Pretty Styling Penrose Tiling

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

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:

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.

The seven starting patterns:

“Star” pattern in the center:

“Sun” pattern in the center:

Printing and assembly

There are two main .3mf files for different sizes. Each file includes one bird tile and one cat tile, so you can print however many of each by copying and pasting them in your slicer app. 

• The “large” file has chord lengths in the scale of inches, so the bird tile is about 3" across the longest dimension. 
• The “small” file has chord lengths in the scale of centimeters, so the bird tile is about 3 cm across the longest dimension. 
• The Golden Ratio diagram above shows what I mean by chord length dimensions. I’ve included a couple of photos of both sizes next to some glasses for scale. 

You can definitely print a lot more small tiles than large ones in a given timeframe, and assemble much more of the pattern in a given space. Personally I find the larger ones more fun and satisfying to play with though. 

You may wish to print the birds and cats in their own colors. I have included two gcode files with five birds and five cats each in the larger scale. The print time shown is to print both sets for a total of 10 large tiles.

Most of the photos shown here are large-scale tiles, from prints on my MK3/S+, using 0.4 mm nozzle, 0.2 mm layer height, 15% infill, with PLA. These are pretty simple, so you don't need supports, and you might be able to go with thicker layers and/or less infill to speed things up. 

One photo shows the smaller tiles. I wanted to try using translucent PETG and that’s what I used for these. Because they are so small, I found them very difficult to remove from the print bed. You might want to add a brim or something to make them easier to remove. 

I found that I was printing many of these tiles, and they needed containment. I could just put them in a bowl, but I've gone ahead and created containers for both sizes. They hold 30 cats and 60 birds each. 

You'll need more birds than cats – an infinite plane filled with these tiles will require φ birds per cat.  

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

Please post your makes!