A fractal pattern generated by a progressive replacement system - each rule has the same relationship to its neighbors. In math, numbers are constructed with just such a system; start with zero and each number is the number before it plus one. It is how we count.
[Broken links: See
Progressive 2 for a description and
Progressive 2x2 rule systems for descriptions of, and links to, related patterns.] The pattern is a 2-D generalization of Sloane's integer sequence
A000120.
A 2K x 2K version.
MATLAB
Here's a
color version.
Also see
~vidthekid's related pattern "
Silver Squares".
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Sta.sh
If X and Y are the two cartesian coordinates of any point in the square and have a range of [0,1), then the brightness of the image at that point can be obtained by expressing X and Y in binary (with an infinite number of fractional bits) and combining them with an exclusive or operation. Using a finite but large number of binary bits (in this case, is it 110?) will produce an approximation; a larger number of bits increases both the spatial and brightness resolution -- the pixel resolution and color depth, in computer graphics terminology.
But you probably already knew that.
But the L-system rules (e.g. the top row is the 10th of the sequence 0, 01, 0112, 01121223 ...) are a much simpler way of specifying this pattern. No need to express the X/Y coordinates in a number system, binary or base 10. A few bits of algorithm generates the whole pattern.
I do know that if the gray values are coded in binary sequentially, and the sum of these number's bits are added modulo 2, the pattern is a 2D Thue-Morse sequence [link]
I now see what you meant about constructing this array using XOR on the binary x and y values. What I didn't understand was that you meant bit-wise XOR (on each bit independently). I'm a bit slow on these things.
BUT, notice that the images are not the same. Every pixel in any sub-quadrant of the "9 Levels" image is brighter or darker than those of an adjacent quadrant. This is not true for this image.
I almost posted another full on B/W tiling, but your comment made me reconsider. Looking for something that looks good in color...