What can you do with an Einstein?

It’s been a year of endless Einsteins. In March, a group of mathematical tilers announced they had discovered an “aperiodic monotile,” a shape that can tile an infinitely flat surface in a pattern that does not repeat itself — “einstein” is the geometric term of art for this entity. David Smith, a shape hobbyist in England who made the original discovery and investigated it with three collaborators with mathematical and computational expertise, called it “the hat.” (The hat tiling allows reflections: the hat-shaped tile and its mirror image.)

Now the results are in from a competition between the National Museum of Mathematics in New York and the United Kingdom Mathematics Trust in London, which asked the public for their most creative renditions of an Einstein. A jury assessed 245 entries from 32 countries. Three winners were chosen and a ceremony will take place at the House of Commons in London on Tuesday. (Each winner will receive a reward of £5,000; nine finalists will receive £1,000.)

Among the judges was Mr. Smith, who said in an email that he was “fascinated by the diversity and high standard of all the participants.”

What would you do with an einstein tile?

Play it

For finalist William Fry, 12, from New York, the answer was: Play Tetris, of course! He called his monotile variant of the game Montris. (Another participant had a similar idea, called Hatris.) His sister, Leslie Fry, 14, received an honorable mention for a collage inspired by Paddington Bear and his famous red hat.

Eat it

Evan Brock, 31, an exhibition designer in Toronto, won one of the three top prizes with his hat ravioli. It’s prepared with custom-made wooden molds and promises “a more geometric dining experience,” according to his entry.

Mr. Brock’s ravioli are filled with a potato-onion filling and are made of yellow (turmeric), orange (carrot) and red (beet) dough for non-reflected hat tiles; and green (spinach) dough for mirrored tiles. Other edible entries included hat biscuits and hat biscuits, hat sandwiches and hat dosas. “But these ravioli made us laugh,” Chaim Goodman-Strauss, one of Mr. Smith’s associates, a judge and an outreach mathematician at the National Museum of Mathematics, said in an email. “They look so delicious too.”

Give out

The finalist Sy Chen, 61, one origami artist in Rockville, Maryland, folded origami hat tiles of one dollar bills without cutting. Like Dr. Goodman-Strauss noted: “This origami construction shows unreflected hat tiles – and the reflected tiles by their absence!”

Fire it

Another, classical approach involved more than 1,500 handmade ceramic tiles assembled into a 7-meter-high frieze to decorate the storefront of a ceramics studio. It was designed by finalist Garnet Frost, 70, from London, a visual artist interested in architectural ornament (he is the subject of the documentary “Garnet’s Gold”), and the Alhambra tile projectan educational non-profit organization in Britain, with ceramicist Matthew Taylor and community volunteers.

Sew it

A hand-sewn quilted patchwork tapestry, 65cm high and 67cm wide, by Emma Laughton, 65, a retired gallery owner and finalist from Colyton, Devon, UK

As Ms Laughton explained in her competition entry: “The design aims to please the eye with a combination of elements of repetition and apparent (almost) symmetry, which contrasts with the unpredictable overall aperiodicity.”

Don’t turn it around

Shiying Dong, 41, of Greenwich, Conn., a homemaker with a doctorate in physics and a master’s degree in mathematics, folded a three-dimensional paper artwork, another winner. “I spend most of my time these days thinking about and making 3D things inspired by math,” she said in an email.

The creation of Dr. Dong used the chiral tile (1,1), a member of the hat tile family that does not need to be reflected to tile the plane.

“The tile was made chiral by placing a pyramid on top of it,” noted Dr. Dong on in her entry. The pyramid physically prevents anyone from flipping the tile, forcing a tile with no reflections.

Fly it

In the scholastic category, winner Devi Kuscer from London, 17, a student at UWC Atlantic College at St. Donat’s in Wales, made a large hat tile kite. Like Dr. As Goodman-Strauss described it, the kite is made from a hat, which is made from kites made from hats – “that’s really it.”

Carry it

For its inventiveness, the hat – actually more of a fascinator – from Nancy Clark, 11, of London, received an honorable mention and special admiration from Mr Smith.

Finalist’s paper hat artwork Pierre Broca33, graphic designer and teacher from Marseille, France.

All in all, Dr. Goodman-Strauss, who is also a professor at the University of Arkansas, said it was satisfying to see “ideas that animated someone’s career enter the popular imagination in this way.” People took the competition seriously, he said, “and made the hat their own – Dave’s discovery will live on into the future.”

The other finalists

The opening of an algorithmically generated environmental treatment, by Tadeas Martinat, 16, from St. Mellons, Cardiff, Wales.

Cookie Characters, by Mia Fan-Chiang, 14, from Abingdon, Oxfordshire, UK She said of her entry: “I chose to make some of these characters as cookies because, just as food is part of our everyday life, so is mathematics.” And she added in an email: “In addition, I wanted to use a fun format to show how creative math can be.”

Verity Langley, 16, from Harpsden, Oxfordshire, UK, created a moody light box. “It displays the hat tile design on your wall in many different, relaxing colors,” she said.

Julien Weiner, 17, of New Orleans, summoned a computer-generated Einsteinian succulent. “People ‘invent’ a new shape in the same way Sir Isaac Newton ‘invented’ gravity,” he said. “There was always an aperiodic monotile. My entry, titled The Einstein Bulb, imagines how, just as being unaware of gravity didn’t stop us from taking in the majesty of the rising moon in the night sky, the hat could exist in nature , waiting to be discovered and explored. ”

Mr. Weiner added, “The Einstein Competition really sparked a love for math that I haven’t felt in years, and reminded me that math doesn’t start or end in the classroom.”