Sierpinski triangles on an icosahedron, inspired by Ghee Beom Kim #sierpinskitriangle #icosahedron #geometricart #fractal #fractalart #iteratedfunctionsystem #3dgraphics #raymarching #pythoncode #numpy #opengl #algorithmicart #algorist #mathart #laskutaide #ittaide #kuavataide #iterati

Another blow-up view of the Apollonian spheres, now dropping on a concave surface to gather all the jetsam together. This is what I had in mind when writing the first drop demo, and the model just needed a bit of refinement for the differently sized balls.

New demo just dropped! Testing a basic elastic model with blocks of FCC, BCC and diamond lattice. #particlesimulation #elasticcollision #hookeslaw #bodycenteredcubic #facecenteredcubic #diamondlattice #lattice #crystalstructure #3dgraphics #pythoncode #numpy #opengl

Deforming Apollonian spheres, again using some interesting math that Bluesky doesn't want me to explain in this limited space.

Making Apollonian gaskets usually follows a key rule of iterated function systems: each iteration should make the thing smaller. With inversions, this means going from the outside to the inside of inverting circles. However, it's possible to make valid gaskets using a lopsided configuration, where

Another look at Apollonian spheres, cutting out the top half and showing a few iteration steps. #apollonianspheres #apolloniangasket #iteratedfunctionsystem #inversion #sphereinversion

Apollonian spheres

New implementation of the Apollonian Gasket iteration

2D circle inversion fractals on the spherical surface. This was a fun offshoot of my recent Apollonian endeavours, again using the Riemann sphere mapping to go from 3D to 2D for the iterations.

Apollonian gaskets via circle inversion. For more explanation, you should see my posts on Instagram, Youtube, Facebook or Mathstodon where the posting limits are more sensible.

Different views of iterating the Apollonian gasket

Is it that time of the year yet? #pumpkincarving #sierpinskitriangle

Truncated octahedron Apollonian gasket

Another style for regular 2D Apollonian gaskets #apolloniancircles #apolloniangasket #relief #emblem #geometricart #fractal #fractalart #raymarching #pythoncode #opengl #algorithmicart #algorist #mathart #laskutaide #ittaide #kuavataide #iterati

Apollonian gaskets based on Platonic solids. The vertices of a tetrahedron/octahedron/icosahedron are used for the centre positions of the initial circles.

Riemann sphere mappings of Apollonian gaskets. Since my 2D gaskets are fitted to the unit circle, they cover exactly half of the Riemann sphere, as seen in the first part. My lazy solution to covering the entire sphere was a simple copy-paste, and the result doesn't seem too bad.

Another look at the Apollonian gasket. This shows that the process doesn't use simple inversion, since that would distort the image within each disc. #apolloniancircles #apolloniangasket #geometricart #fractal #fractalart #biblicallyaccurateangels #weirdart #creepyart #horrorart

Back to basics: I made an Apollonian gasket generator based on Descartes' theorem, as explained on Wikipedia. The method involves some nontrivial details on the choice of +/- signs, which I had to figure out on my own. #gasketcase

Testing the Gosper curve in my variable iteration halftoning setup. So far I've only used the Hilbert curve this way, and things get a bit coarser with the Gosper, so it was harder to find images that make nice results. So here we are with the old Venus again. #halftoneart #gospercurve

Riemersma-like dithering using a Gosper curve and a boustrophedon on a hex lattice

Riemersma-like dithering with various space-filling curves

Yet another optimized Voronoi tiling. Instead of the equal-sum partitions, this uses regular dithering to find the Voronoi seed points, and the tile areas are inversely proportional to the colour values.

I wanted to make the optimized tiling look more organic, and I've already played with Voronoi diagrams quite a lot. But this turned out a bit more messy than expected. As before, there's a roughly uniform tiling with an equal number of cells for comparison.

Another visual refinement to a recent demo: adding seams to the optimized tiling scheme. While it breaks the mosaic illusion to some extent, I think it makes the tile sizing idea much clearer. As before, there's a uniform variant with the same number of tiles for comparison.

How it was supposed to look like in the first place: rounded with Bezier curves. #beziercurve #hilbertcurve #planefillingcurve #spacefillingcurve #lineart #linedrawing #singlelinedrawing #pythoncode #opengl #geometryshader #algorithmicart #algorist #laskutaide #ittaide #kuavataide #iterati

The recent stipplings and partitionings reminded me of this demo from about 2 years ago. In short, it's a Hilbert curve where the iteration level varies by the colour value. I wanted to make some small changes, but I ended up rewriting it completely.

