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GA has stronger types than the vector math taught in school, we just don't have an intuition for it. Here's an example: Traditional vector math gives the outpu

GA has stronger types than the vector math taught in school, we just don't have an intuition for it. Here's an example: Traditional vector math gives the output of the cross-product of two vectors the same type. (That's why it's only meaningful in 3d.) The input and output lives in the same space, you can presumably do meaningful math with them and get a result that's still a vector. The types do not help us. GA treats this operation differently. The cross-product-equivalent-operation returns a completely different type. Sure, you can add the inputs and outputs, but that returns a third type! If we built an intuition for this type system, it would help us.

phkahler 3y ago
In geometry code (from my own experience) I use a cross product specifically to get a vector perpendicular to two other vectors. There are other times like calculating the area of a triangle in 3-space that involve taking the magnitude of the cross product or whatever. I don't see how any of this solves hard problems. You need to know a formula to compute what you want regardless of the underlying representation. I also think the more abstract you get, the smaller portion of the population can understand and work with it. Lots of people understand numbers. Some of those understand vectors. Some of those can understand a transformation matrix. Some of those can understand quaternions. Some of those can understand GA. I'm not sure it's a question of teaching it earlier in school either. Go watch the 3-blue-one-brown video on visualizing quaternion rotation, and try to really appreciate how abstract and hard to understand that is compared to multiplying s set of (ortho-normal) basis vectors by some coordinates to do rotation. Then remember that quaternions are in some sense a subset of the GA abstractions people want us to use. This isn't going to happen any time soon.
- ogogmad 3y ago
  That might partially be true, but GA can mostly be given an API that hides the gory and confusing algebra. This is actually quite a natural thing to do. The idea is pretty much that in 3D you can begin with reflections through planes, which most people do understand, and then represent other Euclidean transformations as straightforward compositions of those. Once you have an API for transformations, then a geometric figure like a point or a line can be represented by the reflection (which is a transformation again) through it.
- aap_ 3y ago
  > Go watch the 3-blue-one-brown video on visualizing quaternion rotation, and try to really appreciate how abstract and hard to understand that is That video was not actually very good unfortunately (surprisingly, I might add). I doubt it helped people understand the matter much but just repeated the old "weird incomprehensible 4d number" story.
- itishappy 3y ago
  I learned the way you described: Numbers -> Complex Numbers -> Vectors -> Matrices -> Quaternions -> SO(2) -> SO(3) -> Lie Algebras -> GA (The last 3 might need air quotes around "learned") But I actually think it could be taught: Numbers -> Vectors -> GA -> Maybe Matrices? Discover complex number and quaternions naturally, by playing with rotations. Discover e the same way! (They taught me e through finance.) This would have explained so many of the weird quirks of math I had questions about. I bet it would have made relativity easier. Indeed, this isn't going to happen anytime soon.
  - at_compile_time 3y ago
    Plugging bivectors that square to -1, 0, and 1 into the exponential power series and seeing rotations, translations and hyperbolic rotations come out the other side was a mind-expanding moment for me.
    - phkahler 3y ago
      Wait what?!?!? Now I'm intrigued.
- at_compile_time 3y ago
  >I don't see how any of this solves hard problems. You need to know a formula to compute what you want regardless of the underlying representation. The thing with geometric algebra is that the function is the same regardless of what you're working with. In projective geometric algebra, joining two points into a line is the same function as joining a point and line into a plane. Intersections, projections, rejections, and transformations all have the same function, regardless of the objects involved. Reflecting across a plane, around a line, about a point, really any transformation, uses the same A * B / A, formula, regardless of whether B is a point, line, plane or transformation. You still need to learn it, but you only need to learn it once. >I also think the more abstract you get, the smaller portion of the population can understand and work with it. That's exactly the point. Geometric algebra is a large step down in abstraction from quaternions, yet explains them fully. The rules of geometric algebra build directly from vector algebra and could be taught in a single lecture, with several semesters worth of material in geometry, calculus and physics that follow from those simple rules. But the people who use quaternions already know quaternions, and the physicists who work with complex matrices already know complex matrices. They think they're standing on isolated towers of knowledge they had to scale by their own hard work, and see geometric algebra as a separate height that offers no advantage on its own, when it was actually a shallower ascent, that once climbed, reveals that what had appeared to be distinct areas of study were actually linked and could be understood together. Projective geometry and special relativity might seem like different subjects from the outside, or from within either, but when viewed from geometric algebra, they're just algebras with different signatures (whether the basis vectors square to 1, -1, or 0). A Lorentz transformation is little different than rotation around a line, and if you can work with one, your knowledge transfers directly to the other. All built from the building blocks of vectors, no complex matrices or generators required.
fiforpg 3y ago
> Traditional vector math gives the output of the cross-product of two vectors the same type. (That's why it's only meaningful in 3d.) That is not really the reason. Relevant trivia: that cross product is defined only in dimensions 3 and 7 (not only in 3) has to do with existence of certain algebraic objects — division algebras — which only exist in dimensions 1 (reals), 2 (complex), 4 (quaternions), and 8 (octonions). https://en.wikipedia.org/wiki/Seven-dimensional_cross_product https://en.wikipedia.org/wiki/Seven-dimensional_cross_produc... https://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(composition_algebras) https://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(compositi... You could say there is cross product in 1d by analogy, but of course it is always equal to 0.
- ComplexSystems 3y ago
  The wedge product (or "outer product" in geometric algebra) generalizes the cross product and works in any number of dimensions, in the sense that the cross product in 3D is just the Hodge star of the wedge product. The cross product gives a vector perpendicular to the plane generated by two vectors, whereas the wedge product gives an object representing that plane directly, and the Hodge star then mapping it to its orthogonal complement.
- itishappy 3y ago
  Thanks, my math's never been particularly rigorous. The 7 dimensional cross product is fascinating, my brain really wants it to be 5 dimensional. I often use the cross product in 2d too, don't tell anybody.

