Geometric Algebra

by agnishom • 1 year ago

200 61 1 year ago

TheCleric• 1 year ago

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.

Jtsummers• 1 year ago

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.

ludwigschubert• 1 year ago

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…

simojo• 1 year ago

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 ;)

spartanatreyu• 1 year ago

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.

colinthompson• 1 year ago

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.

enkimute• 1 year ago

fixed it. mea culpa.

koolala• 1 year ago

This link doesn't really explain anything directly.

phrotoma• 1 year ago

Really needs an about page. WTF am I looking at?

Qem• 1 year ago

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).

wizviper• 1 year ago

A dual vector is just a linear-algebraic concept, no need for GA.

at_compile_time5• 1 year ago

It's also a bivector in 3D GA, no need for linear algebra.

WillAdams• 1 year ago

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

vivzkestrel• 1 year ago

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?

Kaijo• 1 year ago

The 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• 1 year ago

It 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• 1 year ago

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• 1 year ago

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.

Sharlin• 1 year ago

As 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...

matthberg• 1 year ago

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.

[0]: Bivector section timestamp: https://www.youtube.com/watch?v=JpxZQxXxMWY&t=479s

[1] https://marctenbosch.com/ndphysics/

[1]: PDF: https://marctenbosch.com/ndphysics/NDrigidbody.pdf

[2] https://4dtoys.com/

[3] https://miegakure.com/

barrenko• 1 year ago

Would this be useful for e.g. modelling terrains, or is used in something like GIS?

meindnoch• 1 year ago

No.

aeonik• 1 year ago

I think it could be, but there is a lack of good prediction libraries that support accelerated versions of the GA types.

howling11• 1 year ago

An article that argues against GA: 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• 1 year 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.*

howling11• 1 year 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?

adornKey• 1 year ago

Interesting article.

I have been telling people for more than a decade now to use the exterior product (and e.g. get rid of the clutter of strange Minus-Signs and div, grad and rot in Electrodynamics).

And I was really happy to see that people finally start doing that.

But when I saw the Geometric Product, it didn't look like anything I want. If someone says that it looks like a thing that the cat brought in, I'll think about it and will probably agree.

hamish_todd6• 1 year ago

The geometric product does transform composition. It has all the properties you want for multiplication in an algebra, or indeed a monoid or group.

So it is like matrix multiplication, but for transforms represented as multivectors. Multivectors are nicer than matrices because they are made out of the separate (exterior algebra) objects so you can geometrically interpret them. For example, a rotation-reflection (rotoreflection/improper rotation) will have a grade 1 part and a grade 3 part. One of them is the plane you reflect in, one is the point you rotate around.

aap_4• 1 year ago

The article reads more like a trolling attempt. Geometric/Clifford algebra is incredibly useful and by throwing away its product you lose a lot of the power of the algebra. It's like saying matrix multiplication is not useful and you really want to be multiplying and adding numbers in various ways. After all GA/CA elements can always be mapped to elements of a matrix algebra. To get rid of the idea of linear transformations and that they should compose just doesn't sound well thought through.

I don't know what sort of crackpots he's talking about, personally i haven't heard of them, only the accusations. If the author can't separate the math from the people who developed and/or popularized it, too bad. Does GA magically give intuitive explanations for all sorts of weird things? no. Can you formulate a lot of stuff much more efficiently and concisely, and does it help gain new perspective on some things? yes, absolutely. It provides a wonderful framework for expressing geometric ideas.

howling11• 1 year ago

Can you formulate a lot of stuff much more efficiently and concisely, and does it help gain new perspective on some things? yes, absolutely. It provides a wonderful framework for expressing geometric ideas.

Can you elaborate on what stuff does it help to formulate much more efficiently and concisely?

dist-epoch• 1 year ago

As for pure math—it seems like research mathematics readily talks about and uses Clifford Algebra, but is uninterested in or specificaly avoids the terms and concepts that are specific to Hestenes’ “Geometric Algebra”. I can speculate as to why: even by the 90s/00s, GA had gotten a bad reputation because of its tendency to attract bad mathematicians and full-on crackpots.

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

hamish_todd6• 1 year ago

I did two streams where I went through this article and explained the many places it is wrong. The second part of the article has more maths in it, so most of the content is there, you can watch it here: https://www.twitch.tv/videos/2282548167

(it's very long so I plan to edit the two streams into a digestible 10-15m or something. His fault not mine I'd say!)

Probably other commenters have already said, but the biggest giveaway is how he says we should move away from quaternions, and then demonstrates little to no awareness of why quaternions are used in engineering (vital in gamedev for example, your animations will look awful without quaternions). Yes, quaternions are hard if you are completely married to the idea that everything in geometry is ""vectors"". But the games industry put on its big-boy pants and learned to use them - they wouldn't do that if the things weren't useful for something, so it's bit silly to write an article like this if you haven't figured out why that happened.

howling11• 1 year ago

but the biggest giveaway is how he says we should move away from quaternions

I'm sorry I must have missed that part. Can you point me to where did he say this?

asplake• 1 year ago

A quaternion is just an even subalgebra of the Clifford algebra Cl3,0(R), what’s the problem?

buescher• 1 year ago

Without reading either article - yet - I can tell you "crackpot" can mean a lot of things, and one of them that is relevant to this context is "an academic that's more interested in ideas than in being a good research bureaucrat".

nxobject• 1 year ago

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...

https://github.com/enkimute/ganja.js

redeux• 1 year ago

The site is also unusable on iPhone. The page keeps jumping up and down due to the same janky animation.

johnnyjeans• 1 year ago

works 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• 1 year ago

Ganja.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

Geometric Algebra

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