Summer of Math Exposition

This seems to assume prior experience with the problem, and it is overall far too complicated for the target audience of high school students. The animations are well done. A lot of the notation and topics needed more exposition to be accessible without extra research.

The animations look great. It’s very pretty! The pacing and sound could use some work, and it is a bit unclear why this is an interesting problem to start with without getting pretty deep into it. I really like that you made this video and the concept of “I was on a walk and this came to mind!” - sharing mathematical hobby-work like this is a really nice part othe the mathematical community and math in general: it is something you can just think about and come up with cool things! I think you could lean even more into that story, adjust the pacing a bit and work a bit on the sound (I know this is really hard but it makes such a big difference!) - good job!

Colouring graphs is a complex yet beautiful subject which was really well presented in the video. The presence of actual illustration of the colouring methods is a huge plus. I think that, for an even better accessibility, for example for high-schoolers who also are a target of this content, it could be a good addition to explain some of the symbols and notations used that are not usually encountered at this study level. Awesome video nonetheless, thank you!

Perhaps taking a bit more time to appreciate the problem could have been useful, along with some reminders of what the things defined were when they were used later.

I do not understand the problem. The goal is to color every vertex in the graph, but the rules are inconsistent. Every point inside a circle is a vertex, and no points within the unit circle centered at a vertex can be the same color, unless they’re inside the same circle, but the circles are smaller than the unit circle, and some negative space exists between circles that is either uncolored or an eighth color (black). The perimeters of the circles are two half-open intervals but on the curve that defines the circle, not between the inside and outside of the circle as defined by the Jordan-Brouwer intuition, and are drawn in a ninth color, gray. There is no mention of the Four Color Theorem, which has proven every 2D graph can be colored with four or fewer colors so that no two adjacent vertices, and every map can be colored so no two adjacent areas, have the same color. The Chromatic Exclusion section appeared to be building up to four colors, but the examples never went past odd-numbered cycles. It is not explained how transdimensional measure relates to the problem. It is never mentioned in the final section and its section of the video does not explain its purpose, relating to the Hadwiger-Nelson problem or in general, outside of a representation of (1+t)² being able to tile the plane.

This kind of stab at original research seems like it might be better if accompanied by a formal write-up.

The definitions and motivation for CEM, CVM, and pressure were under-explained.

Your section on “transdimensional measures”— is this original? If so, it seems not entirely fleshed out, in terms of which subsets of the plane are measurable, or even just that the subsets you require are measurable. If not, a reference would be nice. It wasn’t clear why Hahn series would be appropriate instead of just formal power series.

It seems like you’re deep in the weeds of this problem, and it’s hard to care about proposals for an argument that might work if you don’t have evidence that the details can be filled in.

I enjoyed this video a lot as I learned a lot.