[An “einstein” is a shape that tiles an infinite flat surface
without repetition. It all began with a hobbyist “messing about and
experimenting with shapes.”]
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MATH AMATEUR SOLVES “EINSTEIN” PROBLEM  
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Siobhan Roberts
March 28, 2023
New York Times
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_ An “einstein” is a shape that tiles an infinite flat surface
without repetition. It all began with a hobbyist “messing about and
experimenting with shapes.” _

,

 

Last November, after a decade of failed attempts, David Smith, a
self-described shape hobbyist of Bridlington in East Yorkshire,
England, suspected that he might have finally solved an open problem
in the mathematics of tiling: That is, he thought he might have
discovered an “einstein.”

In less poetic terms, an einstein is an “aperiodic monotile,” a
shape that tiles a plane, or an infinite two-dimensional flat surface,
but only in a nonrepeating pattern. (The term “einstein” comes
from the German “ein stein,” or “one stone” — more loosely,
“one tile” or “one shape.”) Your typical wallpaper or tiled
floor is part of an infinite pattern that repeats periodically; when
shifted, or “translated,” the pattern can be exactly superimposed
on itself. An aperiodic tiling displays no such “translational
symmetry,” and mathematicians have long sought a single shape that
could tile the plane in such a fashion. This is known as the einstein
problem.

“I’m always messing about and experimenting
[[link removed]] with shapes,” said Mr. Smith, 64,
who worked as a printing technician, among other jobs, and retired
early. Although he enjoyed math in high school, he didn’t excel at
it, he said. But he has long been “obsessively intrigued” by the
einstein problem.

And now a new paper [[link removed]] — by Mr.
Smith and three co-authors with mathematical and computational
expertise — proves Mr. Smith’s discovery true. The researchers
called their einstein “the hat,” as it resembles a fedora. (Mr.
Smith often sports a bandanna tied around his head.) The paper has not
yet been peer reviewed.

“This appears to be a remarkable discovery!” Joshua Socolar, a
physicist at Duke University who read an early copy of the paper
provided by The New York Times, said in an email. “The most
significant aspect for me is that the tiling does not clearly fall
into any of the familiar classes of structures that we understand.”

“The mathematical result begs some interesting physics questions,”
he added. “One could imagine encountering or fabricating a material
with this type of internal structure.” Dr. Socolar and Joan Taylor,
an independent researcher in Burnie, Tasmania, previously found
a hexagonal monotile [[link removed]] made of
disconnected pieces, which according to some, stretched the rules.
(They also found a connected 3-D version of the Socolar-Taylor tile.)

 

[David Smith’s explorations using cut-out paper.]

David Smith’s explorations using cut-out paper.Credit...David
SmithFrom 20,426 to one

Initially, mathematical tiling pursuits were motivated by a broad
question: Was there a set of shapes that could tile the plane only
nonperiodically? In 1961, the mathematician Hao Wang conjectured
[[link removed]] that such sets were
impossible, but his student Robert Berger soon proved the conjecture
wrong. Dr. Berger discovered an aperiodic set of 20,426 tiles, and
thereafter a set of 104.

Then the game became: How few tiles would do the trick? In the 1970s,
Sir Roger Penrose, a mathematical physicist at University of Oxford
who won the 2020 Nobel Prize in Physics for his research on black
holes, got the number down to two
[[link removed]].

Others have since hit upon shapes for two tiles. “I have a pair or
two [[link removed]] of my own,” said Chaim
Goodman-Strauss, another of the paper’s authors, a professor at the
University of Arkansas, who also holds the title of outreach
mathematician at the National Museum of Mathematics
[[link removed]] in
New York.

 

[An example of a Penrose tiling with kites and darts.]

An example of a Penrose tiling with kites and darts.Credit...Craig
Kaplan

He noted that black and white squares also can make weird nonperiodic
patterns, in addition to the familiar, periodic checkerboard pattern.
“It’s really pretty trivial to be able to make weird and
interesting patterns,” he said. The magic of the two Penrose tiles
is that they make only nonperiodic patterns — that’s all they can
do.

“But then the Holy Grail was, could you do with one — one tile?”
Dr. Goodman-Strauss said.

As recently as a few years ago, Sir Roger was in pursuit of an
einstein, but he set that exploration aside. “I got the number down
to two, and now we have it down to one!” he said of the hat.
“It’s a tour de force. I see no reason to disbelieve it.”

The paper provided two proofs, both executed by Joseph Myers, a
co-author and a software developer in Cambridge, England. One was a
traditional proof, based on a previous method, plus custom code;
another deployed a new technique, not computer assisted, devised by
Dr. Myers.

Sir Roger found the proofs “very complicated.” Nonetheless, he was
“extremely intrigued” by the einstein, he said: “It’s a really
good shape, strikingly simple.”

Imaginative tinkering

The simplicity came honestly. Mr. Smith’s investigations were mostly
by hand; one of his co-authors described him as an “imaginative
tinkerer.”

 

[Mr. Smith’s hat tiling made with the Polyform Puzzle Solver by
Jaap Scherphuis.]

