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An article that argues against GA: https://alexkritchevsky.com/2024/02/28/geometric-algebra.html https://alexkritchevsky.com/2024/02/28/geometric-algebra.htm...

An 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?
  - wvlia5 2y ago
    Yes, invertible like if you have a.x=b, then you can find x=b/a if . is the geometric product. Why? Well, solving equations sounds somewhat useful, right?
    - howling 2y ago
      First of all it is only invertible for some non-zero elements, especially if `a` is a linear combination of multivectors or we work in PGA that explicitly adds a basis vector of norm 0. Yes sometimes it is useful but that doesn't automatically makes it more fundamental than the inner product and exterior product.
      - at_compile_time 2y ago
        You make it sound as though multivectors being invertible is a special case, when the opposite is true. In 2D and 3D GA, every non-zero k-vector and versor has an inverse. In PGA, every non-zero, non-ideal plane, line, and point, and versor has an inverse. The inverse is used all the damn time when composing and applying transformations and performing projections and rejections. As to which is more fundamental, I don't think it matters. You could argue that the dot and exterior products are more fundamental because the geometric product is their sum (for vectors). You could also argue that the geometric product is more fundamental because it is simply the Cartesian product of two multivectors, and you derive the dot, exterior and commutator products by filtering that product by grade. Both definitions are true, and "fundamental" is both a matter of perspective and irrelevant to any practical concern.
adornKey 2y 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_todd 2y 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_ 2y 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.
- howling 2y 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?
  - aap_ 2y ago
    I think one of the coolest examples is probably classical mechanics. See the SIBGRAPI 2021 videos on https://bivector.net/doc.html https://bivector.net/doc.html
    - howling 2y ago
      All of these stuff can be done in normal linear algebra. Some (not all) of the operations can be done more efficiently with GA in low dimensions. It is neither more concise nor more intuitive to understand than normal linear algebra.
      - hamish_todd 2y ago
hamish_todd 2y 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 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.
- howling 2y 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?
  - hamish_todd 2y ago
    The second paragraph of the conclusion: "Nor should we be trying to make everything look more like complex numbers and quaternions. Those are already weird and confusing; we should be moving away from them!" 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!
    - howling 2y ago
      You have to read the first paragraph as well. > I have given a lot of reasons why I think GA is problematic: the Geometric Product is a bad operation for most purposes. It really implements operator composition and is not a very fundamental or intuitive thing. Using a Clifford Algebra to implement geometry is an implementation detail, appropriate for some problems but not for general understandings of vector algebra and all of geometry. Giving it first-class status and then bizarrely acting like that is not weird is weird and alienating to people who can see through this trick. If I understand him correctly, he means Clifford algebra is "appropriate for some problems" but we should "move away" from "giving it first-class status" as it is not more fundamental and often does not help students understand geometry better. I also readily admitted that it has some use cases in game physics in my comment.
buescher 2y 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".

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?
- wvlia5 2y ago
  Yes, invertible like if you have a.x=b, then you can find x=b/a if . is the geometric product. Why? Well, solving equations sounds somewhat useful, right?
  - howling 2y ago
    First of all it is only invertible for some non-zero elements, especially if `a` is a linear combination of multivectors or we work in PGA that explicitly adds a basis vector of norm 0. Yes sometimes it is useful but that doesn't automatically makes it more fundamental than the inner product and exterior product.
    - at_compile_time 2y ago
      You make it sound as though multivectors being invertible is a special case, when the opposite is true. In 2D and 3D GA, every non-zero k-vector and versor has an inverse. In PGA, every non-zero, non-ideal plane, line, and point, and versor has an inverse. The inverse is used all the damn time when composing and applying transformations and performing projections and rejections. As to which is more fundamental, I don't think it matters. You could argue that the dot and exterior products are more fundamental because the geometric product is their sum (for vectors). You could also argue that the geometric product is more fundamental because it is simply the Cartesian product of two multivectors, and you derive the dot, exterior and commutator products by filtering that product by grade. Both definitions are true, and "fundamental" is both a matter of perspective and irrelevant to any practical concern.

adornKey 2y 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_todd 2y 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_ 2y 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.

howling 2y 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?
- aap_ 2y ago
  I think one of the coolest examples is probably classical mechanics. See the SIBGRAPI 2021 videos on https://bivector.net/doc.html https://bivector.net/doc.html
  - howling 2y ago
    All of these stuff can be done in normal linear algebra. Some (not all) of the operations can be done more efficiently with GA in low dimensions. It is neither more concise nor more intuitive to understand than normal linear algebra.
    - hamish_todd 2y ago

hamish_todd 2y 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 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.

howling 2y 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?
- hamish_todd 2y ago
  The second paragraph of the conclusion: "Nor should we be trying to make everything look more like complex numbers and quaternions. Those are already weird and confusing; we should be moving away from them!" 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!
  - howling 2y ago
    You have to read the first paragraph as well. > I have given a lot of reasons why I think GA is problematic: the Geometric Product is a bad operation for most purposes. It really implements operator composition and is not a very fundamental or intuitive thing. Using a Clifford Algebra to implement geometry is an implementation detail, appropriate for some problems but not for general understandings of vector algebra and all of geometry. Giving it first-class status and then bizarrely acting like that is not weird is weird and alienating to people who can see through this trick. If I understand him correctly, he means Clifford algebra is "appropriate for some problems" but we should "move away" from "giving it first-class status" as it is not more fundamental and often does not help students understand geometry better. I also readily admitted that it has some use cases in game physics in my comment.

buescher 2y 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".