Summer of Math Exposition

Great video. I wonder if we could generalise these tools to higher dimensions? Could we take Fourier Transforms of eg the Platonic Solids :D

Discrete Fourier Transform is like a subset of Representation Theory, in which we take the trace of matrices acting on vector spaces, forming groups.

I almost did my video this year on the Compound-Of-Five-Tetrahedra, because I noticed the golden ratio in the character table of A_5, and I noticed that it’s a dimension 3 special-orthogonal representation (and a mirror copy) which acts on a dodecahedron. As for the five ‘items’ that A_5 acts on, one can inscribe five tetrahedra inside a dodecahedron, and turn them into a compound. The first person to do that did it in 1876, 150 years ago, so I might have to do a celebratory video at some point for that lol.

Very interesting and fairly clear. I liked the motivation at the beginning. (I liked that there was no distracting music.)

Motivation : 2.5 / 3 Clarity : 2.5 / 3 Novelty: 2.25 / 3

Memorability: I guess I might have in the back of my mind for a while that there exists something called a discrete fourier transform, but from a first watch with no pausing I will be honest I didn’t really understand the core idea. but maybe it’s genuinely difficult to explain it at a high level in simple terms

Novelty: I personally don’t know how to judge common of a topic this is.

Clarity: The production quality is really great! I’m sure that if I wanted to spend some time on it, I’d have no problem piecing the topic together from the diagrams, equations, animations

Motivation: While you provide references to applications in electrical engineering, I found myself wondering what the application actually was. But I get that it could require a lot of background to explain

It’s cool to see the Discrete Fourier Transform in action, represented visually!

I think it could’ve been nice to see how you find the coefficients that give a particular (irregular) polygon of your choice. I think you did in a geometric way for the case of a triangle, but it could’ve been nice to see it in general/for your working example of 12-sided shapes, not necessarily the same type of geometric method. Or if that would be too much math for what you’re going for, just picking 12 arbitrary points and working it out empirically, tuning the coefficients manually until you get something that matches, could work too.

This was a nice overview. A few hiccups:

• I don’t think I understood what the PDN theorem is? They way you presented it it should be that the matrix is invertible; i.e. how to get from an arbitrary triangle to the ZPN vector. But you say that it is another construction implying a forward application of the matrix again.
• In general, there is a dissonance between things you say and show: For example, at 03:38 you say “Multiplying…” but at the same time, plus symbols appear on-screen. Of course, these refer to different things. But it is quite confusing that these happen at the exact same time.
• At around 05:12, you don’t actually show that the triangle areas geometrically behave like you claim. You just claimed it.

I didn’t really know what was happening and I have studied DFT before! Why is there S for even numbers? Why is constructing polygons this way more useful than other methods? Why is the method the way it is? Also audio quality could be improved slightly!

Other than that great video!!