phkahler 3y ago

In geometry code (from my own experience) I use a cross product specifically to get a vector perpendicular to two other vectors. There are other times like calculating the area of a triangle in 3-space that involve taking the magnitude of the cross product or whatever. I don't see how any of this solves hard problems. You need to know a formula to compute what you want regardless of the underlying representation. I also think the more abstract you get, the smaller portion of the population can understand and work with it. Lots of people understand numbers. Some of those understand vectors. Some of those can understand a transformation matrix. Some of those can understand quaternions. Some of those can understand GA. I'm not sure it's a question of teaching it earlier in school either. Go watch the 3-blue-one-brown video on visualizing quaternion rotation, and try to really appreciate how abstract and hard to understand that is compared to multiplying s set of (ortho-normal) basis vectors by some coordinates to do rotation. Then remember that quaternions are in some sense a subset of the GA abstractions people want us to use. This isn't going to happen any time soon.

ogogmad 3y ago
That might partially be true, but GA can mostly be given an API that hides the gory and confusing algebra. This is actually quite a natural thing to do. The idea is pretty much that in 3D you can begin with reflections through planes, which most people do understand, and then represent other Euclidean transformations as straightforward compositions of those. Once you have an API for transformations, then a geometric figure like a point or a line can be represented by the reflection (which is a transformation again) through it.
aap_ 3y ago
> Go watch the 3-blue-one-brown video on visualizing quaternion rotation, and try to really appreciate how abstract and hard to understand that is That video was not actually very good unfortunately (surprisingly, I might add). I doubt it helped people understand the matter much but just repeated the old "weird incomprehensible 4d number" story.
itishappy 3y ago
I learned the way you described: Numbers -> Complex Numbers -> Vectors -> Matrices -> Quaternions -> SO(2) -> SO(3) -> Lie Algebras -> GA (The last 3 might need air quotes around "learned") But I actually think it could be taught: Numbers -> Vectors -> GA -> Maybe Matrices? Discover complex number and quaternions naturally, by playing with rotations. Discover e the same way! (They taught me e through finance.) This would have explained so many of the weird quirks of math I had questions about. I bet it would have made relativity easier. Indeed, this isn't going to happen anytime soon.
- at_compile_time 3y ago
  Plugging bivectors that square to -1, 0, and 1 into the exponential power series and seeing rotations, translations and hyperbolic rotations come out the other side was a mind-expanding moment for me.
  - phkahler 3y ago
    Wait what?!?!? Now I'm intrigued.
at_compile_time 3y ago
>I don't see how any of this solves hard problems. You need to know a formula to compute what you want regardless of the underlying representation. The thing with geometric algebra is that the function is the same regardless of what you're working with. In projective geometric algebra, joining two points into a line is the same function as joining a point and line into a plane. Intersections, projections, rejections, and transformations all have the same function, regardless of the objects involved. Reflecting across a plane, around a line, about a point, really any transformation, uses the same A * B / A, formula, regardless of whether B is a point, line, plane or transformation. You still need to learn it, but you only need to learn it once. >I also think the more abstract you get, the smaller portion of the population can understand and work with it. That's exactly the point. Geometric algebra is a large step down in abstraction from quaternions, yet explains them fully. The rules of geometric algebra build directly from vector algebra and could be taught in a single lecture, with several semesters worth of material in geometry, calculus and physics that follow from those simple rules. But the people who use quaternions already know quaternions, and the physicists who work with complex matrices already know complex matrices. They think they're standing on isolated towers of knowledge they had to scale by their own hard work, and see geometric algebra as a separate height that offers no advantage on its own, when it was actually a shallower ascent, that once climbed, reveals that what had appeared to be distinct areas of study were actually linked and could be understood together. Projective geometry and special relativity might seem like different subjects from the outside, or from within either, but when viewed from geometric algebra, they're just algebras with different signatures (whether the basis vectors square to 1, -1, or 0). A Lorentz transformation is little different than rotation around a line, and if you can work with one, your knowledge transfers directly to the other. All built from the building blocks of vectors, no complex matrices or generators required.

fiforpg 3y ago

> Traditional vector math gives the output of the cross-product of two vectors the same type. (That's why it's only meaningful in 3d.) That is not really the reason. Relevant trivia: that cross product is defined only in dimensions 3 and 7 (not only in 3) has to do with existence of certain algebraic objects — division algebras — which only exist in dimensions 1 (reals), 2 (complex), 4 (quaternions), and 8 (octonions). https://en.wikipedia.org/wiki/Seven-dimensional_cross_product https://en.wikipedia.org/wiki/Seven-dimensional_cross_produc... https://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(composition_algebras) https://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(compositi... You could say there is cross product in 1d by analogy, but of course it is always equal to 0.

ComplexSystems 3y ago
The wedge product (or "outer product" in geometric algebra) generalizes the cross product and works in any number of dimensions, in the sense that the cross product in 3D is just the Hodge star of the wedge product. The cross product gives a vector perpendicular to the plane generated by two vectors, whereas the wedge product gives an object representing that plane directly, and the Hodge star then mapping it to its orthogonal complement.
itishappy 3y ago
Thanks, my math's never been particularly rigorous. The 7 dimensional cross product is fascinating, my brain really wants it to be 5 dimensional. I often use the cross product in 2d too, don't tell anybody.