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3 Nov, 4 tweets, 2 min read
Peter Sarnak on the “best” rotation in the plane. Turns out there are 16 equally good ones, including one that uses the golden angle! Wonderful Math Encounters talk @momath1. More at golden.momath.org
A theorem about the sense in which a golden rotation is “best”
Visualizing free groups as fractal trees
“ Most of this is elementary math. As I like to say, elementary means hard.” – Peter Sarnak

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More from @stevenstrogatz

Steven Strogatz Profile picture
31 Oct
Evolution of a manuscript: Three versions of the introduction to INFINITE POWERS, followed by the published version. Here's version 1:
Version 2:
Version 3:
Read 4 tweets
Steven Strogatz Profile picture
10 Sep
While my wife and I were binge watching #ThisIsUs tonight, we were shocked to hear some new characters discussing math concepts like the discrete cosine transform and the Karhunen-Loève transform. I was personally amazed at how accurate the math was!
By the end of the show, it all made sense. The show was celebrating the real-life mathematician Nasir Ahmed who gave us the means of sending pictures and videos in compressed form. His work helped many folks stay connected during this crazy pandemic. cc @Dan_Fogelman
The backstory of how Nasir Ahmed’s work came to be featured in #ThisIsUs is discussed here: people.com/tv/this-is-us-…. But if you are a fan of the show and haven’t gotten to this episode in season five yet, beware of spoilers in the article!
Read 4 tweets
Steven Strogatz Profile picture
11 Aug
Lots of great tips in this thread from @BettySLai. My own technique is to use the dictation function on my iPhone. I dictate while I’m walking my dog. Just talk and talk. It gives the writing a conversational sound. Most of it is garbage, but occasionally something good comes out
Like many of the commenters in the thread, my own biggest weakness is a tendency to censor myself, and to keep editing the same sentence over and over. But when I dictate, the sentences flow. I pick one small topic and talk to myself about it. It generates a lot of raw material.
Editing that raw material is easy. It’s what I love to do. It’s producing the first draft that kills me.
Read 5 tweets
Steven Strogatz Profile picture
26 May
I’m thrilled about this preprint w Martin Kassabov and Alex Townsend. We prove that a network of identical Kuramoto oscillators synchronizes —regardless of the details of its wiring diagram — if every oscillator is connected to at least 75% of the others. arxiv.org/abs/2105.11406
This puzzle has fascinated a lot of us in my little corner of nonlinear dynamics since 2012. What’s the smallest level of connectivity that guarantees that a homogeneous Kuramoto model (the simplest kind of oscillator system) will always fall into sync?
We still don’t know the answer. The magic level of connectivity was previously proven to lie between 68.38 and 78.89%, and conjectured to be exactly 75%. In this preprint we’ve now reduced the upper bound to 75%. But that’s still far away from the best known lower bound.
Read 6 tweets
Steven Strogatz Profile picture
10 May
Regarding the natural place to introduce e, my preference is to wait until calculus. Once you learn that the antiderivative of x^n is x^[n+1]/(n+1), it becomes fascinating to ask: what happens when n = -1? So define L(x) as the indefinite integral of 1/x and explore it.
Doing it this way, you discover something truly amazing and beautiful: L(x) behaves like a logarithm! For instance, it obeys L(ax)=L(a)+L(x), as you can show by taking d/dx of L(ax). Once you know L is a log function, the natural question is: what is its base? Answer: call it e
Then, once e is in hand, define e^x as the inverse function to L(x). After that, you discover further wonders: e^x is its own derivative! Or as my old HS calc book (by Lynch and Ostberg) put it, e^x is "indestructible" under differentiation.
Read 6 tweets
Steven Strogatz Profile picture
2 Mar
I’ve spent the morning reading this preprint: arxiv.org/abs/1805.11556. It has a story behind it. A finance person named @MarcosCarreira does math for pleasure, inspired by @CutTheKnotMath. While playing with a classic problem, he finds something weird in a famous paper about it.
It seems that Marcos discovered an error in that famous paper (by Gilbert and Mosteller) which nobody noticed until now. But I’m not an expert in probability, and it would be great if those of you who are would take a look at Marcos’s paper. It strikes as a neat piece of work.
Marcos uncovered the error by running numerical simulations and finding that his results didn’t quite match the predictions of the classic analysis. Puzzled, he redid the analysis very carefully himself, helped by Mathematica, and found a subtle mistake in the earlier work.
Read 6 tweets

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