Craig S. Kaplan

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

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Last Notes

2026-06-23 22:32:42 GMT

Following closely on the heels of this year's papers, here's the online gallery for the Bridges 2026 Exhibition of Mathematical Art, Craft, and Design! https://gallery.bridgesmathart.org/exhibitions/bridges-2026-exhibition-of-mathematical-art

2026-06-21 00:10:46 GMT

The Bridges Archive has been updated with 2026 papers: https://archive.bridgesmathart.org/2026. Stop by and enjoy a trove of math-art delights!

2026-05-10 15:25:20 GMT

As promised, here's the Shadertoy page for a GPU shader that draws the hat tiling: https://www.shadertoy.com/view/s3lGD7. Hopefully the code makes sense (I've tried to organize it to be easy to work with), and people will find fun ways to play with it! :monotile:

2026-03-23 01:25:57 GMT

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@nprofile…yxlh The video won’t play for me, but that looks like Arthur Ganson’s “Machine with 27 scraps of paper”, which has long been a favourite of mine.

2026-03-03 02:07:05 GMT

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

2026-02-22 21:23:46 GMT

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@nprofile…yxlh I think there's certainly a place for it! The hard part would be to deliver enough of a GA payload in a tight space to convince the reader of the value of the techniques, without trying to unpack all the math. And of course, your paper should make a concrete connection to real artistic examples, not just imply the possibility of art. (FWIW Charlie Gunn has come to the conference a few times, so there's at least some GA precedent there.)

2026-02-03 03:20:40 GMT

https://media.mathstodon.xyz/media_attachments/files/116/004/535/445/802/361/original/9a7f78a0b5ee2ed0.mp4

2026-02-01 14:07:47 GMT

January 31: GLSL day. Create an artwork using only shaders.
I'm going to cheat a bit and share a work-in-progress image from a forthcoming paper by my PhD student Brian Do. This is a hyperbolic version of Kitaoka's Rotating Snakes illusion. The whole thing is rendered in real time in a browser-based app, using shaders. Pre-print and interactive tool coming soon. #genuary #genuary2026
https://media.mathstodon.xyz/media_attachments/files/115/995/746/399/759/443/original/3e81d7cae06e7c2d.png

2026-01-29 18:08:12 GMT

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@nprofile…yxlh Yes, the notation is hard to beat! The trouble tends to come in the implementation, where a naive representation of geometric objects tends to require an exponential number of coefficients. I assume there's some nice research to be done in compilation methods for GA that perform static type inference on the objects being processed (which are usually quite tame) and emit efficient code. I haven't looked too hard to see what's already been done in this context.

2026-01-29 15:02:29 GMT

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@nprofile…yxlh I previously watched your talk about GA. For my part, I'm still not sure how to exploit a possible connection to patterns and tilings. It's cool that you're making that connection.
Interestingly, my colleague in the computer graphics lab, Stephen Mann, does a lot of research on CGA and PGA, and co-authored a textbook on the subject. I should pick his brain about connections.

2026-01-29 13:30:01 GMT

January 28: No libraries, no canvas, only HTML elements.
Here's a patch of the Penrose P3 tiling rendered entirely as pure HTML+CSS. I used code to generate the HTML, but the end result is self-contained. Each tile is a <div>; the trick is (relatively) easy because rhombs are affinely transformed rectangles.
You can see the HTML at https://cs.uwaterloo.ca/~csk/tmp/penrose.html. #genuary #genuary2026
https://media.mathstodon.xyz/media_attachments/files/115/978/596/815/595/611/original/3a46731ab439613f.png

2026-01-26 15:27:24 GMT

January 26: Recursive Grids. Split the canvas into a grid of some kind and recurse on each cell again and again.
Construct a line through the middle of the canvas, whose orientation is taken from a fixed set of regularly spaced angles. That divides the canvas into two polygons. Recursively apply the same splitting process to those polygons, up to some maximum depth, then draw them. The result is something like an ice ray lattice. Source code at https://editor.p5js.org/isohedral/full/H1Rj2GwgZ. #genuary #genuary2026
https://media.mathstodon.xyz/media_attachments/files/115/962/092/711/997/278/original/ca94514380198c3a.png
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2026-01-26 03:24:29 GMT

