Taking the concept to a 2d path: https://www.myfourierepicycles.com

And, with your own drawing: https://gofigure.impara.ai

Only if it is band-limited.

Back in my uni days I did not get why that works. Why are sine waves special?

Turns out... they are not! You can do the same thing using a different set of functions, like Legendre polynomials, or wavelets.

It makes more sense when you approach it from linear algebra.

Like you can make any vector in R^3 `<x,y,z>` by adding together a linear combination of ` <1,0,0> `, ` <0,1,0> `, ` <0,0,1> `, turns out you can also do it using `<exp(j2pi0/30), exp(j2pi0/31), exp(j2pi0/32)>`, `<exp(j2pi1/30), exp(j2pi1/31), exp(j2pi1/32)>`, and `<exp(j2pi2/30), exp(j2pi2/31), exp(j2pi2/32)>`.

You can actually do it with a lot of different bases. You just need them to be linearly independent.

For the continuous case, it isn't all that different from how you can use a linear combination of polynomials 1,x,x^2,x^3,... to approximate functions (like Taylor series).

This kind of statement is too vague to be useful. I even know all the words in there and I still don't know what you mean.

Hmm I always viewed it as sum over orthogonal bases. Sine/Cosine is choice, while there can be many other bases, like Legendre polynomials. Can you explain more how FT is linked to 'a sum over equivalence classes'?

On your site you make the claim that: "Our thesis is that there is 100 years of physics and math research that has gone unnoticed by the CS/ML communities and we intend to rectify that."

Extraordinary claims require extraordinary evidence. Especially considering that a decent fraction of the CS/ML researchers that I know have solid physics and math backgrounds. Just of the top of my head, Marcus Hutter, David MacKay, Bernhard Scholkopf, Alex Smola, Max Welling, Christopher Bishop, etc. are/were prominent researchers with strong math and physics backgrounds. More recently Jared Kaplan and Dario Amodei at Anthropic also have physics backgrounds, as well as plenty of people at DeepMind.

To claim that you have noticed something in "100 years of physics and math research" that all of those people (and more) have missed and you didn't is pure hubris.

I’ve dabbled in explaining the FT some, and I think there’s an important trait of that video that needs to be highlighted: it’s a superb demonstration of what the Fourier transform does, mechanically, but it does not at all try to explain why it works in the places we usually apply it or how (having, admittedly, a thoroughly anachronistic mathematical background) you could have invented it.

To be clear: it’s a very good video and you should watch it if you don’t have a feel for the Fourier transform. I’m just trying to proactively instil a tiny bit of dissatisfaction with what you will know at the end of it, so that you will then go looking for more.