[HN Gopher] Why can't you multiply vectors? [video]
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Why can't you multiply vectors? [video]
Author : BenoitP
Score : 72 points
Date : 2023-10-30 14:22 UTC (8 hours ago)
(HTM) web link (www.youtube.com)
(TXT) w3m dump (www.youtube.com)
| BenoitP wrote:
| This post was brought to you by the Geometric Algebra gang.
|
| Join us at:
|
| https://www.youtube.com/watch?v=60z_hpEAtD8
|
| https://bivector.net/
|
| https://enkimute.github.io/ganja.js/examples/coffeeshop.html...
| contravariant wrote:
| I never did quite 'get' geometric algebra. I mean the exterior
| algebra gives a couple of clear generalisations, and for the
| ones that require a metric you can typically use the Hodge star
| to find a generalisation.
|
| Geometric algebra then blends all of this together, and I'm
| still not entirely convinced that this improves things. Is it
| actually ever handy to have to deal with mixed degree
| multivectors?
| adrian_b wrote:
| Even without using mixed-degree multivectors, the fact that
| the existence and the properties of all the different kinds
| of multivectors results from a small set of simple axioms is
| very satisfying in itself.
|
| Before all these concepts were unified by geometric algebra,
| the system of physical quantities as taught in most places
| was a huge mess of many different kinds of quantities,
| scalars, polar vectors, axial vectors, pseudoscalars,
| tensors, pseudotensors, complex numbers, quaternions, spinors
| and so on.
|
| It was not at all obvious why there are so many kinds of
| quantities, which are the relationships between them, are
| there any other kinds of quantities besides those already
| studied, etc.
|
| Geometric algebra has brought order in this chaos and it has
| enabled a much deeper and more complete understanding of
| physics, by reducing a long list of seemingly arbitrary rules
| to a much smaller set of axioms, and by deriving all the many
| kinds of physical quantities from the vectors in the strict
| sense, i.e. from the translations of the space, which are
| themselves derived from the points of the affine space (as
| equivalence classes of point pairs).
|
| (Both logically and historically, in physics the vectors are
| more fundamental than the scalars. The scalars are obtained
| by dividing collinear vectors, i.e. they are equivalence
| classes of pairs of collinear vectors. This division
| operation is a.k.a. measurement and what are now named as
| "real numbers" were named as "measures" in the past, for more
| than two millennia. For any Archimedean group it is possible
| to define a division operation using the Axiom of Archimedes,
| generating a set of scalars over which the original group is
| a vector space).
| contravariant wrote:
| Yeah but like all of that is true about the exterior
| algebra as well. Except exterior algebra is a bit more
| explicit about where you use the metric, which is really
| convenient when the metric starts to become important.
| BenoitP wrote:
| I'm not versed at all into the Hodge star operator but it
| feels like Maxwell's equations expressed in both systems[1]
| might locate the answer. With Hodge star:
| dF = 0 d * F = J
|
| Expressed in GA: [?]F = J
|
| The eyeball-difference of which (dF = 0) would be your "how
| GA then blends all of this together", if I understand
| correctly. I'd guesstimate that dF = 0 to be akin to Gauss'
| law; and that maybe GA somewhat incorporates that the curl of
| a gradient is the zero field.
|
| [1] https://en.wikipedia.org/wiki/Mathematical_descriptions_o
| f_t...
| nyssos wrote:
| These are not quite equivalent, since using the hodge dual
| explicitly incorporates the geometry of space in a way left
| implicit by your geometric algebra formulation. The
| appropriate exterior algebra analogue here is simply `F =
| dA`. F is the electromagnetic field tensor, d is the
| exterior derivative, A is the 4-potential.
| nyssos wrote:
| Clifford (i.e. "geometric") algebras are of some mathematical
| interest, but for physics the only real value I've seen from
| them is in offering a fairly nice presentation of spin
| groups, which is ultimately where the Dirac matrices come
| from. The geometric algebra advocacy, so far as I can tell,
| comes entirely from engineers who were never given a proper
| account of tensors in the first place: it's certainly an
| improvement over whatever godawful basis-dependent stuff gets
| taught there.
| itishappy wrote:
| I'd be interested to know your opinion on space-time
| algebras. They seem like they would provide a nice way of
| unifying spatial rotations and Lorentz boosts, but that may
| be blind optimism as your last sentence seems to have been
| written to describe me specifically...
| smaddox wrote:
| Yes. In physics. E.g. Maxwell's equation reduces to a single
| equation in geometric Algebra:
| https://peeterjoot.com/archives/geometric-
| algebra/maxwells_g...
| captainclam wrote:
| Thank you so much for the links! I just happen to be
| endeavoring to seriously learn this stuff at the moment, driven
| by interests in physics, differential geometry, and gamedev.
| I'm seriously jazzed to check these out, thank you.