January 25: Organic Geometry. Forms that look or act organic but are constructed entirely from geometric shapes.
As a fun riff on the prompt, I wrote a sketch that draws generative organic molecules. There were definitely some interesting subtleties involved here in getting things to look vaguely OK. There's still a lot of tweaking I could do (generative chemical names would be awesome), but it's bedtime.
Obviously these aren't even remotely plausible. In my defense, through a quirk of the educational systems of Québec and Ontario in the late 1980s, I never took an organic chemistry course, so I know very little about the subject, and must operate purely on vibes here.
Source code at https://editor.p5js.org/isohedral/full/VTlFRldEP. #genuary #genuary2026
https://media.mathstodon.xyz/media_attachments/files/115/959/252/008/611/881/original/1729ec371a5fe620.png
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2026-01-24 23:27:57 GMT

January 24: Perfectionist’s nightmare.
Draw a random tiling using tactile-js (based on a modified version of my Jan. 29 sketch from Genuary 2023). However, at startup time the sketch hijacks P5.js's path drawing functions so that paths are triangulated and drawn with faint hairlines, with outlines slightly offset, and with unmitered corners.
I suppose this can serve as atonement for my Jan. 20 sketch, in which allowed hairlines to slip through without fixing them. Source code at https://editor.p5js.org/isohedral/full/PvNK4kCKG. #genuary #genuary2026
https://media.mathstodon.xyz/media_attachments/files/115/952/646/828/714/644/original/1fbea704125670fd.png
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2026-01-22 16:35:48 GMT

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@nprofile…jydu Brian Wyvill wrote a fun paper about drawing pictures based on the Gosper curve: https://webhome.cs.uvic.ca/~blob/publications/flowpaint.pdf.

2026-01-22 01:52:17 GMT

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@nprofile…zl3j That's just about perfect, thanks.

2026-01-21 16:41:59 GMT

There's a standard algorithm everyone uses to draw regular hyperbolic tilings on the GPU: given a point in the Poincaré disk, repeatedly reflect it across the edges of a fundamental triangle of the tiling until it lies inside that triangle, and then sample a colour.
Is there a decent reference for that algorithm, i.e., a paper or even a blog post? As far as I know it's a "folk algorithm" that everyone just seems to know about.

2026-01-19 21:18:01 GMT

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@nprofile…tfwq It varies a lot. The 16x16 sketch probably took about an hour. The morphing tilings took more like half a day (and required looking things up in books, doodling ideas, etc.).

2026-01-19 15:02:45 GMT

January 19: 16x16.
It's known to be possible to assign one of four colours to each of the cells in a 16x16 grid in such a way that no axis-aligned rectangle of four grid cells has the same colour at all of its four corners. (It's even possible to do this for 17x17 and 18x18.)
Assign four colours at random to a 16x16 grid and then try (badly) to alter individual cells to reduce the number of monochromatic rectangles. Draw all the rectangles along with the coloured cells. The badness is good (?) here because the keeps the sketch changing all the time (though the sketch might be indistinguishable from one that just changes a few cell colours at random each frame...).
Source code at https://editor.p5js.org/isohedral/full/nwl8RIIrd. #genuary #genuary2026
https://media.mathstodon.xyz/media_attachments/files/115/922/359/674/550/743/original/14d36076e44c4eca.mp4

2026-01-18 19:59:13 GMT

January 18: Unexpected path. Draw a route that changes direction based on one very simple rule.
In this case, the rule is "move in the direction that you're pointing". Watch the user over the webcam and trace out a path by always moving in the direction they're pointing with their index finger. Uses ml5.js, of course. Source code at https://editor.p5js.org/isohedral/full/nPf5BnTIj. #genuary #genuary2026
https://media.mathstodon.xyz/media_attachments/files/115/917/858/547/866/813/original/17dfc07308eaf482.mp4

