> Why shouldn't we? We have a set of object called multipliers that we want to generalize to the 2 dimensional plane. The presented generalization (eg, the multiplier identified by x maps the point at 1 to the point at x) seems natural
Rotating all of a sudden does not feel natural to me at all. Why not start cutting up the 2D plane? Why not fold it? In analytic terms, we would cause discontinuities, whereas rotating is "smooth". But this is definitely nontrivial. Rotating is just so arbitrary. I think that skewing or flipping, for example is just as "natural" as rotating.
> The fact that this function has anything to do with exponential, trigonometry, or analysis is neat, but not important in this context.
It's actually at the heart of why rotation (and not some other geometric operation) is key to e^iπ.
The better idea is to start with vectors. Adding vectors or scaling them is straight-forward. It’s trickier to figure out what it means to multiply or divide two vectors.
The quotient of two vectors v/u should be some kind of operator which transforms one into the other (that is, when you multiply it by one, you get the other, (v/u)u = v(u\u) = v, because we want multiplication to be associative). If those two vectors are the same length but different directions, the natural transformation to use is a rotation. The reason to use a rotation is that we want the transformation to make sense irrespective of any arbitrary coordinate system we decide to impose. If we used some kind of skew, it would break down under change of coordinates. As for reflections: if the quotient of two vectors was some kind of reflection, then we could square any quotient of vectors to get an identity transformation, which would not result in a very useful or consistent arithmetic.
We want some operation that maps the point 1 to the point x, while holding the origin constant. I agree that scale and rotate is not the only such function, but when I personally visualize taking a grid and moving one point while keeping another constant, that is what I visualize. Additionally, this has the added property of maintaining the grid structure.
It is not immediately obvious to me how skewing could define such an operation in the general case; or how flipping could define such an operation in most cases.
In any case, the system of adders and multipliers in 2-dimension he describes, even if not the only reasonable 2D generalization, is certainly a reasonable generalizaion, and one that has proved useful.
>It's actually at the heart of why rotation (and not some other geometric operation) is key to e^iπ.
In this video, e^x is defined as a mapping between multipliers and adders. Multipliers are defined to be a combination of rotation and scaling. Any relationship between exponential, calculus, infinite sums, etc is purely a result of these definitions (except, perhaps, the motivation for choicing i * pi to be the principle multiplier mapping that gets mapped to -1 under e.)
> Multipliers are defined to be a combination of rotation and scaling.
True, my point was only that the definition obscures the fact that rotation (and specifically the stunning relationship between trigonometry and exponential functions) is the key to the answer of "why". Without that, I think the video is just an exercise in indirection.
Rotating all of a sudden does not feel natural to me at all. Why not start cutting up the 2D plane? Why not fold it? In analytic terms, we would cause discontinuities, whereas rotating is "smooth". But this is definitely nontrivial. Rotating is just so arbitrary. I think that skewing or flipping, for example is just as "natural" as rotating.
> The fact that this function has anything to do with exponential, trigonometry, or analysis is neat, but not important in this context.
It's actually at the heart of why rotation (and not some other geometric operation) is key to e^iπ.