It’s been a year of endless einsteins. In March, a troupe of mathematical tilers announced that they had discovered an “aperiodic monotile,” a shape that can tile an infinite flat surface in a pattern that does not repeat — “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 possessing mathematical and computational expertise, nicknamed it “the hat.” (The hat tiling allows for reflections: the hat-shaped tile and its mirror image.)

Now, the results are in from a contest run by the National Museum of Mathematics in New York and the United Kingdom Mathematics Trust in London, which asked members of the public for their most creative renditions of an einstein. A panel of judges assessed 245 submissions from 32 countries. Three winners were chosen, and, on Tuesday, there will be a ceremony at the House of Commons in London. (Each winner receives an award of 5,000 British pounds; nine finalists receive 1,000 pounds.)

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

What would you do with an einstein tile?

Play it

For the finalist William Fry, 12, of New York, the answer was: Play Tetris, of course! He named his monotile variant of the game Montris. (Another entrant 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 exhibit designer in Toronto, took one of the three top prizes with his hat ravioli. Prepared with bespoke wood molds, it promises “a more geometric dining experience,” his submission notes.

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

Spend it

The finalist Sy Chen, 61, an origami artist in Rockville, Md., folded origami hat tiles from one-dollar bills without cutting. As Dr. Goodman-Strauss observed: “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 24-foot frieze to decorate the storefront of a ceramics workshop. It was designed by the finalist Garnet Frost, 70, of London, a visual artist who has an interest in architectural ornament (he is the subject of the documentary “Garnet’s Gold”), and the Alhambra Tiling Project, an educational nonprofit in the U.K., with the ceramist Matthew Taylor and community volunteers.

Stitch it

A hand-sewn quilted patchwork wall hanging, 25 inches tall and 27 inches wide, by Emma Laughton, 65, a retired gallery owner and a finalist from Colyton, Devon, U.K.

As Mrs. Laughton explained in her contest submission: “The design aims to please the eye with a combination of elements of repetition and apparent (near) symmetry, contrasting with the unpredictable overall aperiodicity.”

Flip it not

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 3-D things inspired by math,” she said in an email.

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

“The tile is made chiral by having a pyramid on top,” Dr. Dong noted in her submission. The pyramid physically prevents one from flipping the tile over, thus forcing a tiling without reflections.

Fly it

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

Wear it

For its inventiveness, the hat-cluster hat — really more of a fascinator — by Nancy Clark, 11, of London, won an honorable mention and special admiration from Mr. Smith.

Paper-hat artwork by the finalist Pierre Broca, 33, a graphic designer and teacher from Marseille, France.

All in all, Dr. Goodman-Strauss, who is also a professor at the University of Arkansas, found it satisfying to witness “ideas animating one’s career take off like this into the popular imagination.” People took the competition seriously, he said, “and made the hat their own — Dave’s discovery is going to live on and on into the future.”

The remaining finalists

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

Cookie characters, by Mia Fan-Chiang, 14, of Abingdon, Oxfordshire, U.K. She said of her submission: “I chose to make a few of these characters as biscuits because, just like food is part of our daily lives, so is maths.” And she added in an email: “As well as that, I wanted to use a fun format to demonstrate just how creative maths can be.”

Verity Langley, 16, from Harpsden, Oxfordshire, U.K., manufactured 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. “Humans ‘invent’ a new shape in the same way Sir Isaac Newton ‘invented’ gravity,” he said. “An aperiodic monotile was always there. My submission, titled The Einstein Bulb, imagines how, just as being unaware of gravity didn’t stop us from taking in the majesty of the moon rising in the night sky, the hat could exist in nature just waiting to be discovered and explored.”

Mr. Weiner added: “The einstein contest truly reignited a love for math that I haven’t felt in years and reminded me that mathematics does not start nor end in the classroom.”