Aperiodic tiling of the plane by one tile: A family of Einstein tiles

Typeface designers have been inspired by geometry, and so typographers may be interested in a newly discovered shape.

In mathematics and design, people study and use planar tilings. For example, a periodic tiling may be made using, e.g., an equilateral triangle, a rectangle, and a hexagon. Previously, a periodic tilings have used a small set of distinct shapes. There was an open problem of finding a single tile that could generate a non-periodic tiling, an einstein (one stone, in German). The einstein problem seems to have been solved.

E.Le Morvan, CC BY-SA 4.0, via Wikimedia Commons

What is astonishing is that the solution also specifies an infinite family of shapes.

YouTube video showing the continuous deformation of one einstein tile into another.

Animation at the Computer Science Department of the University of Waterloo (CC BY 4.0)

Above, the hat einstein-tile was shown, which appears in the middle of the animation: The two extreme einstein-tiles are more likely to inspire typographers.