David Smith is an amateur mathematician and retired print technician from

Bridlington Bridlington (previously known as Burlington) is a seaside town and civil parish in the East Riding of Yorkshire, England. It is on the Holderness part (Flamborough Head to the Humber estuary) of the Yorkshire Coast by the North Sea. The town is ...

, Yorkshire, England, who is best known for his discoveries related to aperiodic monotiles that helped to solve the

einstein problem In plane discrete geometry, the einstein problem asks about the existence of a single prototile that by itself forms an aperiodic set of prototiles; that is, a shape that can tessellate space but only in a nonperiodic way. Such a shape is call ...

Einstein tile

Initial discovery

Smith discovered a 13-sided

polygon In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...

in November 2022 whilst using a software package called ''PolyForm Puzzle Solver'' to experiment with different shapes. After further experimentation using cardboard cut-outs, he realised that the shape appeared to

tessellate A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of ...

but seemingly without ever achieving a regular pattern.

Contacting experts

Smith contacted Craig S. Kaplan from the

University of Waterloo The University of Waterloo (UWaterloo, UW, or Waterloo) is a Public university, public research university located in Waterloo, Ontario, Canada. The main campus is on of land adjacent to uptown Waterloo and Waterloo Park. The university also op ...

to alert him to this potential discovery of an aperiodic monotile. They nicknamed the newly discovered shape "the hat", because of its resemblance to a

fedora A fedora () is a hat with a soft brim and indented crown.Kilgour, Ruth Edwards (1958). ''A Pageant of Hats Ancient and Modern''. R. M. McBride Company. It is typically creased lengthwise down the crown and "pinched" near the front on both sides ...

. Kaplan proceeded to further inspect the polykite shape. During this time, Smith informed Kaplan that he had discovered yet another shape, which he nicknamed "the turtle", that appeared to have the same aperiodic tiling properties. By mid-January 2023, Kaplan enlisted software developer Joseph Samuel Myers from

Cambridge Cambridge ( ) is a List of cities in the United Kingdom, city and non-metropolitan district in the county of Cambridgeshire, England. It is the county town of Cambridgeshire and is located on the River Cam, north of London. As of the 2021 Unit ...

and mathematician

Chaim Goodman-Strauss Chaim Goodman-Strauss (born June 22, 1967 in Austin, Texas) is an American mathematician who works in convex geometry, especially aperiodic tiling. He retired from the faculty of the University of Arkansas and currently serves as outreach mathem ...

from the

University of Arkansas The University of Arkansas (U of A, UArk, or UA) is a Public university, public Land-grant university, land-grant research university in Fayetteville, Arkansas, United States. It is the Flagship campus, flagship campus of the University of Arkan ...

in order to help complete the proof. Myers realised that "the hat" and "the turtle" were in fact a part of the same continuum of shapes, which possessed the same aperiodic tiling properties but with sides of varying lengths.

Publication and further proofs

The team published their proofs in a preprint paper called 'An aperiodic monotile' in March 2023. Smith emailed Kaplan less than a week after posting of their preprint informing him of the apparent properties of a new shape. This shape, nicknamed "the spectre", was found at the midpoint of the team's spectrum of shapes published in their paper. It was an anomaly within the spectrum of shapes as it produced a periodic pattern when tiled with its reflection. However, Smith had discovered that it would produce an aperiodic pattern when tiled without its reflection. The team worked on a proof that confirmed the chiral aperiodic tiling property of "the spectre" and published a preprint paper in May 2023. As of July 2024 the first result ("the hat") has been formally published in the journal ''Combinatorial Theory.''

References

External links

An aperiodic monotile
by Smith, Myers, Kaplan, and Goodman-Strauss. {{DEFAULTSORT:Smith, David Amateur mathematicians Recreational mathematicians Tiling puzzles Year of birth missing (living people) Living people