One of the The oldest and simplest problems in geometry have caught mathematicians off guard—and this is not the first time.
Since ancient times, artists and geometers have wondered how shapes could line up an entire plane without gaps or overlap. And yet, “not much was known until fairly recently,” says Alex Iosevicha mathematician at the University of Rochester.
The most obvious tiling repeats: It’s easy to cover the floor with copies of squares, triangles, or hexagons. In the 1960s, mathematicians found strange sets of tiles that could cover the plane completely, but only in ways that never repeat.
“You want to understand the structure of such panels,” says Rachel Greenfeld, a mathematician at the Institute for Advanced Study in Princeton, New Jersey. “How crazy can they be?”
It turned out to be pretty crazy.
The first such non-repeating or recurring pattern is based on a set of 20,426 different tiles. Mathematicians wanted to know if they could bring that number down. In the mid-1970s, Roger Penrose (who would continue Winner of the 2020 Nobel Prize in Physics for the study of black holes) demonstrated that a simple set of just two bricks, named “kites” and “darts”, is sufficient.
It is not difficult to come up with non-repeating patterns. Multiple repeating or recurring cells can be tweaked to form non-repeating cells. Suppose, consider an infinite grid of squares, lined up like a chess board. If you shift each row so that it deviates by a different amount from the row above it, you will never be able to find an area that can be cut and pasted like a stamp to recreate the entire tile. .
The real trick is to find tile sets—like Penrose’s—that can cover the entire plane, but only in non-repeating ways.
Illustration: Merrill Sherman/Quanta Magazine
Penrose’s two bricks raised the question: Would a single, cleverly shaped brick fit the bill?
Surprisingly, the answer turned out to be yes—if you’re allowed to move, rotate, and mirror the cell, and if the cell is disconnected, that means the cell has space. Those gaps are filled by other suitably rotated, mirrored copies of the tile, eventually covering the entire two-dimensional plane. But if you are not allowed to rotate this shape, you cannot tile the plane without leaving gaps.
Real, many years agomathematician Siddhartha Bhattacharya proved that—no matter how intricate or subtle a cell design you come up with—if you can only use shifts or translations of a single cell, it is impossible to create a single cell. can cover the entire plane periodically but not periodically.
