New shape answers a 50-year-old geometry mystery Premium

The aperiodic monotile was theoritical until a shape-hobbyist from the UK put it together

The grey “hat” polykite tile is an aperiodic monotile. | Photo Credit: arXiv.org

What are the facts? A group of four scientists have discovered an entirely new shape which has 13 sides and has named it “the hat”. It is the first ‘einstein’ tile to have been found: a shape that can cover a plane without overlapping, leaving gaps, or repeating patterns. The name bears no connection to Albert Einstein but means ‘one stone’ in German. The shape is a polykite with the edges of eight kites to make it. The study proving the existence of the shape has already generated excitement in the scientific community.

What is the context?

• The discovery has answered a long-standing question in the field of plane geometry where one tries to tile a plane in different ways. While several ways that tiling exists in nature and the real world (think bathroom tiles or the hexagons of honeycomb), these are periodic and have a translation symmetry–the size, shape and angle remains the same, only the location changes.
• The 13-sided ‘hat’ is also a shape that tiles plane, however, it does not possess translational symmetry. These types of shapes are called aperiodic tile sets, according to the release.
• Up until the discovery, an aperiodic monotile was purely theoretical. “You’re literally looking for like a one-in-a-million thing. You filter out the 999,999 of the boring ones, then you’ve got something that’s weird, and then that’s worth further exploration,” said co-author Chaim Goodman-Strauss according to New Scientist.

Why does it matter?

• The discovery of the ‘hat’ shape is the first time a single aperiodic tile has been found to cover a plane.
• There are a limited number of aperiodic tiles that have been found over the years since the first tile sets, consisting of 20,000 shapes, were created in 1960s. Over the years, the number of shapes in a tile has decreased, with British Mathematician Roger Penrose using only two shapes to create an aperiodic tile in the 1970s. As a result, the discovery of the ‘hat’ has been long-awaited. 
• The study was an interdisciplinary effort with the shape first being identified by shape hobbyist David Smith, who is a retired printing technician. The other three authors consisted of Joseph Myers, a software developer, Craig Kaplan, a computer science professor and Chaim Goodman-Strauss, a mathematics professor at the University of Arkansas.