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Actually, that very same website gives it too, and even a 2nd visualization ! https://betterexplained.com/articles/understanding-why-complex-multiplication-wor

Actually, that very same website gives it too, and even a 2nd visualization ! https://betterexplained.com/articles/understanding-why-complex-multiplication-works/ https://betterexplained.com/articles/understanding-why-compl... Is what bothers you that the complex plane has this very specific feature that purely imaginary x purely imaginary = real, so you're "mixing" orthogonal dimensions ? However, I still don't understand your reticence, since imaginary numbers are fundamentally about cycles and rotations (via exponentiation), and rotations fundamentally require having 2 dimensions to represent them : hence a plane ! (Like you say, you also need 2 dimensions to be able to have orthogonality...) And I'm getting a bit rusty here, but I guess that imaginary being orthogonal to real might be why these Fourier basis vectors are orthogonal ? I mean, it's literally Euler's formula : exp(𝛕iθ) = cos(𝛕θ) + i sin(𝛕θ) I don't remember, aren't those 2 orthogonal basis vectors "cos" and "sin" ?

geokon 5y ago
I guess what bothers me is that visualizations are supposed to leverage our intuition and gut feeling. Like in Geometry, the teacher tell you that when the lines are parallel, the angels on on a bisecting line are the same. You "see" it on the image and it immediately makes sense. When you have a weird new operation that doesn't behave in any relatable way then the visualization is effectively useless. I explained a bit further here how you can completely skip this pseduo-2D stuff and things remain very clear: https://news.ycombinator.com/item?id=27246386 https://news.ycombinator.com/item?id=27246386 Maybe the 2D plane is useful in other contexts and it's an analogy that's worth internalizing - I don't wanna knock it entirely - but in the case of Fourier Transform it's a high cognitive load for something that's actually not that important and easy demonstrated from the roots of unity. And the net effect is that the stuff that can't be visualized, and that is arguably more important, is glossed over b/c it's not pretty enough
- BlueTemplar 5y ago
  "How many angels can dance on intersecting lines" ? XD
  - geokon 5y ago
    haha, i shouldn't type replies late at night. So many typoes..

geokon 5y ago

I guess what bothers me is that visualizations are supposed to leverage our intuition and gut feeling. Like in Geometry, the teacher tell you that when the lines are parallel, the angels on on a bisecting line are the same. You "see" it on the image and it immediately makes sense. When you have a weird new operation that doesn't behave in any relatable way then the visualization is effectively useless. I explained a bit further here how you can completely skip this pseduo-2D stuff and things remain very clear: https://news.ycombinator.com/item?id=27246386 https://news.ycombinator.com/item?id=27246386 Maybe the 2D plane is useful in other contexts and it's an analogy that's worth internalizing - I don't wanna knock it entirely - but in the case of Fourier Transform it's a high cognitive load for something that's actually not that important and easy demonstrated from the roots of unity. And the net effect is that the stuff that can't be visualized, and that is arguably more important, is glossed over b/c it's not pretty enough

BlueTemplar 5y ago
"How many angels can dance on intersecting lines" ? XD
- geokon 5y ago
  haha, i shouldn't type replies late at night. So many typoes..