| uxp8u61q wrote:
| Somehow, in your links, there isn't a single explanation
| anywhere (that's not in the form of a video that I don't have
| time or inclination to watch) about what "geometric algebra" is
| supposed to mean. I'm trying to gather what this is supposed to
| be from the catchphrases on the front page, and honestly, this
| just looks like linear algebra.
|
| Can you explain what "geometric algebra" is supposed to be? Or
| just link to written explanations? I'm a mathematician
| (algebraic topology), so don't be afraid to get technical. How
| is this different from linear algebra? The only things I can
| see are some tensor products and exterior products, and a few
| couple of division algebra structures. Can you enlighten me? Is
| there any actually new math behind all this?
| zxexz wrote:
| Oh lovely, can't wait to give this a watch. Freya dives deep, and
| gives wonderful talks. Her video "The Continuity of Splines"[0]
| is my favorite watch of the past year.
|
| [0] https://www.youtube.com/watch?v=jvPPXbo87ds
| postmodest wrote:
| The surprising thing about that video is how it helped me a lot
| with my CAD.
| kubb wrote:
| You can. Thanks for asking.
| mcphage wrote:
| On one hand, this comment saved me from watching this 50 minute
| video where I'd also learn that you can multiply vectors. So
| very efficient usage of time. On the other hand, I'll probably
| watch the video anyway because I expect it to be more
| interesting and informative, and go into a lot more depth. So
| I'm not really sure this comment helped me at all.
| corethree wrote:
| It's just the title bro. In The end of that video they find a
| way that's not the cross product or the dot product.
| dventimi wrote:
| Personally, I found this comment to be helpful. Personally, I
| don't appreciate it when speakers "get cute" with what I regard
| as misleading titles like this one has.
| manchmalscott wrote:
| Freya is such a talented educator? Storyteller? Her stuff is
| always a joy to watch.
| xigoi wrote:
| I love how she roasts programmers who don't like mathematics.
| sdfghswe wrote:
| Saying you don't like maths is like saying you don't like how
| the universe works. It's like... fine, but that's not really
| actionable.
| falserum wrote:
| "I hate math" usually is shorthand for "I hate math lessons,
| and I don't feel any need to learn it".
| sdfghswe wrote:
| That's probably true for the vast majority of things, isn't
| it?
|
| I also hate ballet, but I suspect that if I made an effort
| I would learn to appreciate it. However, I don't make it a
| point to go around loudly telling everyone that I hate
| ballet as if that was a badge of honor.
| Kranar wrote:
| Do you really "hate" ballet though? Like let's be honest
| here.
| sdfghswe wrote:
| Totally honest, I find it revolting.
|
| Why does this surprise you? It doesn't surprise you that
| some people hate math.
| Kranar wrote:
| No it's not. Plenty of people don't like math and it has
| nothing to do with how the universe works. The way that math
| is taught to most people is absolutely atrocious, involving
| rote memorization, following rules for the sake of following
| rules, very little intuition involved.
|
| If anything, most people who express a hatred of math do so
| because it's taught to them in a way that completely divorces
| it from the universe or anything whatsoever.
|
| Most people I know who appreciate math did not learn it at
| school, but learned it either at home from their parents, or
| learned it independently. I have also made it my own
| responsibility to make sure my daughter learns math from me
| and applies math to every day situations and can develop a
| basic mathematical intuition.
| sdfghswe wrote:
| > The way that math is taught to most people is absolutely
| atrocious, involving rote memorization, following rules for
| the sake of following rules, very little intuition
| involved.
|
| The older I get the more I realize the importance of
| memorization if you want to _actually_ apply maths to solve
| problems, as opposed to learning for the pleasure of it.
| When I was a kid I was all about the ideas.
| Kranar wrote:
| So there is definitely a subtlety here, which is not that
| memorization is bad in and of itself, but that learning
| things by memorizing them is a very short term strategy
| for actual understanding.
|
| For example I use math on an almost daily basis working
| as a quant, and so yes I happen to have memorized a great
| deal of math. But that memorization did not happen by
| sitting and explicitly memorizing things, memorizing
| formulas, memorizing rules or procedures or theorems. The
| memorization came over time and through repeated usage
| naturally.
|
| But the way math is taught at school, students are kind
| of pushed into a corner where if they want to do well on
| the test, the quiz, the exams then the path of least
| resistance is to memorize a narrow set of specific
| "material" that will be tested and then hyper fixating on
| that material by employing memorization.
| xigoi wrote:
| You should memorize things by using them, not by
| specifically sitting down to memorize them.
| xboxnolifes wrote:
| Which is easy to say if it's something you use a lot. But
| many things are just stepping stones to, or the
| fundamentals of, the part you do use a lot.
|
| I never gave it much thought in school, but I'm glad we
| did things like memorize multiplication tables. Being
| able to instantly know all multiplications of numbers up
| to 12 makes reasoning about numbers much faster.
| xigoi wrote:
| If you don't use something a lot, then don't memorize it.
