After decades of searching for “einstein tile”—an elusive shape that would never repeat—researchers say they have finally identified one.

Infinitely many copies of a 13-sided shape can be arranged with no overlaps or gaps in a pattern that never repeats.

 
From bathroom floors to honeycombs or even groups of cells, tilings surround us. These patterns cover a space without overlapping or leaving any gaps. Like a rug filled with diamond shapes, where each section looks the same as the one next to it, every tiling ever recorded has eventually repeated itself—until now. The 13-sided figure is the first that can fill an infinite surface with a pattern that is always original. Repeating patterns have translational symmetry, meaning you can shift one part of the pattern and it will overlap perfectly with another part, without being rotated or reflected. 

The shape described in a new paper does not have translational symmetry—each section of its tiling looks different from every part that comes before it.

 Read more