Penrose kite and dart stencil
Posted: January 8th, 2024, 8:27 pm
You have to have a masochistic streak to do this by hand, but I was wondering if a simple Cricut-cut stencil would work for drawing Penrose tilings. The answer is: sort of.
Obviously you could put the shapes themselves on a stencil. You could also have tiles with curved edges to force correct placement. This is a minimal solution in which only 2 out of 4 edges to a tile are given in the stencil, since tiles will share a boundary. It also enforces placement rules, because you can only draw an edge where you are matching the endpoints of a missing edge. To minimize the stencil even further, I have the kite and dart share the long edge.
In practice, there is a certain amount of drift unless you are a lot more careful about placement than I am, and it can get a little confusing to determine what can connect. E.g., the distance between left and right corners of a kite is close enough to the edge length, that you have to watch that you don't try to connect them. Another problem is if you flip the stencil carelessly. I eventually put a notch in the corner to make sure I could tell.
The biggest problem is not really knowing how to layout the tiles ahead of time unless you are starting with a packing. Of course the easiest way to get one is to start with one that's computer generated, which is why I say this "sort of" works as a manual method.
One application I can think of is painting a Penrose tiling on a wall. In that case, it might be easier just to stencil in the outline first. You probably won't be able to print it automatically. It may also make sense to make more elaborate stencils that include common motifs, such as the ace, rather than doing each tile one by one.
Added: Or you could make a larger stencil and draw patterns on the beach to be washed out in the tide. Note that the kite and dart are placed illegally relative to each other on the stencil. This is inevitable because they share an edge. Legal placement requires that only one edge is drawn for any pair of adjacent tiles.
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In practice, there is a certain amount of drift unless you are a lot more careful about placement than I am, and it can get a little confusing to determine what can connect. E.g., the distance between left and right corners of a kite is close enough to the edge length, that you have to watch that you don't try to connect them. Another problem is if you flip the stencil carelessly. I eventually put a notch in the corner to make sure I could tell.
The biggest problem is not really knowing how to layout the tiles ahead of time unless you are starting with a packing. Of course the easiest way to get one is to start with one that's computer generated, which is why I say this "sort of" works as a manual method.
One application I can think of is painting a Penrose tiling on a wall. In that case, it might be easier just to stencil in the outline first. You probably won't be able to print it automatically. It may also make sense to make more elaborate stencils that include common motifs, such as the ace, rather than doing each tile one by one.
Added: Or you could make a larger stencil and draw patterns on the beach to be washed out in the tide. Note that the kite and dart are placed illegally relative to each other on the stencil. This is inevitable because they share an edge. Legal placement requires that only one edge is drawn for any pair of adjacent tiles.