a weird looking dude drawn in MS paint whose eyes have been replaced by "N / N"

N / N

A mysterious purple orb with a coral-like relief texture. Its exterior is subdivided into alternating square and triangular tiles.

A red coral-like orb made of triangular tiles.

A green coral-like orb made of hexagonal and square tiles.

A blue coral-like orb with pentagonal and triangular tiles.

New #MathArt sculptures: reaction-diffusion Truchet tilings of the sphere! These are made from 3D-printed nylon and each individual tile can be removed/replaced.

If you think these look cool or interesting, stick around to learn a little bit about how they were made and the math behind them... 🧵

If you liked this thread, check out my very infrequently-updated coding blog at mzucker.github.io for other #MathArt projects and miscellaneous tinkering.

And feel free to reply with questions – I'll do my best to answer them here.

A gorgeous photo of cliffs along the Irish coastline near Galway with a gray sea, slightly overcast sky, and pink flowers in the foreground. Photo by Chaosheng Zhang

The wooden tiles were exhibited previously at the 2026 JMM art exhibition, and I just learned that the spherical tilings and red squares were accepted for the art exhibition at Bridges Galway 2026 from August 5-8: www.bridgesmathart.org/b2026/

Come check them out if you're in the neighborhood!

A figure from an academic paper about reaction-diffusion Truchet tiles. The caption reads: Figure 7: Screen captures from my Python tiling software: (a) triangle meshing of spherical polygons; (b) initial random seed for reaction-diffusion growth; (c) final texture after growth complete; (d) displacement-mapped sphere; (e) cutaway view of tile CAD models; (f) texture with boundary/vertex clumping artifacts; (g) texture with non-perpendicular gradients at boundaries; (h) alpha mask and (i) symmetric textures used for boundary copy operation.

And after a few extra hacks to mitigate some unpleasant visual artifacts, it all Just Works™.

Of course I wrote an academic paper with all of the gory details – look for it in the upcoming Bridges 2026 conference proceedings. Here's a figure that gives a taste of what the software does.

Finite element method - Wikipedia

I had to write a homemade finite element method (FEM) solver to simulate the Gray-Scott model on this weirdo domain.

FEM solvers tend to throw up their hands and quit when you point them at non-manifold geometry like the folded and glued square, but I hollered at mine until it started behaving.

The same shiny red plastic square reaction-diffusion tiles from upthread.

The same tiles as the previous image, but now rearranged into a pattern with 90° rotational symmetry. Close inspection reveals that there are just two unique tile designs.

You can definitely play with symmetry when laying out the tiles. Here's those same red plastic tiles from before, and again after re-arranging to make a composition with 90° rotational symmetry.

How many unique tile designs do you see here? (Hint, it's the same in both images.)

The same CNC-machined wooden tiles from upthread

A wavy coral-like orb made from 20 copies of the same triangular tile.

A wavy coral-like orb made from 20 unique triangular tiles.

Want more than one tile design? NP, just glue together multiple sheets along their shared edge.

If you look closely at the wooden tiles you'll see there are three distinct tile designs. And here are two different icosahedron tilings, one with a single tile design, and one with every tile unique.

Now all four corners of the square are mathematically considered to be the same point, as are any set of points along a boundary edge that are all the same distance away from the corner.

BTW when solving the PDEs, we don't consider the effects of the folds, just the "glue".

A figure illustrating the paper folding and gluing operation described in this post, kinda resembles origami instructions except I didn't use the correct line style conventions for mountain and valley folds and also I'm pretty sure there is no glue allowed in origami.

Here's what it looks like, starting with a square sheet (a). First, fold it into quarters (b), then fold diagonally to put all of the boundary edges on top of each other (c). Finally, glue all of the edges together (d).

But until I started this project, I don't think anyone was daft enough to make reaction-diffusion Truchet tiles that match up no matter how you reorient them.

The central trick is to solve the Gray-Scott PDEs on an unusual topological domain that collapses all of the edges into a single segment.

Wavy maze patterns that form repeating triangular tiles in the plane. Image produced by SymSim.

Swirly spiral patterns that form repeating rectangular tiles in the plane. Image produced by SymSim.

More interesting RD tilings are possible. Recently, Vladimir Bulatov has been making some really cool patterns with his SymSim software – check it out at symmhub.github.io/SymmHub/apps....

Screenshot of old-school Pac-Man arcade game

Making RD patterns that tile like wallpaper is easy – apply periodic boundary conditions (PBCs, en.wikipedia.org/wiki/Periodi...) when solving the PDEs.

PBCs are like the old-school Pac-Man arcade game: crossing the left side of the "screen" warps you to the right side, and ditto for top/bottom.

The mathematical formula for the Gray-Scott reaction-diffusion model, a system of PDEs. Image from https://mrob.com/pub/comp/xmorphia/index.html

Making reaction-diffusion patterns amounts to solving a system of partial differential equations (PDEs) – the Gray-Scott model – that describe the simulated chemical reaction. The formula might look like Greek, but it's not too exotic from a math/coding standpoint.

Reaction-Diffusion Tutorial Illustrated tutorial of the Gray-Scott Reaction-Diffusion model.

OK, so what's "reaction-diffusion" then? In brief, it's a mathematical model of a chemical reaction that produces patterns similar to many found in nature. For more info, see www.karlsims.com/rd.html and/or mrob.com/pub/comp/xmo....

(BTW if these orbs look familiar, maybe you saw the thread I posted about a year ago about similar work I did.)

That same blue orb from the previous post

Same blue orb again, but showing a star-shaped gap where several tiles have been removed. You can see a black lattice underneath with magnets used to register the tiles to particular locations on the sphere.

"Truchet tiling" means that that each tile can be rotated in place or swapped with another of the same shape without breaking up the pattern.

To get this you need the tiles' edges to all be identical and mirror-symmetric. This is not obvious at first glance when looking at the overall tiling, IMO.

Square planar reaction-diffusion Truchet tiles in walnut and maple. They have wavy maze textures but all meet up nicely at their edges.

Square planar Truchet tiles of 3D-printed nylon. They have a industrial-looking glossy red finish, with similar coral-like textures to the objects in previous images.

Oh, and here are some planar variants I made, too. The left one is CNC-machined walnut and maple, because it seemed like a really cool idea before I realized exactly how much hand-sanding would be involved (too much, it was too much sanding). The right one is 3D-printed nylon.

A mysterious purple orb with a coral-like relief texture. Its exterior is subdivided into alternating square and triangular tiles.

A red coral-like orb made of triangular tiles.

A green coral-like orb made of hexagonal and square tiles.

A blue coral-like orb with pentagonal and triangular tiles.

New #MathArt sculptures: reaction-diffusion Truchet tilings of the sphere! These are made from 3D-printed nylon and each individual tile can be removed/replaced.

If you think these look cool or interesting, stick around to learn a little bit about how they were made and the math behind them... 🧵

My all-undergraduate college employer has offered (but does not require) ALICE training for faculty & staff

Ooooh

Thread derail: what wuxia?

My cooking style after the edible hits:

this is actually brilliant

Exactly this yes

you SLAM the tim tam? you dunk it into hot beverage like the basketball? oh! oh! jail for mother! jail for mother for One Thousand Years!!!!

Sweet - Fox On The Run - 45 (OFFICIAL) YouTube video by Official Sweet Channel

www.youtube.com/watch?v=kRv7...

good ol’ streets department, never stop sucking

Ok but what if you got telefragged at Waffle House, that would really suck