Craig S. Kaplan

Mathematics, Computer Science, Art, Design, Music, Soup, Pedantry. he/him/anything

Academic: https://cs.uwaterloo.ca/~csk/
Personal: https://isohedral.ca/
Bridges: https://www.bridgesmathart.org/

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csk_at_mathstodon.xyz@momostr.pink

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Last Notes

2026-09-18 13:21:12 UTC

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Here are some other preprints by the same author, all posted online since June:
The Cosmological Constant as a Contingent Modulus: A Six Birds Demarcation of Dark Energy
Riemann Hypothesis via Self-Dual Trace Confinement: A Conditional Closure
P ≠ NP under Closure of the Saturated SAT Layer
Navier-Stokes as a Layer-Dissolving Membrane Theorem
A Conditional Proof of the Strong Birch-Swinnerton-Dyer Conjecture
My, he's been busy!
But for all that, the construction of a 3D aperiodic monotile in the paper appears to be valid. @npub1cqn…cqpx is working on a slop-free write-up. Should we be excited? Sigh.

2026-09-12 21:57:38 UTC

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I assume you know this, but just in case, there's an easy way to see that the regular tetrahedron emerges from these coordinates (without jumping to the 24-cell!). Two opposite-facing regular tetrahedra can be inscribed in a cube, each one touching four of the cube's vertices, with the edges running along the cube's face diagonals. This compound of two tets is what Kepler called the "Stella Octangula".
https://media.mathstodon.xyz/media_attachments/files/117/260/299/544/730/276/original/52a9ebea8ad37b1b.png

2026-09-09 13:08:08 UTC

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Presumably the confusion arises because Ireland isn’t part of Schengen, and that’s a key distinction when moving around Europe. Still annoying, though.

2026-09-09 00:43:03 UTC

Everyone's droning on about the ability of AI to solve millennium prize problems in mathematics, etc., etc. But not enough people are talking about using AI to draw Islamic Geometric Patterns.
I created the first version of Taprats as a class project around 1998/1999. I wrote it in Java, which means that it's almost impossible for anybody to use it these days (and it's quite janky in any case). But David Bachmann recently fed my papers into ChatGPT and came out with a completely serviceable web-based app for drawing patterns. Might be fun to play around with: https://davbachman.github.io/Topkapi/.

2026-05-08 14:37:14 UTC

Recently, Sébastien Labbé and Peter Selinger posted a preprint (https://arxiv.org/pdf/2604.20964) that describes a sophisticated construction for the hat tiling, based on what's called a Markov partition (see the thread by @npub1l3c…nw2k at https://mathstodon.xyz/@pieter/116484517078239617 for more details).
One nice side effect of their construction is that it provides a means of rendering the hat tiling on the GPU using a fragment shader. That's cool, because it requires a constant amount of work per pixel, regardless of how many tiles you're drawing, and you can pan around forever without ever worrying about running out of information about the positions of tiles (as you would with a drawing algorithm based on substitution).
I have a slightly glitchy prototype written up in Shadertoy. I'll make the code available, but I want to clean it up first.
https://media.mathstodon.xyz/media_attachments/files/116/539/386/444/825/187/original/546ccaae373f6e76.mp4

2026-04-12 15:37:25 UTC

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Obviously that deserves an explanation.
In my early years of teaching our computer graphics course, we would often make an end-of-term t-shirt featuring student projects. We'd include an image of the prof somewhere. This was Spring 2003; I appear in Neo drag, doing a Bullet Time maneuver (Matrix Reloaded had just come out).
In the early 2020s we were cleaning up the lab and got rid of extras of old t-shirts we still had lying around. We left them in a hallway somewhere. A random first-year engineering undergrad (who was born in 2003, ha) grabbed one. By chance, he wore it to go rock climbing yesterday. I'd like to thank him for making life a little more surreal.
[edit: dug the t-shirt image out of the memory hole]
https://media.mathstodon.xyz/media_attachments/files/116/392/487/046/222/380/original/9b08bc2301c41211.png

2026-04-12 15:31:27 UTC

Yesterday, through a somewhat unlikely chain of events, I met a complete stranger who was wearing a t-shirt with a picture of me on it.

2026-03-03 02:07:05 UTC

In my online undergraduate P5.js course, students are about to begin the module on motion and physics, including a bit of physics simulation using Matter.js. It suddenly occurred to me that I had never seen anybody put together this particular demo before, and I realized it had to be done. Messy source code at https://editor.p5js.org/isohedral/full/vJa5RiZWs.
https://media.mathstodon.xyz/media_attachments/files/116/162/786/689/835/359/original/62d579f72469c1d2.mp4

2026-01-18 03:08:13 UTC

January 17: Wallpaper group
Morph continuously between isohedral tilings representing 16 of the 17 wallpaper groups (the only one missing is p3m1, which cannot be represented isohedrally using unmarked tiles).
This one took some doing. There might have been a way to finesse it using my tactile-js library, but in the end it was probably easier to throw it together from scratch. The code could definitely be cleaned up a lot. Source code at https://editor.p5js.org/isohedral/full/S1H3U1b7E #genuary #genuary2026
https://media.mathstodon.xyz/media_attachments/files/115/913/887/391/572/800/original/49b05d6eb4ebec83.mp4

