Dave Richeson

Mathematician. John J. & Ann Curley Chair in Liberal Arts at Dickinson College. Author of Tales of Impossibility (Princeton University Press, 2019) and Euler's Gem (Princeton University Press, 2008). Columnist at Quanta Magazine. Topology, dynamical systems, history of math, recreational math, math and art, math. Coffee drinker.

Home/blog: https://divisbyzero.com/home
Twitter: https://twitter.com/divbyzero
Books: https://press.princeton.edu/our-authors/richeson-david-s
Quanta: https://www.quantamagazine.org/authors/david-richeson/

Last Notes

2026-04-14 00:32:47 UTC

This is so cool! Given only the "exponential minus log" function, elm(x,y)=exp(x)-ln(y), and the constant 1, you can perform +, -, x, ÷, exp, ln, trig, powers, roots, etc. You can also obtain e and π! See https://arxiv.org/pdf/2603.21852 and https://arxiv.org/src/2603.21852v2/anc/SupplementaryInformation.pdf
https://media.mathstodon.xyz/media_attachments/files/116/400/241/349/939/463/original/9e2c80c41343ecea.png

2026-03-29 20:36:54 UTC

Given a Riemannian manifold that is topologically a sphere, we can talk about "geodesics"—paths as close to straight as possible on the surface. On a true sphere, geodesics are great circles, but on topological spheres, the geodesics may roam around the sphere forever without closing up.
In 1905, Poincaré conjectured that any such sphere must have at least three simple closed geodesics. That is, they are closed curves without self-intersections. For example, for an ellipsoid x²/a²+y²/b²+z²/c²=1, think of the ellipses where the ellipsoid intersects the coordinate planes.
In 1929, Lyusternik and Schnirelmann proved the conjecture. Although the idea of the proof was correct, it contained a flaw that was fixed in the 1980s by Grayson.
Their result was about simple closed geodesics. What about closed geodesics in general? You head off in some direction, you may cross your path multiple times, but you end up back where you started from, heading in the same direction. What can we say about those?
In the early 1990s, Franks and Bangert each published an article covering a special case: the sphere has positive Gaussian curvature (Franks) and all other spheres (Bangert). Together, their work proved the remarkable theorem:
Every Riemannian manifold that is topologically a sphere has infinitely many closed geodesics!
I was telling my topology class about this theorem because we were discussing the annulus, and the key to Franks's result was to show that a certain map on the annulus had infinitely many periodic points.
https://media.mathstodon.xyz/media_attachments/files/116/314/379/174/286/448/original/fd27f4bf747ae52e.png

2026-03-22 22:00:46 UTC

I had the idea of making a Truchet tiling holder that lets me rotate each piece by 90°. There's a little pin in the tray and a slot under each piece that allows each insert to rotate and line up in one of two orientations.
I used Random.org to choose the tile orientations for this tiling:
11010111
11001111
00111100
11111111
11100110
01101111
00000101
00101011
https://media.mathstodon.xyz/media_attachments/files/116/275/067/598/776/320/original/bba90a0845a253da.jpeg
https://media.mathstodon.xyz/media_attachments/files/116/275/067/641/094/930/original/b6f0029b2992035e.jpeg
https://media.mathstodon.xyz/media_attachments/files/116/275/067/682/785/770/original/24569d627e00f45f.jpeg

2026-01-28 18:21:31 UTC

I printed four n=7th iterate dragon curves. You can fit two together to get n=8 dragon curve, and four together to get n=9 dragon curve. You can also fit them together in a way that tiles the plane.
https://media.mathstodon.xyz/media_attachments/files/115/974/100/372/699/480/original/dab00f9e4ca32516.jpeg
https://media.mathstodon.xyz/media_attachments/files/115/974/100/527/302/844/original/2e5a1d1da4a846e7.jpeg
https://media.mathstodon.xyz/media_attachments/files/115/974/100/734/021/621/original/3aae5e91c2ad9347.jpeg
https://media.mathstodon.xyz/media_attachments/files/115/974/100/896/846/638/original/705c8fdf055768d7.jpeg

2025-09-02 23:58:44 UTC

As I was walking home today, I had this idea: Is it possible to turn a QR code into a Truchet tiling that's still a functioning QR code? Yep.
https://media.mathstodon.xyz/media_attachments/files/115/137/412/546/027/629/original/bb64b7de0111ae52.png

2025-08-15 11:41:48 UTC

- reply

Thanks! I made mine "by hand"—Adobe Illustrator + Photoshop.

2025-08-13 13:58:00 UTC

Yesterday at the MoMath MOVES conference, Erik Demaine discussed QR codes and showed examples of how to incorporate images into them so they are still scannable. I had to give it a try. This QR code should take you to my website.
More details: The second image is the QR code I started with (I used this page to create it https://www.nayuki.io/page/qr-code-generator-library). Certain "finder patterns" must be in the code—the nested squares and a row and column of alternating pixels. The rest are black and white pixels encoding the text. But for these, all that matters is whether the center of the pixel is black or white. So, if you shrink the black and white pixels by a factor of three, say, you can use the rest of the area however you want. I used Photoshop to turn my profile photo into a black-and-white pixellated image of the correct size. Then, I placed the QR code on top of it.
https://media.mathstodon.xyz/media_attachments/files/115/021/772/418/613/096/original/f403eb2858bceb2c.png
https://media.mathstodon.xyz/media_attachments/files/115/021/776/104/553/326/original/671516562af17e61.png

2025-06-02 02:12:26 UTC

I got an email from someone asking for my "Essential Trigonometry for Calculus" cheat sheet, which I give to my calculus students. So, I thought I'd put it on GitHub for anyone to download (compiled PDF and source code). https://github.com/divisbyzero/Essential-Trig-for-Calculus
https://media.mathstodon.xyz/media_attachments/files/114/611/337/621/451/424/original/c4389ee8a11e3b95.png
https://media.mathstodon.xyz/media_attachments/files/114/611/337/853/412/386/original/8e48423fdd72d78e.png
https://media.mathstodon.xyz/media_attachments/files/114/611/337/972/695/159/original/7ab19d5a694e0278.png

2024-08-14 16:58:24 UTC

So... my son nerd sniped me! He saw a video on the Fold-and-Cut Theorem by Vsauce and said I should figure out how to fold a piece of paper to cut out the hat and spectre tiles with one cut. Mission accomplished! This blog post has downloadable PDFs: https://divisbyzero.com/2024/08/14/fold-and-cut-hat-and-spectre-tiles/
https://media.mathstodon.xyz/media_attachments/files/112/961/425/876/231/661/original/42b8369a7377a43a.jpeg
https://media.mathstodon.xyz/media_attachments/files/112/961/426/087/948/937/original/b6c898c5f8a4d59b.jpeg
https://media.mathstodon.xyz/media_attachments/files/112/961/428/833/691/075/original/acfffb61383d9717.mp4
https://media.mathstodon.xyz/media_attachments/files/112/961/431/895/041/000/original/b70800586cc4d3d4.mp4