A brickwork fractal • Ross Churchley

Brick pavements and tatami mats are traditionally laid out so that no four meet at a single point to form a ┼ shape. Only a few ┼-free patterns can be made using $1\times 1$ and $1\times 2$ tiles, but the addition of $2\times 2$ tiles provides a lot more creative flexibility.

Bricks laid out in various traditional patterns — Three ┼-free brickwork sections laid out in the stretcher bond, herringbone, and pinwheel patterns, respectively.

When I discussed tatami tilings with my relative Oliver Linton, he suggested applying similar rules to other brick sizes to make beautiful tiling patterns. The tatami condition alone does not provide enough of a constraint to mathematically analyze tilings with arbitrary shapes and sizes, but it is a good starting point when looking for interesting patterns.

With the addition of $2\times 2$ square tiles, it’s possible to construct rectangular blocks that fit together to tesselate the plane while preserving the four-corner rule.

A stretcher bond pattern, with the outline of each brick made of 1×2 and 2×2 tiles — Copies of the same rectangular block can cover the plane without four-corner intersections.

This opens the door to self-similar tilings, which I’m very interested in! The goal is to use $1 \times 1$ , $1 \times 2$ , and $2 \times 2$ tiles to construct $n\times n$ , $n \times 2n$ , and $2n \times 2n$ blocks which can be put together in the exact same way to make increasingly intricate $n^k \times n^k$ tilings that maintain the tatami condition.

The simplest non-trivial example I could find involves a set of $5\times 5$ , $10 \times 10$ , and two $5\times 10$ rectangular tilings.

Two square and two 1×2 rectangular shapes covered by 1×1, 1×2, and 2×2 tiles satisfying the tatami condition — 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 $1\times 1$ , $2\times 2$ , and $1\times 2$ bricks with a corresponding $5\times 5$ , $10\times 10$ , or $5\times 10$ rectangular tiling in the correct orientation. (This will produce a few four-corner intersections, but we can fix these by merging adjacent pairs of $1\times 2$ bricks.)

Square and rectangular patterns made of square and rectangular tiles in tatami arrangements — 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 $\partial T$ (i.e. the set of points on boundaries between two or more tiles). Note that if a point is in $\partial T_i$ , then it will be in every subsequent $\partial T_j$ unless it is one of the few bricks merged in $\partial T_{i+1}$ . So we might sensibly define the limiting object of the tiling sequence $T_i$ as the union

\bigcup_i \left(\partial T_i \cap \partial T_{i+1}\right).

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 $(2n+1)\times (2n+1)$ , $(4n+2)\times (4n+2)$ , and $(2n+1)\times (4n+2)$ tiles with similar boundaries. What’s the prettiest brickwork fractal you can find?