It's been quiet on the demo front as my musician side has recently taken over. This slow return to musicianship started last summer, with the new twist of using the Karelian language. I've also put some of my old and new demos to work for the music videos. Here's the latest piece by me and Noira, you'll find a couple more on the same Salixvelox channel:

https://youtu.be/XI9XJKDMd3k

#karelianlanguage #karelianproper #southkarelian #karjalankieli #karjalankielieläy #varzinkarjala #suvikarjala #opengl #pythoncode #algorithmicart #algorist #creativecoding #artxcode #computerart #ittaide #kuavataide #iterati

From a pile of shit, new life grows. Conway's Game of Life running independently in the 3 colour channels in 64x64 cells with wrap-around topology.

Not much new code here, just added image loading capability to my old pile-up demo of 2D cellular automata.

#cellularautomaton #gameoflife #conwaysgameoflife #pixelart #voxelart #blockart #3dgraphics #opengl #pythoncode #numpy #algorithmicart #algorist #creativecoding #artxcode #computerart #ittaide #kuavataide #iterati

Rule/Yule 30 cellular automaton with a "band-pass" filter every few iterations. Done in the Bash shell using bitwise math, so each row/state is a single number of 63 bits.

#cellularautomata #cellularautomaton #rule30 #bitwiseoperators #textmode #textmodeart #oldskool #retrocomputing #unixshell #shellprogramming #xterm #joulu6 #xmastree #algorithmicart #algorist #mathart #laskutaide #ittaide #kuavataide #iterati

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: properly scaled masses and elastic factors, as well as proper handling of these quantities in each pair collision.

As a recovering science teacher, it's fun to see such physics in action: a simple, linear elastic force is all it takes to keep each body in its place. Well, at least approximately; I've included a basic drag term to help things settle, but it seems it would take a while, as the tiniest balls are easily thrown around by the larger masses.

#apollonianspheres #apolloniangasket #particlesimulation #elasticcollision #hookeslaw #3dgraphics #pythoncode #numpy #opengl #algorithmicart #algorist #mathart #laskutaide #ittaide #kuavataide #iterati

With 2D Apollonian gaskets, it's easy to build arbitrary initial configurations. Simply picking 3 random points means you have to solve for 3 radii to make a kissing setup. Since there are exactly 3 distances between the points, this makes a basic linear system. But not so in 3D: you have 4 points and 4 radii, but 6 different distances, so a linear solution won't cut it. You could start with 3 kissing spheres using the 2D logic, but then you can't put the 4th point just anywhere.

I didn't bother with the messy quadratic system, because there's an easier way: take the symmetric tetrahedral config and deform it using an inversion. Yep, the same tool that's already the bread and butter of gasket-weaving. What's more, we can build the symmetric gasket first and then deform the whole thing. Inversion preserves spheres as spheres and maintains their kissing relations, it doesn't care how many there are.

In other words, the order doesn't matter with inversions. I've used this trick years ago in some 2D inversion demos to simplify things, and this 3D also benefits hugely from it. Besides the problem of initial config, 3D gaskets also have a speed issue due to deduplication (explained in an earlier post). The inversions are very fast as they can be parallelized, and this also applies to the deformations. So it's nice that we need not rebuild the gasket again for every config, we can just deform the same thing again.

#apollonianspheres #apolloniangasket #gasketweaving #iteratedfunctionsystem #inversion #sphereinversion #geometricart #3dgraphics #digitalsculpture #pythoncode #opengl #algorithmicart #algorist #mathart #laskutaide #ittaide #kuavataide #iterati

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.

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

Another look at Apollonian spheres, cutting out the top half and showing a few iteration steps.

#apollonianspheres #apolloniangasket #iteratedfunctionsystem #inversion #sphereinversion #geometricart #3dgraphics #digitalsculpture #pythoncode #opengl #algorithmicart #algorist #mathart #laskutaide #ittaide #kuavataide #iterati

Taking my lastest Apollonian gasket code from 2D to 3D was quite straightforward in principle, though there were a few kinks in the road. A particular difference between 2D and 3D gaskets is that in 3D, the inversion spheres overlap, which can create duplicate spheres.

Viewing detailed 3D structures isn't trivial either. We can only really see in 2D, as one dimension is taken up by the ray of light. Looking from outside, I wouldn't guess this blob contains over 10k spheres, so I blew it up for this clip.

The sheer amount of balls is also heavy on the drawing side, so I used my low-poly "sprites" where each ball is drawn by a geometry shader from a single input point. The low-poly aspect is quite clear in the largest spheres, but I think it's OK for this math demo.

#apollonianspheres #apolloniangasket #iteratedfunctionsystem #inversion #sphereinversion #geometricart #3dgraphics #digitalsculpture #pythoncode #opengl #geometryshader #algorithmicart #algorist #mathart #laskutaide #ittaide #kuavataide #iterati

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.

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