float x2 = sin(5.0*sin(x/5.0));
float y2 = sin(5.0*sin(y/5.0));
return isint(x2+y2, 0.1) ? vec3(0.9) : vec3(0.11);

#TilingTuesday

#TilingTuesday but ðe tiles are √(z + |α> + |β>) IFS fractals🤨🤔

#tiling #gonefractal #mathart #fractal #abstract #art

For #TilingTuesday, 4-colourings of the Penrose P2 and P3 tilings, via refinements of their substitution systems.

Penrose tilings must _admit_ proper 4-colourings, by the Four Colour Theorem and the Erdős–de Bruijn theorem. 4CT says that any finite patch of a given plane tiling must have a 4-colouring; EdB says, in that case, so does the whole plane.

But EdB doesn't provide an efficient algorithm for constructing such a colouring; it only guarantees that one exists in principle. If you wanted a large patch of 4-coloured tiling, you might have to generate a finite patch first, and then run one of the horrible algorithms from proofs of 4CT. And once you'd done it, there'd be no guarantee that _that_ 4-coloured patch would extend to a 4-colouring of the whole plane.

What you want is a refined substitution system, containing multiple types of each Penrose tile shape, each already equipped with a colour. Then you could generate tilings efficiently from that, and you'd know you had not just a finite coloured patch, but a finite patch from a whole coloured plane.

For P3 (rhombs) this is easy. Split each rhomb into a subtype for each of its 10 rotations. The adjacency graph on those 20 classes has no self-edges. There's no reason to expect that graph to be _planar_, but it turns out it can be 4-coloured anyway. So you can 4-colour a P3 tiling by assigning colours based only on tile shape and orientation.

For P2 (kites and darts), if you make the same graph, it doesn't admit a 4-colouring. But as part of my transducers research, I wrote code to refine a substitution system further, by classifying each tile based on its immediate neighbours. If I classify the P2 tiles by orientation and _then_ run a pass of that refinement algorithm, it generates 50 subtypes of kite and 40 subtypes of dart, and the adjacency graph for _those_ classes is 4-colourable. So it takes a bit more work, but it's still a substitution system that directly delivers a 4-coloured P2 tiling.

The pictures here are derived from those two refined substitution systems. In each case, the 4-colouring of the class graph was done by brute force: SageMath's first_coloring() function. So the colours aren't optimised for aesthetics!

What are the rules for tiling with regular pentagons and the parallelogram that fits in the gaps?

Here I've got two patches of tilings.

The one on the right has fivefold rotational symmetry and it looks like it could expand forever.

The one on the left doesn't have rotational symmetry. I think the difference was that I started with one pentagon with 4 others adjacent to it, which has sort of forced everything else. At the moment I've got two pentagons of degree 4, and I think I can get another pentagon of degree 4 by building on the left-hand side.

Is there something like the Aztec diamond theorem for this tiling? @JimPropp ?

#TilingTuesday

Black &| white

#tilingtuesday #geometry #creativecoding #p5js