If you’re into tilings, or just looking to redo your bathroom in the most modern way possible, there’s big news. A team of researchers at the University of Washington-Bothell have discovered a previously unknown way to tile a plane using irregular pentagons.
Attempts to tile a plane with regular pentagons never really got off the ground, but until now there were 14 known ways to use irregular convex pentagons to cover a plane. Five were discovered in 1918 by German mathematician Karl Reinhardt, and it was assumed this was the complete list until more were discovered in 1968 and 1975, including four found by housewife Marjorie Rice upon reading about them in Scientific American (covered, of course, by Martin Gardner).
The Bothell researchers, Casey Mann, Jennifer McLoud and David Von Derau, employed an exhaustive computer search to find the new shape last month, and will shortly be publishing a paper on their work.
The new discovery, which involves a pentagon with angles $60^\circ$, $135^\circ$, $105^\circ$, $90^\circ$ and $150^\circ$, brings the total up to 15, all of which are illustrated below. Aren’t they beautiful? The new one is in the bottom right.
It’s still not known whether this is all the possible pentagonal tilings, or whether more are still out there. The team hope they can continue with their method to find other arrangements. Keep looking, pentagon pickers!
Attack on the pentagon results in discovery of new mathematical tile, by Alex Bellos at The Guardian
A new way to tile your floor (if you like pentagons), by Kevin Knudson at Forbes.com