Random partitionings with colour mapped from uniform partitioning (basic pixelation) #splittingimage #stablegenius

Optimizing pixel sizes for better detail, compared to uniform pixelation. Both pictures have 4096 rectangular tiles. #pixelart #blockart #tiling #pythoncode #opengl #algorithmicart #algorist #mathart #laskutaide #ittaide #kuavataide #iterati

New idea for stippling images using the 1D partitioning logic from last post, alternating in x and y dimensions. I also made the minimum spanning tree, and it turned out quite interesting with the horizontal and vertical lines dominating large areas.

I came across this halftoning idea sometime last week. While the idea was relatively easy to understand and fun to implement, I've spent quite a lot of time trying to make the result look nice. I'm also including a fun glitch from the early tests. #halftoneart #lineart #pythoncode #opengl

Next in line for my polyhedral camera loop demos is a truncated cuboctahedron. I've been skipping a lot of otherwise nice polyhedra, because triangular faces just don't cut it — well actually they cut too much of the square camera view.

I've always found circuit boards aesthetically inspiring, and in recent years this fascination has turned into a number of PCB-styled demos. I think a key inspiration for these has been the "Absolut Intelligence" vodka advert a couple of decades ago.

Another attempt to make the floor tiling fractals more interesting: Pythagorean tiling with a varying size ratio. Also testing a new colouring scheme. #pythagoreantiling #fractaltiling #tiling #fractalart #iteratedfunctionsystem #cameraloop #pythoncode #opengl #algorithmicart #algorist #mathart

Minimum spanning tree of a dithered image, 4096 vertices. Model: Swetlana #spanningtreeyoungandsweet #minimumspanningtree #primsalgorithm #graphtheory #halftoneart #lineart #pythoncode #opengl #algorithmicart #algorist #mathart #laskutaide #computerart #ittaide #kuavataide #iterati

I've finally managed to put my entire "geodesic series" of himmelis side by side, thanks @noira.bsky.social for the suggestion. The edge counts are 6, 12, 30, 36, 42, 48, 84, 90, 120, and 210. #himmeli #puzuri #strawart #geodesicseries #geodesichimmeli #geodesicpolyhedron #polyhedron

The "floor tiling" variant feels kind of dull compared to my other recent camera loop demos. Here I've switched from regular square tiling to the alternating "brickwork" style, but it doesn't get any more exciting. So I thought I'd embrace the dullness and return to monochrome.

I've been asked for more of the tiling camera loop demos — nice! Moving away from the regular Platonic solids, this one uses the truncated octahedron. I think there's a nice balance between the square and hexagonal faces, it doesn't go too far from the regular dodecahedron.

Spanning trees on a geodesic polyhedron. Another example of some rather basic theory that I hadn't yet done, probably because it didn't feel that exciting. But the algorithm is now done and it might turn out handy somewhere. #geodesicpolyhedron #graphtheory #spanningtree #pythoncode #opengl

I got this crazy idea that tiles might belong on a regular plane instead of a polyhedron. So here's a flat variant of my tiling camera loop fractal demo — it even uses the floor() function. #laattalentää #fractaltiling #cameraloop #pythoncode #opengl #algorithmicart #algorist #mathart #laskutaide

Testing a colouring idea for my dodecahedron camera loop. Plain colour filters are not great for iterated work, as they would just lose all light after a few steps. Instead, I use hue shifts, with a different constant shift per each face polygon.

I've been messing around with a new camera loop idea: put the camera inside a polyhedron, and feed the camera view back onto each face polygon. This makes some kind of a fractal tiling for each camera position and set of face orientations. This first example uses a dodecahedron.

I've been thinking of turning my "manual algorithm" for Eulerian himmelis into code, and here's my first working attempt, using the square-augmented snub cube. It's not quite how I'd do it in real life, but it's not very far either.

No new himmelis here, but I thought it might be interesting to compare some of them side by side. In all of these himmelis, the long edges are 4.5 cm. The short edges are not much shorter, and they are in the minority anyway. The total edge counts are 30, 90, 120 and 210.

One more simple geodesic himmeli: a snub cube with the square faces augmented into pyramids. 30 vertices, 84 edges and Eulerian.

Continuing on my geodesic himmeli series, here are two specimens using octahedral/cubic symmetry, which I haven't used much. The vertex/edge/face counts are 14/36/24 and 18/48/32. #himmeli #puzuri #strawart #geodesichimmeli #geodesicsphere #geodesicpolyhedron #tetrakishexahedron

As a by-product of a failed idea, here's another simple geodesic himmeli: a truncated tetrahedron whose hexagonal faces are augmented into pyramids. With 16 vertices, 42 edges and 28 faces, it goes between the icosahedron and the pentakis dodecahedron in my geodesic series.

Another step in my series of geodesic Himmelis: a snub dodecahedron with the pentagons augmented into pyramids. Like my previous geodesics, this too can be made with just 2 different edge lengths. It is also known as {3,5+}_2,1 and has a total of 210 edges.

{3,5+}_2,0 geodesic polyhedron himmeli