An Old Geometry Problem Solved: A Non-Repeating Tiling Pattern Using A Single Shape

For more than 50 years, researchers have tried to find a solution to the "one stone" problem, which asks if a single shape can be used for tiling that has no repeating patterns.

Have you ever wondered if you can tile a geometric shape in a way such that it forms a non-repeating pattern? This is a geometry problem that mathematicians have tried to solve for decades. When a particular two-dimensional shape, say, a square or a triangle is tiled, it does repeat itself. There are several instances of repeating patterns of geometric shapes in real life, such as the tiles of one's bathroom floor, the honeycomb structure of a beehive, a chessboard, or the mosaics of Spain's Alhambra Palace.

What is the “one stone” problem?

For more than 50 years, researchers have tried to find a solution to the "one stone" problem, which asks if a single shape can be used for tiling that has no repeating patterns. This form of tiling is known as aperiodic tiling. 

Now, an international team of mathematicians, which includes David Smith from Yorkshire University in the United Kingdom, Joseph Samuel Myers from Cambridge University in the United Kingdom, Craig S Kaplan from the University of Waterloo in Canada, and Chaim Goodman-Strauss from the University of Arkansas in the United States, has found a geometric shape that does not repeat itself when tiled, solving the decades-long geometry problem. The paper describing their findings appears in a preprint on arxiv, a repository of electronic preprints. 

What is tiling?

Normally, the tiles on a floor have geometric shapes such as squares or triangles arranged periodically. Tiling is a process in which one fits shapes together in a way such that there are no overlaps or gaps. In other words, the plane has to be broken into pieces without gaps or overlaps. An infinite checkerboard is an example of periodic tiling, and one with translational symmetry. When a figure moves from one position to another, with the same orientation in the forward and backward motion, it is said to have translational symmetry. For instance, if we pick up a square from an infinite checkerboard, and place it at some other position with the same orientation, it will fit into that position. 

Has aperiodic tiling been achieved?

While aperiodic tiling, or the tiling in which a pattern does not repeat itself, has been achieved in the past, it has either used multiple shapes, or has been finite, which means that after a certain point, patterns will start repeating themselves. 

In the 1960s, the first aperiodic tile set without any translational symmetry was created, but this was finite, and used 20,426 different shapes. 

In 1974, British mathematician Sir Roger Penrose made an aperiodic tiling set using two different rhombus shapes.

However, all instances of aperiodic tiling in the past used more than one shape.

What is an “einstein” shape?

The four mathematicians claim that the shape found by them can alone be used for aperiodic tiling, and also infinitely. The single shape that can be used for aperiodic tiling all by itself is known as the "einstein" shape. 

The nomenclature is not inspired by Albert Einstein, but comes from the phrase "one stone" in German. The mathematicians claim that they have proved that the shape found by them is indeed an "einstein" shape. 

The authors note in the paper that the einstein shape is the first topological disk that tiles aperiodically with no additional constraints or matching constraints. The shape is a non-convex polygon, and has 13 sides. It is called the "hat". When one or more of the interior angles in a shape is more than 180 degrees, the shape is called a non-convex polygon. 

The researchers tried different possibilities on a computer, and then studied the smaller sets by hand. After this, they performed some more tests, and finally concluded that the "hat" is indeed an "einstein" shape. 

In a statement released by the University of Arkansas, Professor Edmund Harriss from the university described the roles played by the different mathematicians in discovering the "einstein" shape.

David Smith is a retired printing technician who created the shape, and Joseph Myers is a software developer who found the two proofs that led the team to conclude that the "hat" alone can make aperiodic tiling possible.