The task is simple: a tile should have a specific Form have, so they never have one Muster repeated – no matter how much area it is laid. However, the solution to this riddle has so far led to mathematicians headache.
In mathematics, such tiles are called “aperiodic monotiles“, in English also as “EinsteinThe shape has nothing to do with the famous physicist. Rather, it is a pun on the words “one” and “stone”, meaning “a (single) tile”.
So far at least 2 forms necessary
Over the past 60 years, research on these Einstein forms has progressed, the closest being to the pattern in the 1970s. The Physics Nobel Laureate Roger Penrose found pairs of tiles that could form patterns that never repeated. However, the problem could not be solved with a single tile.
Until the English amateur mathematician David Smith took on the problem and tracked down the Einstein form. He christened the 13-page form “the Hut“, in his first attempts he cut 32 pieces out of cardboard and laid them together. When he saw that he was on to the solution, he informed Craig Kaplan, a Informatikprofessor at the University of Waterloo in Canada.
Whole group of Einstein forms discovered
Together with a mathematician and a software developer, they were able to prove that “the Hut” was actually an Einstein shape. And not only that: Smith found another shape, which is based on the hat and fulfills the requirements of an Einstein shape: “the turtle“.
The hat (left) and the turtle (right).
© David Smith
The researchers made their results in one go Paper public, although this has not yet been reviewed by the research community. However, they are certain that they encountered an entire class of Einstein forms.
To what extent these forms have applications outside of the world of mathematics remains to be determined. Kaplan is sure that there are “many applications in art, design and architecture”. Also for research quasicrystals the Einstein forms might be useful.
“First of all, the question arises as to who is the first to have a photo of his bathroom that was tiled with the hats,” Kaplan told the Guardian. After all, cookie cutters already exist.
