Carnival of Mathematics #197

Welcome to the 197th Carnival of Mathematics!

197 is an unseemly number, as you can tell by the Wikipedia page which currently says that it has “indiscriminate, excessive, or irrelevant examples.” How deviant. It’s also a Repfigit, which means if you start a fibonacci-type sequence with the digits 1, 9, 7, and then continue with $ a_n = a_{i-3} + a_{i-2} + a_{i-1}$, then 197 shows up in the sequence. Indeed: 1, 9, 7, 17, 33, 57, 107, 197, …

Untangling the unknot

Kennan Crane et al showcased a new paper that can untangle tangled curves quickly, and can do things like generate Hilbert-type space-filling curves on surfaces. It’s a long thread with tons of links to videos and reading materials, covering energy functions, functional analysis, Sobolev methods, and a custom inner product.

New paper with Chris Yu & Henrik Schumacher: https://t.co/SDlisHgmXu

We model 2D & 3D curves while avoiding self-intersection—a natural requirement in graphics, simulation & visualization.

Our scheme also does an *amazingly* good job of unknotting highly-tangled curves!

[1/n] pic.twitter.com/lee4WmFyPT

— Keenan Crane (@keenanisalive) August 2, 2021

Folding equilateral triangles without measuring

Dave Richeson shows off a neat technique for folding equilateral triangles using just paper and no measurements. Replies in the thread show the geometric series that converges to the right 60 degree angle.

TIL how to fold a row of equilateral triangles in a strip of paper. Start with any fold. Here I’ve done one that is nowhere near 60°. Open it and fold the paper so an edge follows the previous crease. Repeat. The triangles quickly become almost equilateral. Can you justify this? pic.twitter.com/ALUaiVgbXS

— Dave Richeson (@divbyzero) August 7, 2021

Shots fired at UMAP and t-SNE

Lior Pachter et al. study what sorts of structure are preserved by dimensionality reduction techniques like UMAP (which I have also used in a previous article) by comparing it against a genomics dataset with understood structure. They make some big claims about how UMAP and t-SNE destroy important structure, and they show how to contrive the dimensionality reduction plot to look like an elephant even when there’s no elephantine structure in the data.

I’m not expert, but perhaps one best case scenario for UMAP enthusiasts would be that their analysis only applies when you go from very high dimensions down to 2 just so you can plot a picture. But if you stop at, say, $ \sqrt{n}$ dimensions, you might still preserve a lot of the meaningful structure. Either way, they make a convincing pitch for Johnson-Lindenstrauss’s random linear reductions, which I’ve also covered here. Their paper is on biorXiv.

It's time to stop making t-SNE & UMAP plots. In a new preprint w/ Tara Chari we show that while they display some correlation with the underlying high-dimension data, they don't preserve local or global structure & are misleading. They're also arbitrary.🧵https://t.co/XkAOTKlOcs pic.twitter.com/dmFzD5RR6R

— Lior Pachter (@lpachter) August 27, 2021

Studying the Sieve

Ben Peters Jones took up Grant Sanderson’s math video challenge and released a series of videos studying the Sieve of Eratosthenes.

Additional submissions

• Tanya Khovanova remarks on their Number Gossip, that 1331 is the only nontrivial cube that can be written as $ x^2 + x – 1$.
• Ganit Charcha explains the math behind Montgomery Multiplication, i.e., modular exponentiation without computing a division. Super useful in cryptography!
• Anne Schwartz asked about your favorite math teaching articles, and received a big response with lots of interesting articles.
• Colin Beveridge shows off how their favorite bag of tricks to determine which triangular numbers are also squares.
• An oldie, but new to me, was the series of computational notebooks, 12 Steps to Navier Stokes, which teaches the basics of Computational Fluid Dynamics.
• Nisar Khan has been drawing pictures of random number generators

Be sure to submit fun math you find in September to the next carvinal host!