2026-01-18 13:27:24 GMT

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@nprofile…zl3j Three colours are sufficient (and achievable by adding a single line of code). The tilings are all variations on either hexagons or squares, and you can re-use the natural 3-colouring of the hexagon tiling on the squares without any problems.
These aren't "perfect colourings" (namely, a colouring in which every symmetry of the tiling acts as a permutation of the set of colours). Figuring that out would be more complicated. With tactile-js, I could ask the library to assign colours to tiles and then just interpolate between colour assignments.
https://media.mathstodon.xyz/media_attachments/files/115/916/309/477/995/626/original/1b85cf22883d3140.mp4

2026-01-18 03:08:13 GMT

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

2026-01-15 15:52:30 GMT

January 14: Everything fits perfectly.
I had a few different ideas for this prompt but ran out of time to execute them (in fact, I'm a day late). So here, have some looping hexagons. Source code at https://editor.p5js.org/isohedral/full/yMgYSuWm2. #genuary #genuary2026
https://media.mathstodon.xyz/media_attachments/files/115/899/909/932/692/926/original/40b6218ce94f372b.mp4

2026-01-10 23:22:51 GMT

January 10: Polar coordinates
Your basic spirograph, but with two interesting experiments layered on.
First, draw the entire parametric curve using only two calls to sin() and two calls to cos() at the beginning—all subsequent points are computed incrementally using the angle sum formulas. Sweet, sweet trig identities! I've never seen this approach before, and it works remarkably well (too well: it doesn't visibly affect the output).
Second, optionally compute those four trig functions using polynomial approximations. This causes the curve to degrade into something a little looser and more interesting than a standard spirograph curve.
Source code at https://editor.p5js.org/isohedral/full/BGqlOsNAE. #genuary #genuary2026
https://media.mathstodon.xyz/media_attachments/files/115/873/349/701/565/257/original/068a4870f601ea98.png
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2026-01-09 15:18:42 GMT

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

2025-12-21 16:32:54 GMT

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@nprofile…l4xy Well, I mean, I'm the site owner, and I can tell you that I didn't configure this message and have never seen it before. Presumably it's baked into our provider's web server. I suppose I could... just follow their instructions, and send them the request ID...

2025-12-21 02:40:18 GMT

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@nprofile…l4xy Weird. it's working for me, and I've never seen that error before. Are you using anything unusual in terms of browser / OS / extensions?

2025-12-09 21:15:38 GMT

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@nprofile…stct Isn't part of the problem also that ML researchers don't really understand why these techniques work as well as they do? As a result, there aren't really good empirical evaluation methods that go beyond "poke it with a stick and see what happens".

2025-11-11 13:44:12 GMT

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@nprofile…u6hy I talk about a few variations of Kepler’s tiling in an old Bridges paper. https://archive.bridgesmathart.org/2006/bridges2006-465.pdf

2025-10-12 20:38:56 GMT

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 GMT

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 GMT

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@nprofile…yu7x @nprofile…svv0 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).

2025-07-29 14:15:04 GMT

Robert Ammann independently discovered Penrose tilings after pondering the dot patterns using to reproduce colour photographs in newspapers.
Marjorie Rice worked in a printing firm until she got married (and her husband later started a typesetting company). She went on to discover several families of tiling convex pentagons.
Dave Smith, discoverer of the hat, had a long career as a technician in a print shop.
Apparently, if you want to make significant contributions to tiling theory, the best preparation is to study printing and typesetting.

2025-07-28 22:24:06 GMT

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@nprofile…dcr8 @nprofile…749s It would be really interesting if the natural mode of the D12 is to roll around a fivefold axis, kind of wobbling between the two offset families of decagon vertices around the equator.

2025-07-28 14:11:04 GMT

My son informed me of a phenomenon that's well known among D&D players: a D12 (dodecahedron) rolls much farther than a D20 (icosahedron). Is there an obvious reason why this should be? I can imagine a couple of possibilities, but I wouldn't know how to verify them.
(I suppose I should tag @nprofile…dcr8, given his recent work on the physics of rolling dice.)

