In 2007 I thought of a pretty way to paint a square so that all pixels are different, but similar colors are clustered. For each pixel, set x,y to its coordinates; if their sum is odd, set the low bit of one of the color channels to 1. Replace x,y with (-x+y)>>1, (-x-y)>>1; this has the effect of rotating the grid by 3/8 turn and shrinking it by a factor of √2, so that the former even points, which formed a larger oblique grid, now fall on the original grid, and the odd points have their new half-coordinates truncated away. Repeat until a bit has been assigned to each bit of the three color channels.

Colors that match in their higher bits form twindragon fractals, thus:

In 2012, I thought: what if the rotation alternates clockwise and counterclockwise?

A bit on the boring side.

But in 2017, I thought: what if the sequence of left and right turns is randomized? (For the design as a whole, that is, not for each pixel.)

Some of these have a Deco flavor.

These pictures use only 17 bits. The repeat unit is a tilted square whose edge length is 2**(17/2) pixels. I did that so that the neighborhood of the corner is repeated in the middle. The low bit of green, the low two bits of red, and the low four bits of blue are all zero here.