two basis vectors such that e1*e1 = 1, and e1e2 = -e2*e1, then if you multiply two vectors: (a1*e1 + a2*e2)(b1*e1 + b2*e2) = a1*b1*e1*e1 + a1\b2*e1*e2 a2*b1*e2*e1+ a2*b2*e2*e2 = a1*b1 + a2*b2 + (a1*b2 - b2*a1)*e1*e2 = (a . b) + (a ^ b) The first is the dot product. The second is the exterior product that everybody agrees is so useful. Now you know where both concepts came from. They are just from multiplying polynomials. The geometric product is a *a product*, it's the product of two polynomials. Yes, sometimes you just need the dot product, and sometimes you just need the exterior product. If you are coding, or giving the final form of some formula, you don't have to always put both of them in your code or paper. But neither the dot product nor the wedge product are investable by themselves. Having an investable product on vectors is endlessly useful while you are *deriving* the formulas.*
- howling 2y ago> Yes, sometimes you just need the dot product, and sometimes you just need the exterior product. If you are coding, or giving the final form of some formula, you don't have to always put both of them in your code or paper. In my experience 99% of the time you just want the dot product or the exterior product. Even when you want both it is rare that you want to combine them linearly except in some niche physics/mathematics. > But neither the dot product nor the wedge product are investable by themselves. Having an investable product on vectors is endlessly useful while you are deriving the formulas. Do you mean invertible? Why is invertibility is so useful?
- jeerycharls 2y ago[dead]
- jeerycharls 2y ago[dead]
- jeerycharls 2y ago[dead]
- jeerycharls 2y ago[dead]
- nxobject 2y agoAs 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... https://github.com/enkimute/ganja.js https://github.com/enkimute/ganja.js
- redeux 2y agoThe site is also unusable on iPhone. The page keeps jumping up and down due to the same janky animation.
- johnnyjeans 2y agoworks fine on firefox on a 10 year old thinkpad. considering someone is also reporting issues on iphone, might want to file a bug report with apple.
- itishappy 2y agoGanja.js is an amazingly general GA library. The problem with GA in general is that it uses a LOT of numbers. (3DPGA uses 4d multivectors with 16 numbers for every object, so dyadic operations can involve up to 16^2 elementary operations, similar to a matmul!) For most practical applications many of them cancel out and can be optimized away. I don't believe ganja.js does this, but there are algebra generators on the site that produce optimized code. I'm not sure why they chose the general library for their landing page instead of baking the animations, but the fact that it runs as performantly as it does is actually quite impressive to me! For a glimpse at the power of ganja.js, check out the coffeeshop examples: https://enkimute.github.io/ganja.js/examples/coffeeshop.html https://enkimute.github.io/ganja.js/examples/coffeeshop.html