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‘The Specter’: one of the extraordinary new repetitive forms discovered by a British pensioner.

David Smith, a retired print technician from the north of England, was pursuing his hobby of looking for interesting shapes when he came across a shape unlike any other in November.

When Smith shared its shape with the world in March, enthusiastic fans printed it on T-shirts, stitched it into quilts, created cookie cutters or used it to replace the hexagons on a soccer ball. but they even designed some tattoos.

The 13-sided polygon, which Smith, 64, called “the hat,” is the first single shape ever found that can completely cover an infinitely large flat surface without ever repeating the same pattern.

This makes him the first “einstein” named after the German for “one stone” (ein stein), not the famous physicist, and solves a problem posed 60 years ago that some mathematicians had deemed impossible.

After wowing the math world, Smitha an untrained hobbyist who told AFP he wasn’t good at math in school, then did it again.

While everyone agreed that “the hat” was the first einstein, his mirror image was required once in seven to ensure a pattern never repeats itself.

But in a preliminary study published online late last month, Smith and the three mathematicians who helped him confirm the discovery revealed a new shape, “the spectrum.”

It doesn’t require a mirror image, which makes it an even purer einstein.

“It Can Be So Easy”

Craig Kaplan, a computer scientist at Canada’s Waterloo University, told AFP that it was “a funny and almost ridiculous story, but a wonderful one.”

He said Smith, a retired print technician who lives in the East Riding of Yorkshire, emailed him “out of the blue” in November.

Smith had found something “that didn’t match his normal expectations of how shapes behave,” Kaplan said.

If you put a bunch of these cardboard shapes together on a table, you could keep building outward without them ever settling into a regular pattern.

Using computer programs, Kaplan and two other mathematicians showed that the shape continued to do so in an infinite plane, making it Einstein’s first, or “aperiodic monotile.”

When they released their first preprint in March, Yoshiaki Araki was among those inspired. The Japanese tile enthusiast has made art using the hat and another aperiodic shape the team created called “the turtle,” sometimes using upside-down versions.






Graph showing the shapes that solved the “einstein problem”, which was impossible to solve for about 60 years.

Smith was inspired and started playing with ways to avoid having to flip the hat.

Less than a week after their first article came out, Smith emailed Kaplan with a new form.

At first Kaplan refused to believe it. “There’s no way it could be that easy,” he said.

But the analysis confirmed that Tile (1,1) was a “non-reflective Einstein,” Kaplan said.

Something still bugged them while this tile could go on indefinitely without repeating a pattern, that required an “artificial ban” against using an upside-down shape, he said.

Then they added little notches or curves to the edges, making sure that only the unflipped version could be used, creating “the spectrum”.

“Hat Party”

Kaplan said both of their articles were submitted to peer-reviewed journals. But the world of mathematics didn’t wait to express his astonishment.

Marjorie Senechal, a mathematician at Smith College in the US, told AFP the findings were “exciting, startling and startling”.

He said he expects the spectrum and its kin “to lead to a deeper understanding of order in nature and the nature of order.”

Doris Schattschneider, a mathematician at Moravian College in the United States, said both forms were “stunning”.

Even Nobel Prize-winning mathematician Roger Penrose, whose earlier effort reduced the number of aperiodic tiles to two in the 1970s, wasn’t sure such a thing was possible, Schattschneider said.

Penrose, 91, will be among those celebrating the new shapes at the two-day ‘Hatfest’ event at the University of Oxford next month.

All participants expressed astonishment that the breakthrough was achieved by someone with no math training.

“The answer has fallen out of the sky and into the hands of an amateur, and I mean in the best possible way, a lover of the subject who is exploring it outside of professional practice,” Kaplan said.

“This is the sort of thing that shouldn’t happen, but very fortunately for the history of science it happens occasionally, where a flash brings us the answer all at once.”

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