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 the initial circles are bunched up on one side. In that case, the first iteration has to make a larger circle to fill the opposite side. This means an inversion from the inside to outside. But we can also think of this as turning the inversion circle inside out.

This turns out nice both visually and conceptually. An inversion circle is essentially a curved mirror, and we can make a smooth transition from the convex to the concave by passing through the flat stage. I wasn't sure if this would work cleanly in this simple demo, since the flat mirror means a circle with infinite radius; fortunately, the finite time steps mean we can skip over the flat point.

As for IFS rules, the system as a whole is contractive, thanks to the other circles that are now more convex.

The second part gives another look at such initially lopsided gaskets.

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As I keep studying the Apollonian gasket, I've now implemented the inversion approach on the CPU for finding the circle centres and radii. Now I can generate these arrays of eyes much faster, as the inversion is easier to parallelize. It's so fast that the bottleneck is now in the drawing stage.

The colours denote a kind of family tree of inversions: the 4 initial circles each have their own colour, and their inversion images retain the colour. The outer circle is not shown here, but its descendants show the colour that's distinct from the other 3.

I still needed something other than inversions for setting up the initial quartet, but I wanted find my own solution instead of relying on Descartes' theorem. The theorem actually comes in two parts: Rene's original theorem only deals with the radii, while the complex quadratic formula for finding the circle positions was only developed in the late 1990s.

Well, I found an alternative solution to the latter part, and it reduces to a pair of linear equations. It isn't particularly fast to compute, but I think it's easier to understand — it's basically junior high school math. In fact, it seems so basic that I can't be the first one to discover it.

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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.

The inversion circle centres come from a tetrakis hexahedron and a triakis icosahedron, so the circles form approximations of a truncated octahedron and a truncated dodecahedron.

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In the last post, I noted how the incremental iterates of the Apollonian gasket look like the output of an iterated function system. There's indeed such an IFS, and it's a system of circle inversions. It's how I've made a lot of fractal art over the years, but I've usually started directly with the inversion circles/spheres themselves.

Now that I've worked with the "classical" approach to the Apollonian gasket, I thought I'd translate a given Apollonian setup to the language of inversions. It was a fun little exercise and the math was surprisingly simple, just playing with vectors and solving linear equations. I then used my old inversion shaders from the late 2010s to show the results.

The first part shows it all together: the 3 largest coloured circles are the initial Apollonian circles, and the 4 inversion circles can be seen in the darkest grey in the background. (The initial Apollonian circles also include a 4th one, but here we can only see it as the perimeter of the coloured area.)

The second part uses a pointillist process, and it shows essentially the incremental iterates of the previous post. The inversion circles are not seen, but the Apollonian circles are all there as the empty space.

#apolloniancircles #apolloniangasket #iteratedfunctionsystem #inversion #circleinversion #geometricart #fractal #fractalart #pythoncode #opengl #algorithmicart #algorist #mathart #laskutaide #ittaide #kuavataide #iterati

The Apollonian gasket is a bit peculiar as an iterated system. The Nth stage of circles isn't generated solely by generation N-1, but all of the preceding generations. Alternatively, one might say that each circle also regenerates itself for the next level. Either way, it doesn't work like a typical IFS.

As I wonder how it all works, I'm showing you a couple of different views of the Apollonian iteration. The first part is just the regular progression. The second is the same, but only the newest generation of circles is shown. It looks a bit like a regular IFS as the iteration level increases.

The last two parts show a kind of graph view of the process, with the same structure seen through 2 different cameras. The balls and sticks are scaled in proportion to the circles they represent, and each child is connected to its parents. To avoid messing up the view completely, I've left out the outer circle that encompasses all the others, so a lot of circles show only 2 parents.

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The set of polyhedra that can be converted into Apollonian gaskets via sphere-plane mapping is quite limited. The face polygons should be regular, the edge midpoints should all lie on the same sphere, and the vertices should be 3-fold. There are some Platonic solids that work, and I've showed all of these earlier. It turns out that Archimedean solids with 3-fold vertices work too. So here's a truncated octahedron, also showing a progressive view of the gasket iteration.

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Another style for regular 2D Apollonian gaskets

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Apollonian gaskets based on Platonic solids. The vertices of a tetrahedron/octahedron/icosahedron are used for the centre positions of the initial circles. These are Riemann-sphere-mapped to the complex plane, where the gasket is iterated for more circles, and the result is mapped back onto the sphere. It's a little roundabout, but it works for me, and the heaviest part by far is drawing the visuals.

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Riemann sphere mappings of Apollonian gaskets. After the 2D gaskets, I'd been thinking of some kind of 3D versions for a while, but the final inspiration came from Antti Immonen's sculptures I saw on Friday at the opening of his exhibition here in Jyväskylä. As a real-life sculptor who incorporates fractals and other math ideas in his works, he's a pretty rare specimen at least by national standards.

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.

#apolloniancircles #apolloniangasket #riemannsphere #complexmath #geometricart #fractal #fractalart #pythoncode #opengl #algorithmicart #algorist #mathart #laskutaide #ittaide #kuavataide #iterati

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.

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