Trapezoids and diamonds

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Trapezoid and diamonds

Around 1999, after learning about the substitution rules for Penrose tilings, I started experimenting a bit with substitutions and noticed that 60° angled trapezoids and diamonds can naturally be deflated into each other with a scaling factor of 2, thereby generating aperiodic tilings. I didn't expect these tilings to be original, but it seems that they are. The Tiling Encyclopedia was not listing them a few months ago. (The owner now kindly added an entry).

It turns out that 40 distinct substitution rules and tilings can be constructed, if we exclude mirror operations in the deflation rules. This project showcases these 40 tilings.

Substitution rules

Here are the two substitution rules for the diamonds (labeled 0 and 1), and the twenty substitution rules for the trapezoids (labeled from 0 to 19). The diamonds are marked with a dot near the bottom left edge, to make their orientation obvious. Combining each substitution rule for the diamonds with each substitution rule for the trapezoid, we find altogether 40 substitution rules. As far as I can tell, each corresponds to a distinct tiling.

We will label the tilings as "TXDY", where X is the substitution rule for the trapezoids and Y the substitution rule for the diamonds.

Each tiling is shown both with its tiles outlined, and with its diamond colored in dark grey, making the tile arrangement more obvious, and creating interesting emergent patterns.

T0D0