One It’s not the first time that the oldest and simplest problem in geometry has caught mathematicians off guard.
Since ancient times, artists and geometers have wondered how shapes line up across a plane without gaps or overlaps. Still, “it wasn’t very well known until fairly recently,” said Alex Iosevic, a mathematician at the University of Rochester.
The most obvious tiling repetition: It’s easy to cover the floor with copies of squares, triangles, or hexagons. In the 1960s, mathematicians discovered a strange set of tiles that could completely cover a plane, but the method was never repeated.
Rachel Greenfeld, a mathematician at the Institute for Advanced Study in Princeton, New Jersey, said: “How crazy can they get?”
Pretty crazy.
The first such non-repeating or aperiodic pattern relied on a set of 20,426 different tiles. Mathematicians wanted to know if that number could be reduced. By the mid-1970s, Roger Penrose (Black, for his Hall work that would see him win the 2020 Nobel Prize in Physics), was working on his two tiles, called “kites” and “darts”. The set proved sufficient.
It’s not hard to come up with non-repeating patterns. Many repeating or periodic tilings can be tweaked to form non-repeating tilings. For example, consider an infinite grid of squares arranged like a chessboard. If you shift each row and offset it by a different amount than the row above it, you won’t be able to find an area that you can cut and paste like a stamp to recreate the perfect tiling.
The real trick is to find a set of tiles that can cover the whole plane like Penrose, but only in a non-repeating way.
Illustrated by Merrill Sherman/Quanta Magazine
Penrose’s two tiles begged the question: Could there be one cleverly shaped tile that fits the bill?
Surprisingly, if moving, rotating and flipping the tiles is allowed and the tiles are cut, i.e. with gaps, the answer is yes. These gaps are filled by other properly rotated and properly reflected copies of the tiles, eventually covering her entire two-dimensional plane. But if this shape is not allowed to rotate, it is impossible to line up the planes without leaving gaps.
In fact, several years ago, the mathematician Siddhartha Bhattacharya proved that no matter how complex and subtle a tile design may be, it is impossible to devise it if only a single tile shift or transformation can be used. Did. Tiles that can cover the entire plane aperiodically, but not periodically.
