Newly discovered 'einstein' tile is a 13-sided shape that solves a decades-old math problem

The first known instance of the elusive "einstein," a single form that can be tiled endlessly without repeating a pattern, is a novel 13-sided design.

Examine carefully! A unique 13-sided design that can be tiled endlessly without ever duplicating a pattern has been created by mathematicians. It is known as "the Einstein."

Mathematicians have pondered the possibility of finding a single unique form that could properly tile a surface without creating any gaps or overlaps and with a pattern that would never repeat for many years. Of course, repeating patterns are basic to work with; simply take a glance at a bathroom or kitchen floor, which is presumably covered in plain rectangular tiles. You may locate a location where your floor appears precisely the same as it did in the previous position by picking it up and moving it (referred to as a "translation" in mathematics), demonstrating that the pattern repeats itself.

Mathematician Hao Wang postulated in 1961 that it was impossible to create aperiodic tilings, or tilings that never recur. However, Robert Berger, one of his own students, outsmarted him by identifying a collection of 20,426 forms that, when meticulously assembled, never repeated. Later, he reduced that to a set of 104 tiles. It follows that there would never be a repeated pattern if you were to purchase a set of those tiles and lay them on your kitchen floor.

The Penrose tiling, which was discovered by Nobel Prize–winning physicist Roger Penrose in the 1970s, is a set of just two tiles that may be placed in a nonrepeating pattern.

Since then, mathematicians from all over the world have been on the prowl for "the einstein," the mythical object of aperiodic tiling. The term comes from the German translation of the renowned Albert's last name, one stone, rather than from him. Is it possible for a single tile, or a single "stone," to completely cover a two-dimensional space without ever repeating the pattern it makes?

David Smith, a retired printing technician from East Yorkshire, England, has recently learned the solution. How did he discover this brilliant solution? Smith said to The New York Times, "I'm always playing around and experimenting with shapes." "Practical experience is usually pleasant. It has a contemplative quality.

The new design was given the moniker "the hat" by Smith and his co-authors mostly due to its fuzzily fedora-like appearance. The design, which has 13 sides, has long been known to mathematicians, but they had never thought of it as a potential candidate for aperiodic tiling.

Marjorie Senechal, a mathematician at Smith College who was not involved in the study, told The Times, "In a way, it has been sitting there all this time, waiting for somebody to uncover it.

Smith developed two proofs demonstrating that "the hat" is an aperiodic monotile – an einstein — in close collaboration with two computer scientists, a mathematician, and another. One method of demonstrating how the pattern never repeats as the surface area increases included creating larger and larger hierarchical groupings of the tiles. The team's discovery that there were an endless number of similar forms instead of just one of these tiles might serve as the basis for the second piece of evidence. The team's publication is accessible on the preprint service arXiv, but neither the proofs nor it has been peer-reviewed.

These aperiodic tilings are more than just interesting mathematical phenomena. For starters, they show that several medieval Islamic mosaics used comparable nonrepeating patterns, as seen in the Penrose tiling at the Salesforce Transit Center in San Francisco.

Aperiodic tilings are particularly useful for explaining the formation and behavior of quasicrystals, which are ordered but non-repeating crystal structures.