Mr. Smith’s hat tiling made with the Polyform Puzzle Solver by Jaap
Scherphuis.Credit...David Smith

To begin, he would “fiddle about” on the computer screen
with PolyForm Puzzle Solver
[[link removed]], software developed
by Jaap Scherphuis, a tiling enthusiast
[[link removed]] and puzzle theorist in
Delft, the Netherlands. But if a shape had potential, Mr. Smith used a
Silhouette cutting machine to produce a first batch of 32 copies from
card stock. Then he would fit the tiles together, with no gaps or
overlaps, like a jigsaw puzzle, reflecting and rotating tiles as
necessary.

“It’s always nice to get hands-on,” Mr. Smith said. “It can be
quite meditative. And it provides a better understanding of how a
shape does or does not tessellate.”

When in November he found a tile that seemed to fill the plane without
a repeating pattern, he emailed Craig Kaplan, a co-author and a
computer scientist at the University of Waterloo.

“Could this shape be an answer to the so-called ‘einstein
problem’ — now wouldn’t that be a thing?” Mr. Smith wrote.

“It was clear that something unusual was happening with this
shape,” Dr. Kaplan said. Taking a computational approach that built
on previous research, his algorithm
[[link removed]] generated larger and
larger swaths of hat tiles. “There didn’t seem to be any limit to
how large a blob of tiles the software could construct,” he said.

With this raw data, Mr. Smith and Dr. Kaplan studied the tiling’s
hierarchical structure by eye. Dr. Kaplan detected and unlocked
telltale behavior that opened up a traditional aperiodicity proof
[[link removed]] — the method
mathematicians “pull out of the drawer anytime you have a candidate
set of aperiodic tiles,” he said.

The first step, Dr. Kaplan said, was to “define a set of four
‘metatiles,’ simple shapes that stand in for small groupings of
one, two, or four hats.” The metatiles assemble into four larger
shapes that behave similarly. This assembly, from metatiles to
supertiles to supersupertiles
[[link removed]], ad infinitum, covered
“larger and larger mathematical ‘floors’ with copies of the
hat,” Dr. Kaplan said. “We then show that this sort of
hierarchical assembly is essentially the only way to tile the plane
with hats, which turns out to be enough to show that it can never tile
periodically.”

“It’s very clever,” Dr. Berger, a retired electrical engineer in
Lexington, Mass., said in an interview. At the risk of seeming picky,
he pointed out that because the hat tiling uses reflections — the
hat-shaped tile and its mirror image — some might wonder whether
this is a two-tile, not one-tile, set of aperiodic monotiles.

Dr. Goodman-Strauss had raised this subtlety on a tiling listserv:
“Is there one hat or two?” The consensus was that a monotile
counts as such even using its reflection. That leaves an open
question, Dr. Berger said: Is there an einstein that will do the job
without reflection?

Hiding in the hexagons

Dr. Kaplan clarified that “the hat” was not a new geometric
invention. It is a polykite [[link removed]] — it
consists of eight kites. (Take a hexagon and draw three lines,
connecting the center of each side to the center of its opposite side;
the six shapes that result are kites.)

“It’s likely that others have contemplated this hat shape in the
past, just not in a context where they proceeded to investigate its
tiling properties,” Dr. Kaplan said. “I like to think that it was
hiding in plain sight.”

Marjorie Senechal, a mathematician at Smith College, said, “In a
certain sense, it has been sitting there all this time, waiting for
somebody to find it.” Dr. Senechal’s research explores the
neighboring realm of mathematical crystallography
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and connections with quasicrystals
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“What blows my mind the most is that this aperiodic tiling is laid
down on a hexagonal grid, which is about as periodic as you can
possibly get,” said Doris Schattschneider, a mathematician at
Moravian University, whose research focuses on the mathematical
analysis of periodic tilings
[[link removed]], especially
those by the Dutch artist M.C. Escher.

Dr. Senechal agreed. “It’s sitting right in the hexagons,” she
said. “How many people are going to be kicking themselves around the
world wondering, why didn’t I see that?”

The einstein family

Incredibly, Mr. Smith later found a second einstein. He called it
“the turtle” — a polykite made of not eight kites but 10. It was
“uncanny,” Dr. Kaplan said. He recalled feeling panicked; he was
already “neck deep in the hat.”

 

[The hat, left, and the turtle.]

The hat, left, and the turtle.Credit...David Smith

But Dr. Myers, who had done similar computations
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discovered a profound connection between the hat and the turtle. And
he discerned that, in fact, there was an entire family of related
einsteins — a continuous, uncountable infinity of shapes that morph
one to the next.

Mr. Smith wasn’t so impressed by some of the other family members.
“They looked a bit like impostors, or mutants,” he said.

But this einstein family motivated the second proof, which offers a
new tool for proving aperiodicity. The math seemed “too good to be
true,” Dr. Myers said in an email. “I wasn’t expecting such a
different approach to proving aperiodicity — but everything seemed
to hold together as I wrote up the details.”

Dr. Goodman-Strauss views the new technique as a crucial aspect of the
discovery; to date, there were only a handful of aperiodicity proofs.
He conceded it was “strong cheese,” perhaps only for hard-core
connoisseurs. It took him a couple of days to process. “Then I was
thunderstruck,” he said.

Mr. Smith was amazed to see the research paper come together. “I was
no help, to be honest.” He appreciated the illustrations, he said:
“I’m more of a pictures person.”

_A version of this article appears in print on March 28, 2023,
Section D, Page 8 of the New York edition with the headline: This
‘Einstein’ Was a Math Puzzle to Be Solved.  Subscribe
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