Not long after we went online, we received a note from Tim Lexen of Cumberland, Wisconsin. Tim had created an elegant, simply defined shape he calls the Tricurve. Since the there is only one tile shape, the tilings are referred to as monohedral tilings. Here are a couple of the examples Tim sent us:
The pieces are oriented, meaning that when you flip a piece over, you can still use the piece but in different ways. You can use both orientations in the same tiling, which makes it different from most of our other sets of tiles. The flexibility of the Tricurve comes from it's spare geometric and arithmetic properties. The edges are made from a semicircle cut into 3 pieces in the ratio 1:2:3.
Now we rearrange the 3 pieces into this shape.
Because the pieces came from the same circle, all edges have matching curvature. The 1:2:3 ratio allows combining edges that add up to other edges. Moreover, the angles in the tile are also in a 1:2:3 ratio, with 30, 60, and 90 degree angles. This enables several kinds of rotational symmetry: 12-fold, 6-fold, 4-fold, 3-fold, and 2-fold. While the Tricurve can be used to create wallpaper patterns, it seems everyone who plays with it also likes to explore these rotational symmetries. Here is an example from Tim Lexen:
and some we created…
If you want to learn more, read about Tim Lexen and Paul Bourke's explorations with the Tricurve: