If you love pentagons then 2015 was a pretty good year for you, because this year a new pentagon was discovered! To be more precise, what mathematicians Mann, McLoud, Von Derau discovered this year was a previously unknown convex pentagon that can tessellate the plane. You can read about it at Evelyn J. Lamb’s article There’s Something About Pentagons. And Ed Pegg has a fantastic Wolfram Demonstration for Pentagon Tilings that contains vertex data for all of 15 known families of convex tessellating pentagons. So… what can you make out of this?
Turns out we can make a picture, a pattern, a puzzle, a texture, some wallpaper, a desk ornament, and some cookie cutters. With the Pentomizer model on Thingiverse you can make all of those things, based on any of the known tessellating pentagon families.
Use the Pentomizer on Thingiverse: http://www.thingiverse.com/thing:1008750
Here are two 3D-printable designs that were made with the Pentomizer:
Ultimate Math Geek Wallpaper
This design is based on the new 15th pentagon, which has a 12-tile primitive unit; this means that you can combine 12 copies of this pentagon to form a chunk that can tessellate the plane only with translations. To make things more interesting we used different heights for each of the 12 pentagon tiles in the primitive unit. Then we put four of those units together to make a larger chunk for 3D printing. To get a bumpy, three-dimensional wallpaper, print a lot of these and then tile them together and glue them to your favorite wall.
Download the wallpaper on Thingiverse: http://www.thingiverse.com/thing:1009644
Pentagon Geek Desk Organizer
This is a desk organizer based on one of five tessellating pentagons discovered in 1918 by mathematician Karl Reinhardt. The four compartments of the organizer each have the same irregular pentagon base, and the resulting four-pentagon pattern can be used to tile the plane. That means that if you print more than one of these desk organizers, that they will fit together!
Download this organizer on Thingiverse: http://www.thingiverse.com/thing:101060
UPDATE: If you’d like to be able to set some of the desk organizer heights to zero, check out SpoonUnit’s remix.