One of the oldest and simplest problems in geometry have taken mathematicians by surprise, and never for the primary time.
Since antiquity, artists and geometers have wondered how shapes can cover your complete plane without gaps or overlaps. Yet “little was known until quite recently,” he said Alex Iosevichmathematician from the University of Rochester.
The most evident tiles repeat: It’s easy to cover the ground with copies of squares, triangles or hexagons. In the Sixties, mathematicians discovered strange sets of tiles that may cover the plane completely, but only in a way that never repeats itself.
“You want to grasp the structure of such slopes,” he said Rachel Greenfeld, a mathematician on the Institute for Advanced Study in Princeton, New Jersey. “How crazy can they be?”
Turns out it’s pretty crazy.
The first such unique or non-periodic pattern was based on a set of 20,426 different tiles. Mathematicians desired to know in the event that they could lower that number. In the mid-Seventies, Roger Penrose (who went on to win the 2020 Nobel Prize in Physics for working on black holes) proved that a straightforward set of two tiles, called “kites” and “darts”, is enough.
It’s not hard to give you patterns that do not repeat themselves. Multiple repeating or periodic slopes may be modified to create unique ones. Consider, say, an infinite grid of squares arranged like a chessboard. If you progress each row in order that it’s offset by a definite amount from the previous one, you may never have the option to seek out an area that may be cut and pasted like a stamp to recreate full tiles.
The real trick is to seek out sets of tiles – like Penrose’s – that may cover your complete plane, but only in a way that does not repeat.
Illustration: Merrill Sherman/Quanta Magazine
The two Penrose plates raised the query: could there be one cleverly shaped plate that fit the bill?
Surprisingly, the reply is yes – when you are allowed to maneuver, rotate and mirror the tile and the tile is disconnected meaning it has gaps. These gaps are filled by other appropriately rotated, appropriately reflected copies of the tile, eventually covering your complete two-dimensional plane. But when you cannot rotate this shape, it’s inconceivable to put the plane without leaving gaps.
Actually, A number of years agomathematician Siddhartha Bhattacharya proved that – regardless of how intricate or subtle a tile design you give you – when you are only capable of use offsets or translations of a single tile, then it’s inconceivable to give you a tile that may cover your complete plane non-periodic but non-periodic.