2026-01-09 15:18:42 UTC

January 9: Crazy automaton
Build a big fullerene (the dual of an order-6 icosahedral geodesic sphere, I think). Paint its pentagons black and its other faces white (like a soccer ball). Use a simple rule to evolve face colours: re-colour a face black if the subset of the face and its neighbours that are currently black has odd size, white otherwise. The result is a symmetrically evolving series of soccer ball patterns (that appears to repeat after 31 generations). Source code at https://editor.p5js.org/isohedral/full/TN15VTag-. #genuary #genuary2026
https://media.mathstodon.xyz/media_attachments/files/115/865/796/058/422/975/original/04dc7daf6aafc6d8.mp4

2025-10-12 20:38:56 UTC

The delusion that we can advance AI to the point where it rescues us from the mess we’ve created for ourselves is pure millenarianism. The CEOs of the big AI companies are a cult who believe that if they pray hard enough, AI Jesus will come down and grant us infinite abundance and prosperity.

2025-08-24 02:30:40 UTC

Over the summer I thought a bit about a few 2D geometric problems that I hadn't seen before. Here's the first one.
Let 𝑃 and 𝑄 be 2D points and let 𝑛 be an integer ≥ 3.
Let 𝑆 be a shape (say, a topological disk) with 𝑛-fold rotational symmetry, and place two non-overlapping copies of 𝑆 in the plane, one centred on 𝑃 and the other centred on 𝑄. The copies may be rotated and/or reflected differently from each other.
What shape 𝑆 should I choose, and how should I orient the two copies, so as to maximize the area of 𝑆?
I'm assuming the answer is two congruent regular 𝑛-gons, each with an edge on the perpendicular bisector of 𝑃𝑄. But I'm not sure I have a convincing proof of that fact.

2025-08-13 15:04:09 UTC

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I just read it yesterday. The ideas are lovely, though unfortunately I still struggle mightily with the discrete exterior calculus and discrete differential geometry that powers this work (and so much other beautiful contemporary research in computer graphics).

2025-07-26 14:43:16 UTC

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Dave Swart wrote a nice survey of symmetries of soccer balls: https://archive.bridgesmathart.org/2015/bridges2015-151.pdf. Mind you, a lot of the designs in that paper acquire their exotic symmetries through graphics printed on a standard 32-panel ball.
If you want balls on the far more experimental side, check out the beautiful work of Jon-Paul Wheatley, e.g., https://12p.com/pages/lab, https://www.instagram.com/jonpaulsballs. Of course, there's one design of his that I'm particularly attached to: https://12p.com/pages/the-hat-trick.

2025-07-20 16:31:56 UTC

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OK, sure, not actually a glitch, but from a typical zoom level it looks like there are some weird black pixels in the middle of nowhere. I had to zoom in to see what was clearly going to be a place with an interesting history.
https://media.mathstodon.xyz/media_attachments/files/114/886/510/137/788/705/original/3779bcb68743b74d.png

2025-07-20 07:44:56 UTC

In planning my travels around The Netherlands, I noticed a funny glitch on Google Maps near the souther border. It turns out that the town of Baarle-Nassau has 22 "holes" in it, self-contained enclaves belonging to Belgium (in which the town is called Baarle-Hertog). And more deliciously, a couple of them contain "counter-enclaves" that are Dutch again! I'm sure many people have seen this before, but it was new to me. https://en.wikipedia.org/wiki/Baarle-Nassau

2025-06-16 18:00:32 UTC

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Ignoring the fact that promises of AGI are empty marketing hype, it requires a profound failure of the imagination to envision a magical technology of infinite abundance and then assume that its only use is as a means of control.

2025-05-23 11:17:31 UTC

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I ended up singing Oasis to myself for a lot of yesterday (this song, but also Don’t Look Back in Anger) which was not altogether unpleasant. I’ll look forward to the next time you post about a Supermassive Black Hole.

2025-04-29 13:29:02 UTC

As a Canadian, I want to express my thanks to the US president for his tireless efforts to personally shift the tide of federal politics in my country. He truly is a crusader for social justice.
https://media.mathstodon.xyz/media_attachments/files/114/421/466/616/480/284/original/d64777c4292c6aa9.png

2025-02-28 13:10:04 UTC

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This reminds me of a rather nice piece of installation art from 2013, in which (from a mathematical viewpoint) a film sequence is projected onto the subspace spanned by a set of frames from another film. It’s titled “Theorem 8”, presumably based on whatever textbook the artist used as a reference. http://theorem8.concept-script.com/

2025-01-20 20:09:27 UTC

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Can't resist adding another one, if only to use All The Hashtags. #GiantOrangeBaby #Cats #CatsOMastodon
https://media.mathstodon.xyz/media_attachments/files/113/862/485/550/561/070/original/9ce3ae5b12f641ea.jpg

2025-01-20 16:11:20 UTC

On this day, I encourage you to post a photo of your Giant Orange Baby if you have one around the house, to take our minds off of any other giant orange babies who might be mentioned in the news. Here's mine. His name is Otis. #GiantOrangeBaby
https://media.mathstodon.xyz/media_attachments/files/113/861/541/817/490/076/original/888540ea46b1eebe.jpg