The Homemaker Who Helped Solve One of Geometry's Oldest Puzzles

V via Flickr // CC BY-NC 2.0
V via Flickr // CC BY-NC 2.0

The next time you find yourself staring at your bathroom floor tiles, thank Marjorie Rice. The San Diego homemaker helped solve one of the oldest problems in geometry: figuring out which shapes could "tile the plane," or seamlessly cover a flat surface in an endless, repeating pattern. Rice's hand-drawn doodles in the 1970s led to major discoveries in the last few years, finally answering the puzzle that had stumped classical thinkers.

Ancient Greek mathematicians believed that certain shapes could tile the plane, without overlapping or leaving any gaps, in a pattern called a tessellation. They proved that all triangles and quadrilaterals, and some convex hexagons (six-sided shapes), could tile the plane. But for centuries, no one knew how many tiling convex pentagons (irregular five-sided shapes) were out there.

The hunt for tiling pentagons began in 1918 when German mathematician Karl Reinhardt described the first five types of tessellating pentagons. For 50 years it was believed that he had found them all, but in 1968, physicist R. B. Kershner discovered three more classes. Richard James, a computer scientist in California, found another in 1975, bringing the total to nine.

That year, Rice read a column by Martin Gardner in Scientific American about the research and began experimenting to find more tiling pentagons. "I became fascinated with the subject and wanted to understand what made each type unique," Rice wrote in an essay about M.C. Escher's use of repeating patterns. "Lacking a mathematical background, I developed my own notation system and in a few months discovered a new type which I sent to Martin Gardner. He sent it to Doris Schattschneider to determine if it truly was a new type, and indeed it was."

Schattschneider, a mathematics professor at Moravian College in Bethlehem, Pennsylvania, deciphered Rice's notation and realized she had found four new types—more than anyone other than Reinhardt. Schattschneider helped formally announce Rice's discoveries in 1977.

"My dad had no idea what my mom was doing and discovering," her daughter Kathy Rice told Quanta Magazine.

It took another eight years for the next type of tiling pentagon to be found, this time by University of Dortmund mathematician Rolf Stein. Then the trail went cold for 30 years.

In 2015, mathematicians Jennifer McLoud-Mann, Casey Mann, and David von Derau at the University of Washington, Bothell, found the 15th class of tessellating pentagon using a supercomputer. Then, in July 2017, French mathematician Michaël Rao completed the classification of all convex polygons, including pentagons, that can tile the plane. He confirmed that only the 15 known convex pentagons could tessellate [PDF].

The immense amount of research and the scale of the recent discoveries makes the achievements of Marjorie Rice all the more impressive. Though she lacked more than a high-school education and access to supercomputers, Rice remains the most prolific discoverer of tiling pentagons to emerge in the century since Reinhardt first attempted to crack the problem.

11 Masks That Will Keep You Safe and Stylish

Design Safe/Designer Face Covers/Its All Goods
Design Safe/Designer Face Covers/Its All Goods

Face masks are going to be the norm for the foreseeable future, and with that in mind, designers and manufacturers have answered the call by providing options that are tailored for different lifestyles and fashion tastes. Almost every mask below is on sale, so you can find one that fits your needs without overspending.

1. Multicolor 5-pack of Polyester Face Masks; $22 (56 percent off)

Home Essentials

This set of five polyester masks offers the protection you need in a range of colors, so you can coordinate with whatever outfit you're wearing.

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Brio

The breathable, stretchy fabric in these 3D masks makes them a comfortable option for daily use.

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This cotton mask pack is washable and comfortable. Use the two as a matching set with your best friend or significant other, or keep the spare for laundry day.

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RipleyRader

Don’t let masks get in the way of staying active. These double-layer cotton masks are breathable but still protect against those airborne particles.

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Avoid the accidental nose-out look with this cotton mask that stays snug to your face.

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Elicto

With this 12-pack of protective masks, you can keep a few back-ups in your car and hand the rest out to friends and family who need them.

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7. Reusable Dust-Proof Mask with 5 Filters; $22 (45 percent off)

Triple Grade

This dust-proof mask can filter out 95 percent of germs and other particles, making it a great option for anyone working around smoke and debris all day, or even if you're just outside mowing the lawn.

