‘Annoying’ geometry breaks a decades-old tile guess

One of the oldest and simplest problems in geometry have taken mathematicians by surprise – and not for the first time.

Since ancient times, artists and geometers have wondered how shapes can tile the entire plane without gaps or overlaps. And yet “not much was known until quite recently,” he said Alex Iosevicha mathematician at the University of Rochester.

The most obvious repetition of tiles: It’s easy to cover a floor with copies of squares, triangles or hexagons. In the 1960s, mathematicians found strange sets of tiles that can completely cover the plane, but only in ways that never repeat.

“You want to understand the structure of such tiles,” he said Rachel Greenfield, a mathematician at the Institute for Advanced Study in Princeton, New Jersey. “How crazy can they get?”

Pretty crazy, it turns out.

The first such non-repeating or aperiodic pattern was based on a set of 20,426 different tiles. Mathematicians wanted to know if they could lower that number. By the mid-1970s, Roger Penrose (who would go to win the 2020 Nobel Prize in Physics for work on black holes) proved that a simple set of only two tiles, called “kites” and “darts”, was sufficient.

It’s not hard to come up with patterns that don’t repeat. Many repeating or periodic tiles can be modified to form non-repeating tiles. For example, consider an infinite grid of squares aligned like a chessboard. If you shift each row so that it is offset by a certain amount from the row above, you will never be able to find an area that can be cut and pasted like a stamp to recreate the entire tiling.

The real trick is to find sets of tiles, such as Penrose’s, that can cover the entire face, but only in ways that don’t repeat.

Illustration: Merrill Sherman/Quanta Magazine

Penrose’s two tiles begged the question: Could there be a single, cleverly shaped tile that just fits?

Surprisingly, the answer turns out to be yes – if you are allowed to slide, rotate and reflect the tile, and if the tile is disconnected, meaning there are gaps. Those gaps are filled by other appropriately rotated, appropriately reflected copies of the tile, eventually covering the entire two-dimensional plane. But if you are not allowed to rotate this shape, it will be impossible to tile the plane without leaving gaps.

Indeed, several years agothe mathematician Siddhartha Bhattacharya proved that – no matter how intricate or subtle a tile design you come up with – if you can only use shifts or translations of a single tile, it’s impossible to come up with a tile that can cover the entire plane occasionally, but not periodically.

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