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Fractal Penrose tiles

Every distinct set of tiles has certain features, and we are always interested in those that introduce something new or combine features in a new way. Dylan Thurston, who helped us determine the boundaries of the Twin Dragon introduced us to the decomposable fractal Penrose tiles first described by Bandt and Gummelt. By decomposable, we mean every tile can be represented as a combination of other tiles. The boundaries are fractal, like the Dragon and Twin Dragon tiles. And finally, like Penrose P3 tiles, there are two shapes, and they only tile the plain non-periodically (i.e. they don’t repeat like wallpaper).

There are two essential shapes. We were at a loss what to call them. Their boundaries evoke the dragon tiles. But the dragon tiles have rotational symmetry: you can rotate them 180 degrees, and they are unchanged, and that gives them a very different look and behavior when tiling. Turned a certain way, I thought these new tiles look a bit like dogs, so we came up with terriers and poodles:

Each shape can appear in any number of sizes. To get to the next larger size, you scale the dimensions by the golden ratio \(\phi \approx 1.618\).

Our prototype tiles below show how a terrier can be decomposed into two smaller terriers and a poodle. Can you also see how a poodle can be decomposed into a terrier and a poodle?

These tiles are decidedly more challenging when it comes to creating an uninterrupted tiling. At craft shows, we often encounter people who are torn between fractal tilings and Penrose tilings. Now you can have it both ways!