(Probably far from an original idea.)
I was thinking about texture generation today, and in particular what you can do in a fragment shader, where deciding which pixels to affect is not a thing you can do. This seems like it could be a real problem, since many real-world textures are the result of a lot of different objects moving around; for example, the exposed stones in a sawn concrete surface are in some sense scattered randomly, as are the leaves on a forest floor. But in a fragment shader you can’t just generate some random points and place leaves at them, at least not in a way that scales when you zoom out.
I was reminded that when people look at independent identically distributed random points, they generally think they don’t look very random, because clusters of points randomly occur, and so the density of the points varies even at fairly large scales. Many natural textures — the stones in the concrete I mentioned before, for example, but also hairs on the skin, cones on the retina, bubbles in a foam — break up such clusters by a sort of “relaxation” in which points move away from one another, evening out the medium-scale density variations, and eventually the large-scale ones too.
But maybe another approach would be to start with a very even dot distribution and perturb it enough that it looks random. You could have some perfectly regular lattice of cells, with a dot at the center of each — a square or hexagonal lattice — and generate a two-dimensional value of Perlin noise by which to perturb the dot at that center. As long as the dot doesn’t overlap the next cell, the algorithm to determine the color of a pixel is very simple; if z is the pixel coordinate:
c = round(z)
fragColor = hypot(c + r1 * noise2d(c) - z) > r2 ? bg : fg
If we expand r2 or especially r1 to the point that the dots start to wander into adjacent cells, that simple seven-instruction algorithm starts to fail; if we run it for all the adjacent cells, though — 5, 7, or 9 of them, depending on how many candidates there are — we can determine which of the neighboring cells’ dots are overlapping us, at the cost of a work multiplier of 5, 7, or 9.
In the same way, you can draw an almost-Voronoi diagram by having no dot-radius threshold, just coloring the pixel according to which dot’s center it’s closest to. This will occasionally depart from the real Voronoi diagram because long, sharp projections will occasionally be truncated early by cell boundaries; perturbing the cell boundaries slightly with more noise may be a good way to keep that subtle.
Well, what’s granite? Granite consists of a lot of crystals of minerals of different colors which grew as the magma cooled, each one nucleating with some random position and orientation and growing around that center, faster along some planes of their crystal structure than along others. At first only zircon and calcium-rich plagioclases can crystallize (or forsterite, but it doesn’t occur much in granite), but as the temperature drops, other minerals like pyroxenes, more sodic plagioclases, micas, and eventually even quartz can crystallize. As the crystals grow, they deplete the local magma of their own mineral, which means that when the leftover magma eventually does crystallize, it will be of a different color.
This is precisely the kind of prolonged multivariate field dynamical process that’s hard to simulate in a fragment shader†, but perhaps we can generate a similar-looking result by perturbing the Voronoi distances according to a random skew matrix. That is, before computing the magnitude of the displacement from the pixel to a dot center, multiply that displacement vector through a skew matrix particular to that dot (generated from yet another call to noise). This should make the “crystal” tend to be longer in some apparently-random directions, and shorter in others. If the skew matrix also has a random determinant, some “crystals” will be larger and others smaller.
This is a terribly goofy way to generate this image, though, as you can solve precisely where the grain boundaries are going to be. You don’t really need to do all this computering for every pixel.
† Could be harder — at least crystallizing magma isn’t turbulent.