Summer of Math Exposition

That was lovely! Nothing outstanding, but very well written and explained!

I’m really interested in regular polyhedra, and I actually have that exact product from Polydron! I recommend that you try to try make all of Johnson solids (without consulting the internet) with the kit - it can be a fun challenge. However, I felt a bit disconnected from the motivation, especially at the part:

“Anyone who has read this far is probably having the same brilliant idea: can we build the snub-ball while arranging the four triangle colors in some orderly way?”

Truthfully, that had not occurred to me. What does it even mean to arrange the colors in an orderly way?

There are an odd number of triangles surrounding a pentagon (5 touching it along an edge, plus another 10 touching it only at a vertex), while the number of colors is even.

This seems like a strange constraint. Why is the number of colors even? Sure - the box came with 4 colors. But in theory you could use less than 4. You could use 3 colors or 1 color. The more relevant constraint (in my understanding) is that there are 15 triangles surrounding a pentagon and 4 is not a factor of 15.

how many ways are there to build a snub-ball/rhombi-ball with this kind of nice color arrangement?

This is great - but again - what do you mean “this kind of nice color arrangement?” What is nice about it?

I did like the graph theory part but it does assume that the reader has a graph theory background. I did, but not all readers might.

I liked it overall, but I felt that the motivation was a bit weak. The interactive that transforms the more complex polyhedrons into a dodecahedron was very smooth. And the tetrahedron hiding inside the dodecahedron was an awesome unexpected surprise. I recommend working on the ending though - it felt a bit sudden.

Motivation - This was nicely motivated by the author’s own curiosity and makes the reader interested from the very start on how graph theory can be used to nicely construct the snub ball colors.

Clarity - This was very nicely presented! I enjoyed reading through, going through the exercises and playing with the interactive features. I liked the author’s friendly and light writing style and the ability to connect the toy to graph theory principles by naturally building foundations and curiosity. This is great for a variety of audiences, as ideas are introduced clearly and without any assumptions of audience knowledge.

Novelty - The author has a personal writing style that makes the reading fun and this was a unique way to see math in the everyday.

Memorability - I will think of this when I come across everyday objects and think about how they can relate to mathematical concepts and I will remember this post if I ever come across this toy!

Nice article! Not much to add other than I really liked the terminal art. I can only imagine that took sometime to program.

I liked the background to your article, that you were inspired to look into this because you had a Polydron set. And there were some lovely pictures in your piece.

The fantastic interactive graphic of moving from the rhombi-ball to the snub-ball should have been right after the paragraph starting “More precisely, starting from the rhombi-ball, split every square into two right triangles, then twist the entire ball…” It was not clear to me what “twisting” meant, but the interactive graphic further down made everything clear.

I found the article engaging throughout, and I particularly liked the connection between edge colouring the dodecahedron and your colouring problem.

Your post is great. There are good diagrams about all of these hard-to-visualize shapes.

The applet under “different colorings” was critical to understanding the post.

I initially found the first few paragraphs under “Graph Theory Steps in” to be confusing. It would be more clear if you explicitly referred to the applet at this point, or if the applet were nearby.

It seemed strange to say that the dodecahedron graph can be drawn as a planar graph, when every polyhedral graph is planar.

First off, “I bought a toy and ended up in graph theory” is hilarious. The coloring question is well-motivated and easy to understand. The graphic that transforms between the three balls is genuinely beautiful. Something I really liked about this article are the questions you ask as you go, “Is there anything special about the 16:4 ratio?”. I don’t work in pure mathematics, but this is the kind of thought process I imagine real mathematicians experience, and it was nice to go along for the ride. Graph theory is not introduced out of nowhere, its a tool to solve a problem, a problem I want to see solved! I had to read the vertex transformation part a few times, but I think I understand, and what a clever way to look at the problem. The payoff for me was the explanation of the 16:4 ratio, and once you see it I thought “of course, it must be that way.” All-in, an excellent explanation, some of the intermediate steps explained in text were a little hard to follow, but the figures were placed so I was never lost for too long.

clearest deficiency to me is skipped steps in the three-case dodecahedron colouring proof. how do you “propagate constraints around the equator”? but besides that, your article is great, introduced very tangibly and advanced intuitively, with a fitting interactive exactly where i wished for one

It was good fun reading about building the ball. Insights stated could have been a bit clearer. But the animation within the article was very good. Overall a good read

Motivation: 5/9. It is implicit, but this article demonstrates how seemingly miscellaneous problems can be simplified to graph theory

Clarity: 7/9. Excellent visuals (especially the snub-ball to rhombi-ball to dodecahedron slider) with good explanations, except for a few steps. In particular, “count the two winding directions over all triangular faces” is not explicitly defined. (Also, I don’t understand the simplest representative code but I’m not sure if that should count).

Novelty: 7/9. Technically a combinatorics problem, but the graph theory structure is very different from introductory combinatorics. Includes both brute force computation and a satisfying proof from the results.

Memorability: 8/9. Pretty visuals in addition to the interesting math.

Cool blog post! Really interesting and fairly easy to follow! I really loved the slider that transitions smoothly between the different shapes!

good use of a “toy” to illustrate and explore a topic

I think it’s a nice puzzle to play with. I liked that the beginning introduced the story that made it a bit easy to understand for its target audience without too much jargon. Some of the visualizations of the different edge cases to reduce to were also quite useful to help to understand the topic. I think some of the findings and claims that it gives such as why the total windings forms a conjugation invariant isn’t that obvious, and may need a bit more justification, or some picture to illustrate why this is true. Transitioning to graph theory may also be difficult to visualize for a middle/high school audience, so it’d be good to have some animations here so they can see this in action.

The introduction felt, to me, like a simple question to wander into, but it led to a lot of complexity and yummy math. The visuals were helpful, especially the one with the slider.