A 13-sided shape known as “the hat” has mathematicians tipping their caps. It’s the first true example of an “einstein,” a single shape that forms a special tiling of a plane: Like bathroom floor tile, it can cover an entire surface with no gaps or overlaps but only with a pattern that never repeats.
Note that it's got nothing to do with the great "Einstein" but derives it's name from German ein Stein, meaning “one stone,” referring to the single tile. Though the tiles fit neatly together and can cover an infinite plane, they are aperiodic, meaning they can’t form a pattern that repeats.
D. SMITH, J.S. MYERS, C.S. KAPLAN AND C. GOODMAN-STRAUSS
The same rendered as shirts and hats. The hat tiles are mirrored relative to the shirt tiles.
Mathematicians previously knew of nonrepeating tilings that involved multiple tiles of different shapes. In the 1970s, the famous mathematician Roger Penrose discovered that just two different shapes formed a tiling that isn’t periodic. (Do a search for Penrose tiles)
Also there was another one which was close. The Taylor-Socolar tiles are aperiodic, but they are a jumble of multiple disconnected pieces — not what most people think of as a single tile.
Note that it's got nothing to do with the great "Einstein" but derives it's name from German ein Stein, meaning “one stone,” referring to the single tile. Though the tiles fit neatly together and can cover an infinite plane, they are aperiodic, meaning they can’t form a pattern that repeats.
D. SMITH, J.S. MYERS, C.S. KAPLAN AND C. GOODMAN-STRAUSS
The same rendered as shirts and hats. The hat tiles are mirrored relative to the shirt tiles.
Mathematicians previously knew of nonrepeating tilings that involved multiple tiles of different shapes. In the 1970s, the famous mathematician Roger Penrose discovered that just two different shapes formed a tiling that isn’t periodic. (Do a search for Penrose tiles)
Also there was another one which was close. The Taylor-Socolar tiles are aperiodic, but they are a jumble of multiple disconnected pieces — not what most people think of as a single tile.
