A brickwork fractal

On my way to work in Yaletown, I walk along some sidewalks with interesting brick patterns. The three above happen to satisfy a nice mathematical property: the bricks are arranged so that no four corners meet at the same place.

Stretcher bond, herringbone, and pinwheel brickwork patterns

As my relative Oliver Linton has pointed out, paving bricks come in a variety of shapes and sizes, which allows for many more beautiful tilings. For example, the addition of 2×2 square tiles make it possible to construct rectangular tilings that fit together to tesselate the plane while preserving the four-corner rule.

Copies of the same rectangular tiling can cover the plane without four-corner intersections.

This is a very exciting observation — at least, to anybody who likes recursion! We might be able to use the same trick to construct an infinite sequence of increasingly intricate tilings that converge to a self-similar “fractal tiling”. The simplest non-trivial example I could find involves a set of 5×5, 10×10, and two 5×10 rectangular tilings.

Four tilings of rectangles with the same aspect ratios as the bricks they comprise

Starting with any of these four layouts, we can replace each of the 1×1, 2×2, and 1×2 bricks with a corresponding 5×5, 10×10, or 5×10 rectangular tiling in the correct orientation. (This will produce a few four-corner intersections, but we can fix these by merging adjacent pairs of 1×2 bricks.)

The first recursive iteration of our tiling sequence

Repeatedly performing this operation gives an infinite sequence of tilings, but can we say they converge to anything? A tiling T can be identified with its outline ∂T (i.e. the set of points on boundaries between two or more tiles). Note that if a point is in ∂Ti, then it will be in every subsequent ∂Tj unless it is one of the few bricks merged in ∂Ti+1. So we might sensibly define the limiting object of the tiling sequence T_i as the union i(∂Ti⋂∂Ti+1). This self-similar dense path-connected set satisfies the topological equivalent of the “four corners rule” — a pretty interesting list of mathematical properties!

The same strategy could be applied to other sets of (2n+1)×(2n+1), (4n+2)×(4n+2), and (2n+1)×(4n+2) tiles with similar boundaries. What’s the prettiest brickwork fractal you can find?