| xboxnolifes wrote:
| I'd have never learned anything in math past arithmetic
| if that were the case.
| Koshkin wrote:
| Programmers who do not like mathematics solve problems in
| polynomial time; those who do solve them in constant time.
| mo_42 wrote:
| Was expecting something about vector spaces and multiplication
| operations or the lack of them.
|
| Watched a quick ride towards quaternions. Also nice.
| chpatrick wrote:
| I don't think it's really about explaining quaternions, but
| more how quaternions are just a special case that arise from
| this general framework.
| whacked_new wrote:
| clicked out of curiosity and was hooked to the end. fantastic,
| masterful talk, thanks.
| ewgoforth wrote:
| I guess I'm unclear on what physical concept she's getting at by
| multiplying vectors. Depending on which concept you're trying to
| calculate with your multiplication, you have dot products like
| work and cross product like torque.
| itishappy wrote:
| Or rotations like phasors or whatever the heck spinors do or...
|
| (Spoiler: it's all of them)
| gpm wrote:
| Defining "multiplication" to be "any function that takes two
| arguments and outputs one" isn't a very interesting definition
| of multiplication. The word is a lot more useful if you put
| constraints on it like saying for something to be a
| multiplicative operator it has to respect (a + b) * c = a * c +
| b * c (the distributive property).
|
| Once you put a "reasonable" set of constraints on it... you
| discover that you can't actually multiply vectors (no function
| exists that satisfies the properties you want). Though the talk
| isn't about proving that (or justifying the set of constraints
| that mean you can't multiply vectors) and instead goes off in
| another direction of extending your vector space to a bigger
| space (like how the complex numbers are a bigger space than the
| reals) where you can define a reasonable multiplication
| operator.
| dventimi wrote:
| Doesn't the scalar product have the distributive property?
| gpm wrote:
| Yes, that was an example of one property you probably want,
| not a set sufficient to make it such that no such operator
| exists.
|
| Another property you want (and the talk uses) is that the
| operator is that the operator is from V x V to something.
| I.e. we are multiplying two vectors (because that's what we
| asked for in the title) not a scalar and a vector. That
| excludes your counter example, but still isn't nearly
| enough to make it so that no multiplication operator
| exists.
|
| I'll be honest and say I'm not listing properties here
| because I don't remember what properties are needed to make
| it so you can't define the operator... hopefully someone
| who has studied this a bit more recently or thoroughly than
| me can chime in.
| empath-nirvana wrote:
| Basically you want all the properties of a ring:
|
| https://mathworld.wolfram.com/Ring.html
|
| Scalar and dot products don't stay within the group of
| vectors and component wise multiplication doesn't always
| have inverses.
| gpm wrote:
| You need slightly more than being a ring. It's possible
| to make a ring (even a field) over R^n provided you don't
| care about interactions with scalars. For example: Just
| take any one-to-one map from R^n to R and then apply the
| operations in R before mapping the results back. It won't
| make any geometric sense, but it will be a ring.
|
| I've been blanking on what exactly the interactions with
| scalars that we need to preserve are...
| bashmelek wrote:
| If I remember correctly multiplication requires a "zero,"
| an "identity," and for something to be a field each item
| needs an inverse. I imagine we can define multiplication
| in R2 just as we do for C. So by that logic we ought to
| be able to define such an operation on any R(power of 2).
| m3kw9 wrote:
| Why not if the vector is 0,2 and then multiply it by 0,2 you get
| double the magnitude 0,4?
| raphlinus wrote:
| This is known as the Hadamard product and is covered in the
| video. The tl;dr is that, while it certainly has uses, it
| doesn't represent multiplication of vector spaces in any
| reasonable way (in particular, it gives different results when
| there's a change of basis, while other notions of "product",
| including dot product, are invariant).
|
| There's a deeper explanation here:
| https://math.stackexchange.com/questions/185888/why-dont-we-...
| joshlemer wrote:
| Maybe that's okay? If you look at regular multiplication, it
| seems vulnerable to choice of "basis" as well. For example:
|
| 6 x 6 = 36
|
| But if we choose 2 as basis, rather than 1, then we have
|
| 3 x 3 = 9 (aka 3(2) x 3(2) = 9(2))
|
| But 9(2) != 36
|
| So even regular multiplication isn't invariant under chosen
| "basis"
| ryangs wrote:
| Highly recommend Freya's various deep dives into various game
| development contents. She also has twitch vods for developing
| those videos which are also fascinating.
|
| https://www.youtube.com/@Acegikmo
| calderwoodra wrote:
| I liked the talk and found the jokes funny but the crowd was
| eerily silent.
| mcphage wrote:
| It might be that the audio was pulled from the mic directly,
| and so the audience reactions either weren't loud enough, or
| maybe were removed?
| noman-land wrote:
| Pleasantly surprised by this talk. Clicked knowing nothing and
| stayed to the end. Great talk and really great presenter.
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