#TWO IMPOSSIBLE SHAPES ONTOP OF EACHOTHER SOFTWARE#
He liked to play around with the PolyForm Puzzle Solver, a piece of software that lets users design and assemble tiles, according to the New York Times. But David Smith, the 64-year-old retiree, hadn’t given up. Many mathematicians have since searched for the single-tile solution, the “einstein,” but none had succeeded-including Penrose, who eventually turned his attention to other puzzles.
The famous, elegant Penrose tiling uses just two types of tiles to make an aperiodic pattern. Credit: Inductiveload/Wikipedia Three years later mathematician Raphael Robinson found a variant with only six tile types-and finally, in 1974, physicist Roger Penrose presented a solution with only two tiles. Then, in 1968, computer scientist Donald Knuth found an example with 92. First, Berger found one with 104 different tiles. In the decades that followed, mathematicians found smaller and smaller sets of tiles that can create aperiodic mosaics. This discovery raised another question that has dogged mathematicians ever since: What is the minimum number of tile shapes that together can create an aperiodic tessellation? How Low Can You Go? And even better, it is physically impossible to form a repeating pattern with that set of tiles, no matter how you lay them.
In his work, Berger found an unbelievably large set of 20,426 differently colored tiles that can pave a plane without the color pattern ever repeating itself. The culprit: the unpredictable, nonrepeating nature of aperiodic tilings. There will always be cases where you can’t predict whether you can cover the surface without gaps. A mathematician would look at your game and ask, “Can you determine whether you will hit a dead end just by looking at the types of colored tiles you were given at the start? This would certainly save you a lot of time.” An aperiodic mosaic. Credit: Claudio Rocchini/Wikimedia (CC BY-SA 3.0) For example, maybe you needed just one tile in which all the edges were the same color. Game over.īut certainly, if you had the right tile with the right color combination, you could have gotten out of your pickle. There’s a gap you just can’t fill with the tiles you have available, and you are forced to place mismatched edges next to each other. You find a strategy you think is going to work, but at some point, you run into a dead end. With infinite tiles, you begin laying down pieces. You have to follow one rule, however: the edges of the tiles are colored, and only same-colored edges can touch. Suppose you want to tile an infinite surface with an infinite number of square tiles. Can you create an infinite mosaic with only same-color edges touching? Credit: Anomie/Wikimedia But such beauty harbors unanswerable questions-ones that are, as mathematician Robert Berger stated in 1966, provably unprovable. From Beautiful Patterns to Unprovable QuestionsĪnyone who has walked through the breathtaking mosaic corridors of the palace Alhambra in Granada, Spain, knows the artistry involved in tiling a plane.
The team recently reported its results in a paper that was posted to the preprint server and has not yet been peer-reviewed. Even better, they found that Smith had discovered not only one but an infinite number of einstein tiles. Together with software developer Joseph Samuel Myers and mathematician Chaim Goodman-Strauss of the University of Arkansas, Kaplan proved that Smith’s singular tile does indeed pave the plane without gaps and without repetition. When he told Craig Kaplan, a computer scientist at the University of Waterloo in Ontario, Kaplan quickly recognized the potential of the shape. He discovered a 13-sided, craggy shape that he believed could be an einstein tile. Then, last November, retired printing systems engineer David Smith of Yorkshire, England, had a breakthrough. Many mathematicians had already given up hope of finding a solution with one tile, called the elusive “einstein” tile, which gets its name from the German words for “one stone.” Until now, aperiodic tilings always required at least two tiles of different shapes. No matter how you chop up the mosaic, each section will be unique. In these special cases, called aperiodic tilings, there’s no pattern that you can copy and paste to keep the tiling going. Specifically, mathematicians are interested in tile shapes that can cover the whole plane without ever creating a repeating design. For centuries, experts have been studying the special properties of tile shapes that can cover floors, kitchen backsplashes or infinitely large planes without leaving any gaps.
It is also one of the hardest problems in mathematics. Creatively tiling a bathroom floor isn’t just a stressful task for DIY home renovators.
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