66 karma · joined July 12, 2017
But note that Geometric Algebra is significantly a tool for graphics/game developers, who tend to prefer videos (even the older generation I think)
> in some cases
What cases do you have in mind?
> why are these your operators in the first place?
This is a perfectly reasonable question but I want to point out that it's a bit philosophical and not the kind of thing physics undergrads tend to enjoy hearing about. For example Clifford algebras like matrix algebras are associative algebras over commutative fields (real or complex numbers) - why? Why does the universe like that? I can hazard guesses but it's mostly above my paygrade, possibly above anyone's paygrade. But if it's to be asked of GA it should be asked of matrices too, I'm sure you can agree.
There's something that distinguishes Clifford algebras from matrix algebras. They start from the assumption that vectors anticommute when they're orthogonal. That's easy to explain. It says that if A and B are orthogonal vectors then A->B is the opposite (negation) of B->A.
Aside from those the last thing is that you say your basis vectors have a certain "metric" on them. There's deep philosophical questions about why the universe cares so much about metrics, but they're not at all specific to Clifford Algebra.
Personally I find that very intuitive, much more intuitive than any other tensor algebra I'm aware of.
Leading up to classical mechanics, you have the sibgraphi tutorials: https://youtube.com/playlist?list=PLsSPBzvBkYjxrsTOr0KLDilkZ... (I also recommend the bivector discord if you want a community)
And from there, sudgy's videos are good, and the tome "geometric algebra for physicists" packs in a huge amount of stuff.
For example Cl(4,2) can represent the spacetime conformal group and Cl(3,0,1) can represent the Euclidean group. If A and B are elements of those groups represented by multivectors then AB will be their composition.
That is an extremely fundamental operation. Almost as, if not more, fundamental than inner and wedge.
I mention this every time this article is posted on hacker news and nobody wants to talk about it :`(
(Apologies to folks who have seen me repeat this a million times; but it's very important folks be aware of it)
TRANSFORM COMPOSITION!
(sorry, it's not your fault or even his that you didn't know this - GA textbooks should have it as the first thing they teach but they don't)
https://m.twitch.tv/videos/2282548167
TLDR it is quite a bad article. One of the closest thing he has to a real argument is "I don't like it when geometric objects are identified with operators, I want those to be separate things". But this is both anti-GA and anti-Lie-Theory. As he says, he is critical of mathematics as conventionally practiced. So be warned that if you find yourself disliking GA for anything like the reasons he dislikes GA, there's a lot of other (mainstream/prestigious) fields you dislike too.
On the other hand, from what you've posted you're clearly experienced with computer graphics - it sounds like you have at some point interpolated with dual quaternions?
This hackernews thread may not be sustainable, so perhaps join the bivector discord? https://discord.gg/bBvkuTrM I would be very interested to find out the precise things you care about, on the off chance that there is some not-yet-known-to-you way that GA can help you.
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.
(A.B)/B
Projects any A onto any B, in any number of dimensions and with any signature (eg hyperbolic/Euclidean/elliptic). A and B can be lines, planes, points, and with a conformal or anti de Sitter metric a sphere or hyperboloid etc ("blades").
It works because A.B is dimension independently the object "orthogonal to A and containing B or vice versa". And division by B will intersect that orthogonal object with B.
Concise, intuitive, and powerful. What's the linear algebra formula you'd consider to be comparable?
You're correct that you can construct a quaternion with the exponential map - but the most common way to make a quaternion is with a pair of vectors. Every game engine will have that function. GA will tell you how that function works - the vectors are planar reflections, you compose those to get a rotation by twice the angle, and you add (average with) the identity rotation to get a rotation by the precise angle.
> how to detach the two notions from each other
Can you say why would you want to do that? A plane always defines a planar reflection, a point always defines a point reflection, a line always defines a line reflection (assuming we're in euclidean space, which engineering is). To me this doesn't seem to be "happen to have" territory, this seems fundamental.
It's also implicit in the thing he says throughout: "bivectors and trivectors are good, but there's no reason to add a scalar to a bivector or a trivector to a 1-vector, nor is there a reason to multiply such objects". A quaternion is a scalar and a bivector added together!
(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.
They include a visual code editor I worked on - it's very janky (no proper errors) but you can play with it here https://hamishtodd1.github.io/ga/ed - ALT+ENTER to compile
And if you're still with me I'll add that I'm working on a Geometric Algebra-based animation program in VR https://www.youtube.com/watch?v=Tf5RUJP-4oE
You need the log quat representation. Exp( Tlog(q1/q0) ) q0
In fact, because it is SIMD across the shader cores already, you can't necessarily do this. Some GPUs do, some don't
A quaternion with w=1, x,y,z=0 is just the identity.
A quaternion with w=0, x=1, or perhaps w=0, x=y=0.7, those would only ever be rotations by 180 degrees.
If you want arbitrary rotations, you need some combination of the two: "a little bit of 180 around this line, and a little bit of 0deg rotation/identity". That's what it means to have scalar and bivector.
If you "being careful" with wedge and inner to avoid mixtures you are doing it wrong. Geometric product is the boss, and makes excellent mixtures!
It's amazing because that's the diametric opposite of what I'd consider to be true this is what the non-GA approach does, and which we avoid!
The modern approach to GA, eg https://en.wikipedia.org/wiki/Plane-based_geometric_algebra, distinguishes planes-through-the-origin, points, lines-through-the-origin, and lines-at-infinity. These four different types all get called "vectors" in the ordinary mindset. The cross product is said to take in two "vectors" and give you a "vector", but in reality it's generally taking two planes and giving you a line. This is talked about in the fourth paragraph of the wiki article :)
The confusing thing is that, today, John Sharp, also an elderly recreation mathematician, also died.
Currently the only source is the tweet above. We are not sure whether Colm has made a mistake. He did know John Horton Conway, but some people are mistaking his scientific american article, and the later guardian article, for an obit...
Terrible news if JHC is dead, and also terrible that John Sharp is dead. If not... will be weird for him waking up tomorrow to let people know that reports of his death are greatly exaggerated...