New #MathArt sculptures: reaction-diffusion Truchet tilings of the sphere!
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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... 🧵
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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... 🧵
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.
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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.
"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.
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"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.
RE: https://mastodon.social/@matt_zucker/114389429207956266
(BTW if these orbs look familiar, maybe you saw the thread I posted about a year ago about similar work I did.)
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RE: https://mastodon.social/@matt_zucker/114389429207956266
(BTW if these orbs look familiar, maybe you saw the thread I posted about a year ago about similar work I did.)
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 https://www.karlsims.com/rd.html and/or https://mrob.com/pub/comp/xmorphia/index.html.
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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 https://www.karlsims.com/rd.html and/or 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.
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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.
Making RD patterns that tile like wallpaper is easy – apply periodic boundary conditions (PBCs, https://en.wikipedia.org/wiki/Periodic_boundary_conditions) 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.
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Making RD patterns that tile like wallpaper is easy – apply periodic boundary conditions (PBCs, https://en.wikipedia.org/wiki/Periodic_boundary_conditions) 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.
More interesting RD tilings are possible. Recently, Vladimir Bulatov has been making some really cool patterns with his SymSim software – check it out at https://symmhub.github.io/SymmHub/apps/symsim/gray_scott/index.html.
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More interesting RD tilings are possible. Recently, Vladimir Bulatov has been making some really cool patterns with his SymSim software – check it out at https://symmhub.github.io/SymmHub/apps/symsim/gray_scott/index.html.
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.
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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.
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).
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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).
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".
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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".
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.
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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.
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.)
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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.)
I had to write a homemade finite element method (FEM, https://en.wikipedia.org/wiki/Finite_element_method) 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.
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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... 🧵
@matt_zucker I am in awe!
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I had to write a homemade finite element method (FEM, https://en.wikipedia.org/wiki/Finite_element_method) 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.
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.
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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... 🧵
@matt_zucker cool
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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.
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: https://www.bridgesmathart.org/b2026/
Come check them out if you're in the neighborhood!
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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: https://www.bridgesmathart.org/b2026/
Come check them out if you're in the neighborhood!
If you liked this thread, check out my very infrequently-updated coding blog at https://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.
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If you liked this thread, check out my very infrequently-updated coding blog at https://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.
@matt_zucker
So to test my understanding, in order to make them tileable you have to define - for one tile - a surface where all three or four edges loop back on each other (torus geometry for four edges). And for multiple patterns you need multiple surfaces that still all share the same single edge.Now, it should matter which way they're flipped. Is there more than one way to arrange the flips of the common edges? Or there is only the one?
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