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8. Reusable Fun Face Cover / Neck Gaiter (Flamingo); $20

Designer Face Covers

Channel some tropical energy with this flamingo fabric neck gaiter. The style of this covering resembles a bandana, which could save your ears and head from soreness from elastic loops. Other designs include a Bauhaus-inspired mask and this retro look.

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9. Seamless Bandana Mask; $8 (52 percent off)

Eargasm Earplugs

This seamless gaiter-style mask can be worn properly for protection and fashioned up into a headband once you're in the car or a safe space. Plus, having your hair out of your face will help you avoid touching your eyes, nose, and mouth before washing your hands.

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Design Safe

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Its All Good

This mask will definitely come in handy once winter rolls around. It features a fleece neck, face, and ear covering to keep your mask secure and your face warm.

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Prices subject to change.

This article contains affiliate links to products selected by our editors. Mental Floss may receive a commission for purchases made through these links. If you haven't received your voucher or have a question about your order, contact the Mental Floss shop here.

10 Fascinating Facts About Herman Melville

Hulton Archive/Getty Images
Hulton Archive/Getty Images

Born in New York City to a wealthy and socially connected family, Herman Melville (1819-1891) chose a life as exciting as that of his Moby-Dick narrator Ishmael. He spent years at sea on whaling ships and traveled to far-flung places, but also struggled to make it as a novelist while supporting a large extended family. To celebrate his birthday on August 1, we’re diving into Melville’s adventures and fishing for some surprising facts.

1. Herman Melville's mother changed the spelling of their last name.

Despite his family’s wealth and pedigree—his mother Maria Gansevoort descended from one of the first Dutch families in New York, and his father Allan Melvill came from old Boston stock—young Herman had an unstable, unhappy childhood. Allan declared bankruptcy in 1830 and died two years later, leaving Maria with eight children under the age of 17 and a pile of debt from loans and Allan’s unsuccessful businesses. Soon afterward, Maria added an "e" to their surname—perhaps to hide from collection agencies, although scholars are not sure exactly why. "It always seemed to me an unlikely way to avoid creditors in the early 19th century," Will Garrison, executive director of the Berkshire Historical Society, tells Mental Floss.

2. Herman Melville struggled to find employment.

Thanks to a national financial crisis in 1837, Melville had difficulty finding a permanent job, but it wasn’t for lack of trying. He served as a bank clerk, teacher, land surveyor, and crew member on a packet ship before signing on, in 1841, to the whaler Acushnet of New Bedford, Massachusetts, then the whaling capital of the world. He served aboard a few different whalers and rose to the role of harpooner. His adventures at sea planted the seeds for Melville’s interrogation of man, morality, and nature in Moby-Dick. In that novel, Melville (in the voice of Ishmael) says, "A whale-ship was my Yale College and my Harvard."

3. Herman Melville jumped ship in the middle of a three-year voyage. 

Melville and the Acushnet’s captain didn’t get along, so when the ship reached the Marquesas Islands, Melville and a friend, Richard Tobias Greene, hid in the forests until the ship departed. They spent a month living with the Pacific Islanders. Melville was impressed with their sophistication and peacefulness; most Europeans believed that Polynesians were cannibals. He also found reason to criticize European attempts to "civilize" the islanders by converting them to Christianity. Melville drew on his South Pacific experiences in his first two novels, which became runaway bestsellers: Typee (1846) and Omoo (1847).

4. Herman Melville was inspired by a mountain.

Herman Melville's home, Arrowhead, in Pittsfield, MassachusettsDaderot/Wikimedia Commons/CC BY-SA 3.0

Melville moved to Arrowhead, his charming mustard-colored home in Pittsfield, Massachusetts, with his wife Elizabeth and their son in 1850, after he achieved fame as a popular adventure novelist. In the upstairs study, he set up his writing desk so he could look out the north-facing window, which perfectly framed the summit of Mount Greylock, Massachusetts’s tallest mountain. Gazing at the peak on a sunny day, Melville was struck by how much the horizontal apex looked "like a sperm whale rising in the distance." He arranged his desk so he would see the summit when he happened to glance up from his work. In that room, in early 1851, Melville completed his manuscript of Moby-Dick.

5. Herman Melville fictionalized an actual whaling disaster.

While on the Acushnet, Melville had learned about an infamous shipwreck from the son of one of its survivors. In November 1820, a massive sperm whale had attacked and sunk the whaleship Essex of Nantucket in the middle of the Pacific Ocean. Its crew, stranded in three small boats with little food or water, chose to drift more than 4000 miles to South America instead of 1200 miles to the Marquesas Islands—where Melville had enjoyed his idyll—because they thought they’d be eaten by the natives. Ironically, some of the castaways ended up eating their dead shipmates to survive.

