Pretty Penrose tessellation cookie cutters

Whether you're celebrating Phi (φ) Day or any other holiday, these organic Penrose tiles will be a conversation starter! Seriously, how many others will bring non-periodic tessellating cookies with fivefold rotational symmetry to your office potluck?

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. 

About my version

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 (the superimposed kite and dart diagrams are by Geometry guy at English Wikipedia, CC BY-SA 3.0):

This design includes arcs that enforce the non-periodic pattern. The chord lengths of the arcs follow the Golden Ratio:

Printing and assembly

The .3mf file includes both cookie cutters. The cookies are about 3 inches across the longest dimension, with the chord lengths in the Golden Ratio diagram above being in units of inches.

No supports are needed.

I recommend printing in PETG or polypropylene. PLA and ASA are not great for food.

Making 3D prints truly food-safe is not trivial, as the prints are somewhat porous and can harbor bacteria. But you know, so can wood. But to minimize any risk, I recommend using the ironing print setting on the uppermost top layer to seal up any gaps (the print file .gcode attached to this model has ironing the top layer set). Clean carefully between uses and if you want to learn more about the topic of food safety in 3D printing, check out this article. 

Baking notes

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. Some have suggested that dipping the cutter in water may work better than dusting with flour. I haven't tried this. Update 2023-03-17: a model of punches has been uploaded so you can “punch out” the dough and retain its shape!

Moving the dough to 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).
• This recipe for gingerbread cookies purports to yield non-spreading cookies. I haven't tried it. 
• Any other ideas or suggestions for this? Please comment!

If you have any trouble that I can help with, please let me know, and I'll be happy to make modifications.