I saw Steven Strogatz share a quote from Rota earlier in the week:
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This passage both struck me and annoyed me – not totally sure why:
It is certainly not the case that everything that is interesting to mathematicians is going to be interesting to either the general public or to kids learning math. Nonetheless, there are many ideas beyond the (maybe overshared) “usual pap of Klein bottles, chaos, and colored pictures” that are indeed worth sharing.
So my challenge for everyone in math is to write one post – just one post – sharing an idea that shows something that might be interesting to both mathematicians and to students learning math. For every post that gets shared with me, I’ll write another one 🙂
Here’s my first idea. I saw it originally from Ann-Marie Ison and then more recently from Burkard Polster aka “the Mathologer”. The idea shows some fascinating geometry hiding in modular arithmetic.
Our two projects with this type of modular arithmetic are below:
Using Ann-Marie Ison’s incredible math art with kids
Extending our project with Ann-Marie Ison’s art
I think the best way to play with the modular arithmetic patterns is via this Desmos activity written by Martin Holtham:
Also, a good way to dive a little deeper into what’s going on in these pictures is this video from Polster:
I used this modular arithmetic idea in lectures at two math camps last year and both times the kids were just blown away. The connection between geometry and arithmetic here is so fun and so surprising that it is hard to imagine anyone seeing it for the first time not being blow away!
So, no gimmicks, no super abstract math, but something that I think would be both enjoyable for students learning math and something that mathematicians will not throw back on the shelf in disgust.
What are some other ideas 🙂
Mike's Math Page 
I’ll just mention here a recent book I’ve been touting, Ed Scheinerman’s “The Mathematics Lover’s Companion,” which I think is a fabulous collection of topics of interest to both mathematicians & students (…several are a bit well-worn, but still Scheinerman’s treatment is engaging and fairly fresh).
A lot of the “usual pap” are things, like say the Monty Hall problem, that really are quite fascinating the first time you hear about them, but have, because they were initially so interesting, now been explored ad nauseam.
I’m enjoying reading “Things to make and do in the 4th dimension” by Matt Parker recently.
The whole chapter on the Lucas and Fibonacci series is really interesting. The throw out observation that any series built on the recursive rule of adding the previous 2 terms approaches the golden ratio totally challenged my previous assumptions.
I really enjoyed that book as well.
The specific example you are talking about is something that was one of the very first bits of advanced math that I ever saw. It was part of a lecture at my high school when a former student (who’d won a full ride math scholarship) came back and lectured about a few things she was learning at college.
Turns out that I wrote about it in the last part of this blog post:
https://mikesmathpage.wordpress.com/2015/12/15/math-that-made-you-go-whoa/
Well, “in and around mathematics” is my beat (as the author of the monthly Mathematical Enchantments essays), so I guess I’d better reply to Mike’s challenge! I’m not going to write a new essay just for Mike (writing is hard!), but I’ll share one essay I’m especially proud of: https://mathenchant.wordpress.com/2016/03/16/believe-it-then-dont-toward-a-pedagogy-of-discomfort/
I think I’d be on pretty thin ice if I didn’t allow previously written blog posts in!
Thanks for the reply.
Eugenia Cheng’s work is fantastic in this regard, I think. For instance, “How to Bake Pi” gets into category theory via recipes. Here’s a list of posts including “The Maths of Pizza” and “The Maths of Doughnuts”: http://eugeniacheng.com/math/non-specialists/
OK Mike, I took the challenge, and re-ran a simple demonstration I’ve always enjoyed and thought clever regarding prime number gaps:
https://math-frolic.blogspot.com/2017/04/no-largest-prime-gap.html