Pentomizer - Every known tessellating convex pentagon

by mathgrrl, published

Pentomizer - Every known tessellating convex pentagon by mathgrrl Sep 11, 2015


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With this Customizer you can print ALL OF THE PENTAGONS. All of the good ones, that is. That we know of. Fifteen types of pentagons, including one discovered by mathematicians only this year!


  • "All of the good ones" means all of the irregular convex pentagons that tile, or tessellate, the plane.
  • "That we know of" means that since 1918, mathematicians have been trying to find all possible tessellating pentagons, and this Customizer can make all of the ones that have been found so far.
  • "Fifteen types" 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.
  • "Only this year" means that THIS YEAR, 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 one!

But what can you MAKE with this design, and how?

  • Press the "Open in Customizer" button to get started
  • Choose one of the fifteen types of pentagon
  • 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 (most sliders only work on certain classes); use restraint if you want to preserve convexity or five-sidedness
  • Press the "Create Thing" button and then download and print your pentagon design
  • Take a photo, click "I Made One" on this page, and upload your photo to show off your pentagon creation

UPDATE: If you'd like to be able to set some of the desk organizer heights to zero, check out SpoonUnit's remix.


  • Vertex and offset data for each pentagon class were adapted from Ed Pegg's fantastic Wolfram Demonstration Pentagon Tilings
  • Mathematical facts and history about the fifteen pentagon classes can be found on Wikipedia at Pentagonal tiling


Twitter: twitter.com/mathgrrl
Hacktastic blog: www.mathgrrl.com/hacktastic
Shapeways geekhaus store: www.shapeways.com/shops/mathgrrl

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].

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Love the idea of using this as a desk organizer. I couldn't figure out how to get the customizer to alter the number of tiles used for it. I tried using a 0 to remove one but that still seems to draw something. Did I miss something in the customizer or should I download the .scad file already and start tinkering?

Right now you can only make desk organizers that include all of the pentagons in the pattern chunk, but maybe in the future I'll put in the "0" capacity; that would be cool. Or try out the .scad if you don't want to wait for me!

OK. Finally got around to doing this. By the way, the primary reason is that my printer is a little smaller and scaling down just made a little too small to be useful. If you decide to take the change, test, and update yours, I'll just delete mine. Let me know :)

I just came here to say that I enjoyed the energy used to describe this 'thing'.

I am totally nerding out over this one! You do great work!

This is cool! Truly inspirational. And I am glad to see that you got the new one too.