Newfound Mathematical ‘Einstein’ Shape Creates A Never-Repeating Pattern

[papasmurf]

I just find the middle of the area to be tiled and work out from there. (My home is four inches out of square. Makes life "interesting." )

[Borg Refinery]

Creatively tiling a bathroom floor isn't just a stressful task for DIY home renovators. It is also one of the hardest problems in mathematics. For centuries, experts have been studying the special properties of tile shapes that can cover floors, kitchen backsplashes or infinitely large planes without leaving any gaps. Specifically, mathematicians are interested in tile shapes that can cover the whole plane without ever creating a repeating design. In these special cases, called aperiodic tilings, there's no pattern that you can copy and paste to keep the tiling going. No matter how you chop up the mosaic, each section will be unique.

Until now, aperiodic tilings always required at least two tiles of different shapes. Many mathematicians had already given up hope of finding a solution with one tile, called the elusive "einstein" tile, which gets its name from the German words for "one stone."

Then, last November, retired printing systems engineer David Smith of Yorkshire, England, had a breakthrough. He discovered a 13-sided, craggy shape that he believed could be an einstein tile. When he told Craig Kaplan, a computer scientist at the University of Waterloo in Ontario, Kaplan quickly recognized the potential of the shape. Together with software developer Joseph Samuel Myers and mathematician Chaim Goodman-Strauss of the University of Arkansas, Kaplan proved that Smith's singular tile does indeed pave the plane without gaps and without repetition. Even better, they found that Smith had discovered not only one but an infinite number of einstein tiles. The team recently reported its results in a paper that was posted to the preprint server arXiv.org and has not yet been peer-reviewed.

Scientific American