Summer of Math Exposition

That was really nice!

This is a good video that describes a lot of interesting stuff. I especially like the quote from Fermat about “numerous and abstruse mysteries.”

The pacing on some of the early arguments is a bit slow for me, but it may be right for an audience that saw the meme and wanted to learn more.

I’d be interested to hear about interpretations of these identities in terms of arrangements. The theorems of Pythagoras and Plutarch at 7:00 have obvious geometric interpretations. Do the other formulas have known geometric interpretations, especially the 3-triangle Eureka theorem? I get the impression that there isn’t, but it would be good to say.

This was a excellent video! I love how you took the time to explain all of the math and history behind this one simple picture. You had a lot of engaging moments throughout the video. Good job! You should be proud of yourself!

This video starts with a very simple problem and then works up to some much more complex topics. Many viewers may get lost along the way. Yet, I think the topic is really well presented and motivated, in a way that is both informative and entertaining. To me, this is a great example on how to make a really good math video: Start with a simple yet engaging problem, slowly work your way up to some more complex statements and refer back to history along the way. Pacing and presentation are very good, in my opinion, the presenter is clearly a gifted teacher.

This video was already in my personal you-tube feed so I saw it before voting.

I especially like your animations. How you narrate by having your torso to the side and putting the math on the wall. The colors for each step are clear and you picked colorblind friendly colors. The narration was sometimes hard to follow however due to your accent.

Nicely presented. And this person is not AI generated which is a big plus.

I liked the video overall, I like how you build from just the picture of the cube up to introducing more specific concepts!

This video was very informative and it intuitively showed why sqrt(7) is missing while adding rigor in places to support. Very good and clean visuals used as well. There was a bit of bouncing around but overall a very good video.

Lovely theorem! I’ve seen the answers for sums of two squares (3Blue1Brown’s “Pi hiding in prime regularities” is a classic!) and sums of four squares, but never the case of sums of three squares! Or the full polygonal number theorem, for that matter!

Minor point, but I think you could’ve made it a bit clearer that Cauchy showed the pentagonal and higher cases. You do say it, but it’s not clear from what you show onscreen, and by the time you get back to that and say Fermat’s polygonal number theorem is finally complete, you again only show up to the pentagonal case and don’t make it clear that the higher cases were also proved.

Interestingly, the four square theorem also quickly reduces to the case of the three square theorem: if a number cannot be written as a sum of four squares, then it cannot be written as a sum of three squares either, so must be 4^a (8n + 7). But then by the three square theorem, 4^a (8n + 6) can be written as a sum of three squares, and 4^a (8n + 7) = 4^a (8n + 6) + (2^a)^2.

It could also be interesting to note that the polygonal number theorem is in fact optimal, i.e. for every n, there is a natural number which cannot be written as the sum of n - 1 n-gonal numbers, which is easily seen since 2n - 1 works.

In any case, I loved this!

Watched it, loved it. Can’t rememember much as it was a few days ago and I can’t be bothered to watch it again, sry. keep up the great work.

Great video! Clear explanations, nice mix of theory and mathematical history, good production values.

This was quite neat. It gave a self-contained intro to an interesting area of math, with clear proofs and engaging explanation. It is not an area I am familiar with, so it was cool to see some elegant but powerful results explained and put in historical context.

Two comments:

• The story has many twists and different sections with different points to be made. It was not always clear where you were heading. A greater degree of outlining and motivation would help the viewer keep all the pieces in their head and take mental notes for what will matter later.
• It takes a bit of quick mental arithmetic to keep up with the pace of the proofs. The video would end up quite long if you spelled out every step, but I imagine some viewers having to pause and think frequently. Maybe that is your intention anyway though.

Fantastic video, great structure and beautiful storytelling. Maybe some derivations deserve a bit more time to fully grasp them. Still, very well done, animations elevate the video without being invasive.

Motivated.

It’s always a good day when modular arithmetic steps into the ring.

I really liked the hook of one of those math puzzles, the type you would see online with a bunch of confused people in the comments. It made for a really smooth transition into number theory.

Also, the execution of the historical timeline was great. I felt like I could really understand the progression of the topic. A lot of the time, it’s hard to get both the historical and explanatory parts of a problem portrayed in a single video, but this was done really well. I think you did an amazing job in choosing what parts to show, and what to skip for time and narrative.

The visuals were good throughout. I liked the imagery of the joined cubes at the beginning especially. The text and timelines were also nice.

I think that maybe the example from the beginning could have been brought up again after being expanded into an algebraic problem? There’s just not much I can find to critique with this haha. It’s really nice when you find someone that you can tell is passionate on a topic and has enough knowledge on it to explain it in a clear and concise manner, no matter the complexity of said topic.

Best of luck on your PHD!

This was an excellent video not only introducing a topic that seems relatively simple, but also introduces math history in a digestible and interesting way. There’s clearly a progression about this history that doesn’t get covered often.

I felt this was a very relaxing video that didn’t demand too much from the watcher while ALSO enjoying the information. I felt the math was easily digestible and showed how each step makes sense. I wasn’t bogged down too much with the notation or examples.

The history was the side plot of the video which made the discussion even more interesting. It took stress away from all the math and wanted to make the characters in history relevant to the story. Since these theorems names come from these mathematicians, it makes sense to cover them in greater detail and show the collaborative effort between them all.

Also breaking down a popular math meme is always a fun way to engage a video in the beginning. Great animation style, non manim, and good use of the historical timeline of mathematicians.

A truly excellent video. A nice touch would have been a way to go back to the original prompt and show 3 non-natural numbers that do work. For example if you used complex numbers, 2^2 + 2^2 + i^2 would give you 7 although there would be no way to visualize this… or would there?

I really liked the script and the animation style. Good work.

I’ve had a really good surprise watching that one. The topic seems so simple but ultimately has deep implications. The pace is good, the visuals are too, and the voice is pleasant (though probably not native english). I encountered this theorem while searching for the magic square of squares, which made it particularly interesting. In the end, I somehow didn’t get any astonishing moment which is why I didn’t push the mark all the way. Keep going!