Penrose Infill Tiles - Infinite Non-Repeating Tessellation

Penrose Tiles are named for their creator, Roger Penrose, a mathematician and physicist. The Penrose Tiling is an infinite aperiodic tiling, meaning the pattern can be tiled over the entire plane (to infinity) but never repeats the exact same pattern (aperiodic). 

I was inspired to make these tiles after watching “The Infinite Pattern That Never Repeats” by Veritasium (Dr. Derek Muller), and I highly recommend spending a few minutes to watch! You can also read more about Penrose Tiles on Wikipedia. 

To help my kids and myself understand and experience the mathematics and geometry, I created these sets of tiles, and to make them more fun to work with, I printed each tile with a different infill pattern. 

In addition to being a great educational tool, perfect for any mathematics classroom, they also make fantastic and fun infill swatches to show off your infill patterns, densities and filaments. Works with any infill pattern and nozzle size!

Tile Sets

I've created three sets of tiles, to show the three types of Penrose tiles. 

The first is made up of a pentagram, a rhombus (or diamond), a star and a “boat”. To properly tile these in a non-periodic fashion, the pentagram should be printed in three different colors/infills. 

The second version uses two shapes called “kite” and “dart”, which can be combined into seven possible shapes, but may never be used to form a rhombus. 

The final version uses two different rhombuses (or “rhombs”). Because the rules to tile the rhombs can be confusing, they are often shown with lines, notches or curved sides. I've designed my examples with notches. 

Printing / Slicer Settings

Print with any size nozzle, layer height, etc. I used a 0.6mm nozzle, I bet they would look fantastic with a 0.8mm as well, especially if you want to scale them up. I suggest printing in a different color for each shape (including 3 colors for the pentagons in version 1)

• Walls: 2-3, depending on your nozzle size and desired effect.
• Bottom Layers: 4-6, depending on nozzle size
• Top Layers: 0 - this is how you get the great infill effect!
• Infill Pattern / Density: Go wild! Changing the infill pattern and density will of course affect your total print time.

Examples

My examples are all printed in PolyMaker's PolyTerra PLA, with a 0.6mm nozzle. I've included the gcode files for these as well, however these are specific to my printer. 

I printed Version 1 with the following Infill Patterns & Densities:

• Pentagon A - Cross @ 20%
• Pentagon B - Gyroid @ 10%
• Pentagon C - Concentric @ 25%
• Rhombus - Triangles @ 30%
• Boat - Tri-Hexagon @ 45% 
• Star - Cubic @ 25%

Using the Tiles

Print a bunch of the tiles for whichever version you've chosen, being sure to use different colors / infill patterns. 

Version 1 Tiling:

You should have 6 different tiles, three of which are pentagons. 

1. Place Pentagon A (blue)
2. Surround with Pentagon B (green), touching each side.
3. Fill holes with Rhombus (purple)
4. Add Pentagon C (yellow) to Diamond's exposed sides. 
5. Fill outside holes between Pentagon C with Boat (pink)
6. Add Pentagon A (blue) on outside edge, touching vertices with boats
7. Add three of Pentagon B (green) around Pentagon A (blue). 
8. Add Pentagon C (yellow) on Boat's remaining exposed side.
9. Add Star (orange) touching Pentagon C (red)
10. Surround star with Pentagon C (yellow)
11. Repeat!