Aperiodic tile set with only one tile shape.
Mathematical Sciences Professor Chaim Goodman-Strauss is part of a multidisciplinary team that recently published the solution to the ‘Einstein’ or ‘one stone’ problem that has been asked for over 50 years.
This question asks about how to tile planes. Intuitively, tiling consists of dividing a plane into pieces without gaps or overlaps. Examples of tiling abound in the real world and in nature.
For example, an infinite checkerboard provides tiling of a plane by squares. Other tiles can be found in honeycomb hexagons and in the tile mosaics of the Alhambra. However, these tiling examples are periodic. That is, it has translational symmetry. You can imagine picking up an infinite checkerboard, sliding it up one square, and then putting it back where each piece fits exactly into the checkerboard pattern.
Surprisingly, there exists a finite collection of shapes that tile the plane, but none of those tilings have translational symmetry. These are called aperiodic tile sets. The first example was created in the 1960s and required over 20,000 different shapes. This number dwindled, and in the 1970s British mathematician Sir Roger Penrose demonstrated an aperiodic tileset that used only two geometries. The question remained: are there aperiodic tile sets with a single shape?
Such shapes were just discovered by this interdisciplinary team of researchers, including Professor Goodman Strauss. New evidence emerges that this shape is indeed an aperiodic tile set. preprintThe announcement caused a lot of excitement both within and outside the world of mathematics and computer science, new york times article.
“This is something I never thought I’d see in my lifetime,” said Professor Edmund Harris of the Department of Mathematical Sciences. After many years of serious quest for ways to tile planes, software developer Joseph Myers, who created the geometry and found two proofs, Craig Kaplan, a professor of computer science at the University of Waterloo, and Worked in collaboration with Chime Goodman. — Strauss.”