Every convex tessellating pentagon - for boxes, bracelets, cookie cutters, and wallpaper

With this OpenSCAD code you can print ALL OF THE PENTAGONS that can tile, or tessellate, the plane. Fifteen families of pentagons, including the last possible type, discovered by mathematicians in 2015.

What can you MAKE with this design, and how? Download the sample models to 3D print pentagons, tessellations, boxes, puzzles, wallpaper, thermaform bracelets, and even cookie cutters. 

Or download the OpenSCAD code and make whatever you want! Here's how to use the code:

• Download free OpenSCAD and open the file
• Choose one of the fifteen types of pentagon in the code
• Decide whether you want to use your tessellating pentagon to make a pattern, a puzzle, a picture, a container, a desk organizer, or even a cookie cutter!
• Change side lengths and angles within the restrictions of the pentagon class, if possible; use restraint if you want to preserve convexity or five-sidedness
• In OpenSCAD, pressing F5 will compile your new design. Press F6 when you are ready to export and then use File/Export as STL

References and links:

• Vertex and offset data for each pentagon class were adapted from Ed Pegg's fantastic Wolfram Demonstration Pentagon Tilings
• Read about these models at Hacktastic: 3D-Printable Pentagon Tessellations
• And about bulk SLS printing these models: Print all the Pentagons
• You can also order 3D prints of each class of pentagon at Shapeways: mathgrrl shop

Math facts:

• Since 1918, mathematicians have been trying to find all possible tessellating pentagons. It took about a hundred years for Reinhardt, Kersher, James, Rice and Stein to identify 14 families of convex pentagons that can tessellate the plane: Wikipedia article on pentagonal tilings
• In 2015, after a thirty-year dry spell in which no new tessellating pentagon types had been found, mathematicians Mann, McLoud, and Von Derau of the University of Washington Bothell found a new fifteenth family of convex tesselating pentagons: With Discovery, 3 Scientists Chip Away At An Unsolvable Math Problem 
• Just a couple of years later, Rao proved that the fifteen known families of convex tessellating pentagons were the ONLY possible tessellating families: Pentagon Tiling Proof Solves Century-Old Math Problem
• "Fifteen families" means that there are fifteen classes of tessellating pentagons, where each of these classes might include just one pentagon, or might include infinitely many somewhat similar-looking pentagons. This model can make them all!

This design and all associated pictures and files are licensed under the Creative Commons Attribution Non-Commercial Share Alike license. If you want to use designs, images, or files outside of the terms of this license, please email [email protected].