October 10th 2009

Stars from squares

This is simply a regular tiling by squares such that one tile out of four has been colored differently. Note that around each vertice of the square lattice, there are three squares of one color, and one of the other color, what makes this coloring uniform. Let's pile this pattern with a magnification step of 3. Here is what we get:

Consider the red squares in the basic pattern as tiles and the yellowish region as "empty". Now start piling the pattern. Each time a square tile of the copy of the pattern being added touches an existing tile (ie. their closure intersect), incorporate it to the tile. It is easy to see that up to rescaling, we get only one type of tile. The latter has a fractal boundary and tiles the plane (with arbitrarily small tiles...).

I do not remember having seen this fractal before. Its star shape might be vaguely reminiscent of the Koch snowflake, but it is definitely not one. (In an older blog post, I showed how the piling of hexagons arranged roughly in the same way did produce true Koch snowflakes.)

Here is a zoomable image based on this fractal.