This article will explain “Hobbyist Finds Maths Elusive Einstein Tile”. David Smith, a retired print specialist and enthusiast of jigsaw puzzles, fractals, and road maps, was toying with forms in the middle of November of last year.
He had created a tile that resembled a hat using the PolyForm Puzzle Solver, a piece of software. He was currently attempting to fill as much of the screen as possible with duplicates of that tile, without overlaps or gaps.
He frequently produced tiles that either fell into repetitive patterns or didn’t cover a large portion of the screen. The hat tile, however, appeared to accomplish neither. On cardstock, Smith cut out 30 copies of the hat, assembled them, and placed them on a table.
Then he eliminated another 30 and carried on. “I noticed that it was producing a tessellation that I had never seen before,” he stated. It’s a small tile that’s tricky. Craig Kaplan, a friend and computer scientist at the University of Waterloo in Canada, was provided a description of his tile, and he started looking into its characteristics right away.
The hat tile is a single tile whose copies can occupy the entire plane, but only in patterns that don’t consist of a repeating block of tiles, according to Smith and Kaplan and two additional academics who published their findings on March 20. In contrast to shapes like squares or hexagons that can fill the plane in a repeating (or periodic) fashion, mathematicians refer to such a tile, or collection of tiles, as “aperiodic.”
The hat tile possesses “enough complexity to forcibly disrupt periodic order at all scales,” according to the researchers’ article. In addition, they discovered that there are an endless number of possible tiles of this type.
Since Robert Berger created a collection of 20,426 forms in the 1960s that when combined aperiodically tile the plane, mathematicians have been looking for a tile like a hat. Berger’s research sparked a competition to create smaller aperiodic tile sets, which resulted in Roger Penrose’s discovery of sets with only two aperiodic tiles in the 1970s.
In work that won him the 2011 Nobel Prize in chemistry, Dan Shechtman revealed in 1982 that symmetries similar to those in Penrose tilings appear in nature in the form of formations called quasicrystals.
Since then, mathematicians have been searching for a single tile that can continuously fill the two-dimensional plane with no voids or overlaps. Such a tile was jocularly referred to as a “einstein” by the German geometer Ludwig Danzer, a play on the word “ein stein,” which means “one piece.”
In order to cover the plane periodically, two groups developed techniques in the 1990s for overlapping nearby copies of a single 10-sided tile. A decade or so later, Tasmanian math enthusiast Joan Taylor made the discovery of a shape made of numerous disconnected components.
In a 2010 study, she and Joshua Socolar, a physicist at Duke University, demonstrated that the plane is tiled aperiodically. And just last year, Terence Tao of the University of California, Los Angeles and mathematicians Rachel Greenfeld of the Institute for Advanced Study found a high-dimensional structure that aperiodically tiles space without even needing to be rotated or reflected.
However, no one was able to discover a real Einstein, which is a straightforward, periodic, two-dimensional structure. Mathematicians eventually began to question whether such a tile really exists, according to Marjorie Senechal, an expert in tiling and emerita professor at Smith College. “Just mind-boggling,” she exclaimed, “that an Einstein as simple as Smith’s hat was out there all along.”
She reasoned that perhaps the hat had remained undiscovered up to this point because many mathematicians have concentrated on designs with “forbidden” symmetries—ones that can’t arise in periodic tilings.
For instance, Penrose’s tilings exhibit fivefold symmetry, similar to that of pentagons and five-pointed stars. Fivefold symmetries are a logical place to explore for tilings that can’t be periodic because regular pentagons cannot tile the plane.
In comparison, the hat lacks symmetry and is “almost mundane in its simplicity,” according to the writers. Its tilings do, however, have a close connection to a specific periodic tiling: the honeycomb lattice of hexagons.
Connect the midpoints of the opposing sides of the hexagons first to create a hat pattern from a hexagonal pattern. Every hexagon is therefore divided into six “kites.” Eight adjacent kites assembled from nearby hexagons make up each hat. Anyone with a magic marker and a bathroom floor tiled with hexagons may easily trace out a hat tiling.
The hat tile, according to Senechal, demonstrates a closer connection between periodic and aperiodic tiles than previously thought.
Mathematicians and tiling enthusiasts have hurried to obtain the new tiles in the days following the announcement, producing paper cutouts, 3D printing them, as well as hat quilts and biscuits out of them.
Smith, who resides in the northern English seaside town of Bridlington, described the excitement the tiles have caused as feeling “a little surreal.” I’m not used to this sort of thing, she said.
However, this is by no means the first time a layperson has achieved a significant advancement in tiling geometry. One set of Penrose’s tiles was independently found in the 1970s by mail sorter Robert Ammann.
In 1975, Marjorie Rice, a housewife from California, discovered a brand-new family of pentagonal tilings. The Socolar-Taylor tile was then uncovered by Joan Taylor. Perhaps amateurs are “not burdened with knowing how hard this is,” Senechal remarked, in contrast to mathematicians.