2025-07-26 14:43:16 GMT

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@nprofile…yu7x 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 GMT

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@nprofile…fa4k 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 GMT

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

2025-07-20 07:38:52 GMT

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@nprofile…saev Exactly so. I'm spending a couple of extra days in Oskar's region before heading home. Hopefully I'll visit the Escher Museum in The Hague later today.

2025-07-20 07:36:18 GMT

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@nprofile…saev Ha ha, I literally played with these puzzles at Oskar's house yesterday. When you get the correct motion, the way that the elements slide around each other is really cool.

2025-06-26 12:51:57 GMT

Font nerds: have you noticed that ChatGPT seems to have learned a "default font" when rendering images? A slightly rounded sans-serif, maybe a little narrow. It's immediately recognizable (which is nice, because it allows me to identify AI slop). It reminds me of Avería, an old project to generate an "average font": http://iotic.com/averia/.
(Image and alt-text from Lucy Rycroft-Smith https://bsky.app/profile/honeypisquared.bsky.social/post/3lsizpkfiuc27)
https://media.mathstodon.xyz/media_attachments/files/114/749/748/229/857/541/original/2cb6a9d3f3601654.png

2025-06-18 21:54:40 GMT

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@nprofile…saev Years ago I saw a similar error in an exhibit at the British Museum. I contacted them to point out the problem. They weren’t particularly concerned.

2025-06-16 18:00:32 GMT

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@nprofile…fa4k 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 GMT

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@nprofile…aet9 @nprofile…fa4k 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.

2025-03-29 15:52:45 GMT

There are plenty of practical examples of rendering two-dimensional hyperbolic tilings interactively using GPU programming. Does anyone know of an academic paper on the topic? Or something like a GPU Gem? I'd like to have a piece of writing I can give to students to read.

2025-02-28 20:43:13 GMT

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@nprofile…59k6 For reading / reviewing / annotating papers, I'm quite happy with a full-size e-ink tablet (https://www.techradar.com/reviews/onyx-boox-max-lumi-2-review). Lack of colour is occasionally an issue for me, but great for papers otherwise.

2025-02-28 13:10:04 GMT

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@nprofile…fa4k 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/

2025-01-15 13:40:57 GMT

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@nprofile…cvt4 I missed the local theatrical screening, but I’m signed up for the streaming event on the 24th. Looking forward to it! https://www.ohyouprettythings.com/new-products/eno24

2025-01-06 17:41:38 GMT

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@nprofile…lypn OMFG I have no words.

2024-12-05 19:18:03 GMT

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@nprofile…3lr2 Sigh. Similarly, I recently ran up against the Digital Services Act, which basically requires me to publish my home address and telephone number in order to sell mobile apps in the EU. I have the luxury of making things free or ignoring EU customers, but surely this will make life so much more complicated for independent developers who rely on the income.

2024-12-03 23:25:14 GMT

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@nprofile…tgw9 @nprofile…lypn Covenant Cafe is taking orders for batches of Christmas cookies. I haven’t tried them, but their other baked goods sure are delicious. I think The Grove is taking orders too.

2024-07-03 11:24:24 GMT

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@npub1knz…c73p Many congratulations! I look forward to seeing what you end up doing to promote public engagement.

2024-01-01 22:26:57 GMT

Let's kick off Genuary 2024!
January 1: "Particles, lots of them"
Each particle has one random friend and one random enemy. In each time step, a particle takes a step towards its friend, a slightly larger step away from its enemy, and a teeny step towards the centre of the screen. The result is nicely chaotic behaviour with no real physics. Source code at https://editor.p5js.org/isohedral/sketches/zSZ0-Omzx #genuary #genuary2024 #genuary1
https://media.mathstodon.xyz/media_attachments/files/111/683/027/053/844/798/original/16b6b02356843243.mp4

2023-08-02 15:23:47 GMT

It is an old adage that all mathematical art will eventually end up being worn on one's head. Here I am at Bridges 2023 wearing one of Laura Taalman's eminently huggable knitted knots as a hat (or maybe as a poorly conceived hairpiece). Thanks to Susan Gerofsky for the photo.
https://media.mathstodon.xyz/media_attachments/files/110/820/699/030/300/153/original/6bfbfbd3fdfd0ff7.png