Melville used the disaster to form the climax of Moby-Dick, in which the Pequod of Nantucket is destroyed by the white whale. Melville visited Nantucket for the first time only after the novel was published. He personally interviewed the Essex’s captain, George Pollard, who had survived the terrible ordeal and become the town’s night watchman. Later, Melville wrote, "To the islanders he was a nobody—to me, the most impressive man, tho’ wholly unassuming, even humble—that I ever encountered."

6. Moby-Dick was a flop.

Readers who were expecting another rip-roarin’ adventure like his earlier novels Typee or Redburn were sorely disappointed when Melville’s masterpiece was published in November 1851. The British edition of Moby-Dick, or The Whale received some positive reviews in London newspapers, but American reviewers were shocked at its obscure literary symbolism and complexity. “There is no method in his madness; and we must needs pronounce the chief feature of the volume [the character of Captain Ahab] a perfect failure, and the work itself inartistic,” wrote the New York Albion. The reviewer added that the novel's style was like "having oil, mustard, vinegar, and pepper served up as a dish, in place of being scientifically administered sauce-wise."

7. Herman Melville was very fond of his chimney.

Arrowhead became the locus of Melville’s family life and work. Eventually, he and Lizzie, their two sons and two daughters, his mother Maria, and his sisters Augusta, Helen, and Fanny all lived in the cozy farmhouse. For a couple of years, Nathaniel Hawthorne was such a frequent guest that he had his own small bedroom off Melville’s study. After Moby-Dick, Melville wrote the novels Pierre and The Confidence-Man, his collection of works called The Piazza Tales, short stories including “Bartleby the Scrivener,” and many other pieces there. Melville grew very attached to the house, especially to the massive central chimney, which he immortalized in his 1856 short story “I and My Chimney.” Yet his financial struggles after Moby-Dick failed to find an audience led Melville to sell Arrowhead to his brother Allan in 1863. As an homage, Allan painted a few lines from “I and My Chimney” on the chimney's stonework, which are still visible today.

8. Herman Melville finally got a day job.

Melville’s chronic money woes prompted a return to New York City, into a brick townhouse at 104 East 26th Street in Manhattan, where the family benefited from being back in the bustle of civilization. Melville finally found regular employment as a district inspector for the U.S. Customs Service and maintained an office at 470 West Street. At the same time, he mostly abandoned writing short stories and novels in favor of poetry. In between inspections he wrote Clarel: A Poem and Pilgrimage in the Holy Land, based on his visit to the Middle East in 1857. Because of its length—at more than 18,000 lines, it's the longest poem in American literature—and unconventional approach to its subject, Melville once called it "eminently adapted for unpopularity."

9. Herman Melville's last major work was discovered by accident.

The centennial of Melville’s birth renewed interest in his novels and poems, most of which were long out of print by then. Raymond Weaver, a literature professor at Columbia University working on the first major biography of Melville, collaborated with Eleanor Melville Metcalf, Melville’s granddaughter and literary executor, who gave him access to the author’s papers. In 1919, while poking through letters and notes, Weaver discovered the unfinished manuscript of Billy Budd in a tin breadbox. Melville had started to write the short story about a tragic sailor in 1888 but, by his death in 1891, had not completed it. Weaver edited and published the story in 1924, but initially considered the tale "not distinguished." Other scholars asserted that Billy Budd was Melville’s final masterpiece.

10. You can see Herman Melville's personal collection of knick-knacks.

Just a short drive from Arrowhead, the Berkshire Athenaeum in Pittsfield holds the world’s largest collection of Melvilliana in its Melville Memorial Room. Along with first editions of Melville’s work and a full library of books about him, there are priceless objects owned by or associated with the author. Fans can geek out over the earliest known portrait of Melville, painted in 1848; carved wooden canoe paddles that he collected in Polynesia; his walking stick; his favorite inkstand, quills, and other desktop tchotchkes; a collection of scrimshaw, maps, and prints; and Elizabeth Melville’s writing desk. There's a section of the first successful transatlantic cable, which Melville valued as a prized souvenir, and even the actual breadbox in which Billy Budd had been hiding.