“Unpleasant” geometry refutes decades-old tiling hypothesis

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

Since antiquity, artists and geometers have wondered how figures can tile the entire plane without gaps or overlaps. And yet, “not much was known until relatively recently,” said Alexey Iosevichmathematician at the University of Rochester.

The most obvious tilings are repetitive: the floor can be easily covered with copies of squares, triangles or hexagons. In the 1960s, mathematicians discovered strange patterns of tiles that could completely cover a plane, but only in ways that never repeat.

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

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 reduce that number. By the mid-1970s, Roger Penrose (who later win the 2020 Nobel Prize in Physics for working with black holes) proved that a simple set of just two tiles, called “kites” and “darts”, is enough.

It’s not hard to come up with patterns that don’t repeat. Many repeating or periodic tiles can be modified to form non-recurring tiles. Consider, say, an infinite grid of squares aligned like a checkerboard. If you move each row so that it is offset by a certain amount from the one above it, you will never find an area that can be cut and pasted like a stamp to recreate a complete mosaic.

The real trick is to find sets of tiles like Penrose’s that can cover the whole plane, but only in such a way that they don’t repeat themselves.

Illustration: Merrill Sherman/Quanta Magazine

The two Penrose tiles raised the question: could there be one smartly shaped tile that fits the bill?

Surprisingly, the answer is yes if you are allowed to move, rotate, and reflect the tile, and if the tile is detached, which means it has gaps. These gaps are filled with other appropriately rotated, appropriately reflected copies of the tile, eventually covering the entire 2D plane. But if you’re not allowed to rotate this shape, it’s impossible to tile the plane without leaving gaps.

Actually, a few years agomathematician Siddhartha Bhattacharya proved that – no matter how complex or intricate the tile design you come up with – if you can use shifts or displacements of only one tile, then it is impossible to design a tile that could cover the entire plane aperiodically, but not periodically.

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