Over the holiday break, I took a little time for a side quest that engaged my creativity while involving mathematics, 3D modeling, and baking. I designed a pattern based on Penrose tiles and made it into cookies.
I didn’t have it planned out at the outset, and instead followed my inspiration from one step to the next. First, I made an original design for a set of Penrose tiles with a more organic appearance. Then, I created 3D models of the tiles and extended the model to be a cookie cutter. Finally, I printed the cookie cutters and baked cookies that tesselate nonperiodically in a pattern with fivefold rotational symmetry.
Do you have a 3D printer? You can download and print them yourself at these links:
About Penrose tiling
Penrose tiling is a non-periodic tessellation using two shapes to create intricate and interlocking designs. There are a few Penrose tile types, with the “P2” type being the most commonly discussed. The two tile shapes in P2 are often called “kites” and “darts.” The tiles are arranged in a pattern that is not periodic but can appear to be when viewed from a distance, leading it to be called a “quasiperiodic” pattern.
Penrose tilings were first discovered/invented by Nobel Prize laureate physicist and mathematician Roger Penrose in the 1970s. The patterns have been significantly studied in the fields of mathematics and physics, leading to a better understanding of physical materials that form quasicrystals. Learn more about Penrose tiling at Wikipedia.
There are also a lot of great videos about Penrose tiling. I think the best one out there is this one by Numberphile. But there are many others, including this one by MinutePhysics, and this one by Veritasium.
About my version of Penrose tiles
This rendition is intended to give a natural appearance to the pattern so that it might suggest a wreath or cluster of plants.
The resulting tiles no longer resemble “kites” and “darts,” so I have been referring to these as “birds” and “cats,” respectively, although I didn’t plan it that way.
The diagram below shows how I modified the kites and darts by replacing their straight lines with circular arcs.
This design includes arcs that enforce the non-periodic pattern. The chord lengths of the arcs follow the Golden Ratio:
There are seven “starting patterns” that you can use to begin a tiling design, shown below. If you start with the “star” or “sun” central patterns, the resulting design will not only be non-periodic, but it will also have fivefold rotational symmetry.
Notes on baking tessellating Penrose tile cookies
When making cookies, bear in mind that for tiling purposes, you’ll need more birds than cats – an infinite plane filled with these tiles will require φ (~1.618) birds per cat. Consider also the fact that individually, the birds have more area than the cats, so for a given amount of dough, you’ll be able to cut out more cats than birds, which is the opposite of what you want for tessellating. Perhaps the best strategy here is to make more than you need of both and eat the excess.
I found that the cat’s forepaw and the bird’s foot have a tendency to get a little stuck when the dough is soft (i.e., when it is close to room temperature). Be sure to cool your dough, and if you have trouble extricating these parts, a little flour dusting on the interior of the cutters can help.
It might go without saying, but of course, placing the dough on the cookie sheet and baking will change the cookie tiles’ shape somewhat, diminishing their ability to tessellate. Some possible ways to mitigate this problem:
- Tessellate the cut-out cookies on the cookie sheet to ensure a good fit, then move them apart slightly to give them space to bake.
- Same as above, but maybe don’t move them apart at all and leave them in contact. Depending on the recipe, this might lead to cookie fusion during baking, but perhaps they will be separable afterward (maybe while still warm).
- Any other ideas or suggestions for this? Please comment!