Summer of Math Exposition

Very clear explanation, especially with the file analogy and such. Never thought about fractal dimensions that way and it’s nice.

There are a lot of jargons about Information Theory that if I have no background about it, I couldn’t fully appreciate what you’re trying to say. Since the clarity takes a large proportion of points, I have to give you a low score. However, the visual presentation is pretty well-made!

Thank you for adding captions. In the future, keep in mind that on mobile your captions may collide with text you put on the bottom of the screen.

The title cards disappearing from the right makes them harder to read. They should disappear from the left to give the viewer more time to read.

You start using point-set topology terminology at around 1:40 without proper introduction. I’d guess that calling a cube a set isn’t intuitive, even for many young math enthusiasts. At the very least, throw in a paragraph toward the beginning about how it’s useful to think of geometric shapes as sets.

The transition into compression is genuinely jarring—I thought I was watching a video about dimensions! You should take more care when introducing an entirely new topic to prevent your viewers from getting whiplash. Put compression in the title and the thumbnail if the connection is that important, and acknowledge that this connection is surprising in the video before you start talking about it.

I did not find the analogy about squares of decreasing size helpful. I did sort of understand the Cantor set example immediately following it.

Overall, this video bites off more than it can chew. The point-to-set principle seems to involve multiple disparate fields of mathematics. If your audience really extends as young as high school students, you need a lot more scaffolding to establish what’s going on.

wow very cool. I saw the punchline coming from a minute away :)

That’s a super interesting theorem that I had no idea about. I liked the visuals. does describing an algorithm like the ternary to binary one techincally only upper bound the dimension, because there could be a more efficient compression? technically more work needs to be done to show that there does not exist a more efficient compression algortithm (which is easy in that case but maybe not in general)

Really nice idea, and something I’ve never seen before! This would be a great video to show to students who are planning to present a talk at a maths society or something similar, to showcase how to connect ideas together in an interesting way.

I think some explanations went a little fast, especially if people haven’t encountered fractal dimension or compression algorithms before. I liked your example-first approach, but you could spend a little more time breaking down each example just to make it clear precisely what you’re trying to do.

I learned that by finding an algorithm to compress the digits of a point in a set (at least, in a set of real numbers), you can find the compression factor, revealing the dimension of the set.

Beautiful link between geometry and information theory / compression. Video quality is good. What I found lacking is a bit more explanation, especially around 6:00 which is the crux of the video. Would’ve been nice to see a longer, more detailed video, with more visual intuition behind the result (the Cantor set example at the end helped, but it would’ve been nice to see it for the general S shape too).

As a general feedback, you might want to slow your pace a little. I quite like the idea of the dimension being a part of a compression process, since I never really heard about this before. It would have been nice to have an immediate intuitive connection to the sense of global compressibility (I assume there is one, but I couldn’t come up with it myself), which I did not see coming out. Even though, this video is especially nice to have on the novelty side. At last, you might want to avoid having too much text on a single frame (as eg 5:12), distracting the viewer from what you’re telling them, as reading consumes a lot of attention.

1:55 Text definitely not on screen for long enough. Maybe add a pause for contemplation, leave it on screen for the transition, …

The switch from fractal dimensions to compression is too random. I’d add a longer transition, a sentence or two perhaps stating clearly that we’ll come back to it but are first starting an entirely new topic — compression.

3:17 I think there is a missing step here that should be made explicitly: That sequences with some ones and some zeroes will be more compressible if for example there is a greater imbalance between 0s and 1s. Really anything that helps us understand where at 4:30 actual values are coming from when all we have seen previously is some “hand written code compression”. Naturally this is especially inconsistent when for example a program spitting out digits of pi and other computable numbers is basically constant size, yet those are the only examples shown at 4:00 The example at 7:00 give a better idea of how our complexity works, so maybe just leave it open and state it will be explained later.

7:48 nice sleight of hand disappearing the log base there. Not sure if that’s gonna confuse anyone, I’d at least animate it cancelling in the calculation.

Individual criticisms aside, I’m not sure I agree on the point of the video, about dimension not just being a macroscopic property. You do kinda make it sound like you could calculate the dimension of a set from a few of its point, maybe just one well-chosen one. Of course this is completely untrue, since the compression algorithm has to be specifically designed to handle all the points in the set and non outside of it, to work most efficiently and achieve the desired highest compression. In that way it just turns back into a problem of enumeration, where the algorithm has to map the points at a given resolution to as few “indices” as possible, so the compression algorithm itself is still based on the total number of distinct points at a given resolution. It is still a macroscopic approach, thought definitely interpreted very differently.

Great video!

Really awesome topic! I haven’t every seen this connection before and you do a great job motivating it. I think your intuitive examples are great, but could have used more time to set in. I doubt there are many high schoolers that could follow this. You go pretty fast and get a little symbol heavy at times. I’ve taken graduate courses in formal language theory and geometry, but I found myself having to pause and rewind a number of times throughout the video. Your second implication at 7:58 took me a little while to fully understand. You’re implicitly using the fact that there exists arbitrarily long incompressible strings, which I don’t think you state before. Overall, I really enjoyed this but wished you spent a bit longer going through your definitions (particularly of fractal dimension) and explanations.

This is an interesting idea, and some fun animations.

I would have liked some explanation for how/why this is related to other notions like Hausdorff dimension.

Curious for mathematicians who have not explored this type of topic, but too specialized for high-school students. Try not to fill the screen with too much text, otherwise it becomes difficult to follow what you are saying.

fantastic animations! very clear and offers interesting perspectives

Nice video that addresses a reasonably novel topic with innovative blends between geometry and information theory — I never thought fractal dimensions could connect with compression algorithms; very interesting! The video, however, could benefit greatly if it would contextually explain and justify the relevance of some of the jargon used, most notably “fractal dimension,” “effective dimension,” “normalized complexity,” and notations (say, what does "$x \upharpoonright n$" really denote? What is the purpose of "$\max_{x \in S}$"?) The unfortunate lack of clarity made the video very inaccessible, especially to the video’s choice of audience. Also, consider explaining the purpose of the video’s use of math: it is hard to understand the presence of logarithms in the context of fractals, likely because of the lack of definitions. I also suggest establishing a clear motivation on how this subject matter may be of interest and also allocating more time in the video for reading and thinking. A strong effort.