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.

Winston Churchill Once Got a Doctor's Note So That He Could Drink Alcohol in Prohibition-Era America

 Fox Photos/Getty Images
Fox Photos/Getty Images

Winston Churchill never went long without pouring himself a drink, even while traveling throughout Prohibition-era America. As producer and photographer Meredith Frost pointed out on Twitter recently, the future British prime minister and World War II leader got a doctor’s note in January 1932 which claimed he could drink an “indefinite” quantity of alcohol—federal laws be damned—to facilitate his “post-accident convalescence.” He had been struck by a car while on a speaking tour in New York in December 1931, which caused him chest pain in the immediate aftermath. He also suffered bouts of depression amid the aftershock, and it reportedly took him two months to fully recover.

Unfortunately for Churchill, Prohibition didn’t end until 1933. In fact, last week (December 5) marked the 85th anniversary of the repeal. He didn’t let that stop him, though. He admitted he once went to a speakeasy—"as a social investigator," of course.

This wasn’t the only time that Churchill refused to play by the rules insofar as alcohol was concerned. Once, after being told he shouldn’t drink or smoke during a meeting with a Muslim king, he replied through an interpreter, “My rule of life prescribed as an absolute sacred rite smoking cigars and also the drinking of alcohol before, after, and if need be during all meals and in the intervals between them.”

However, several historical accounts have argued that Churchill's drinking was for show and that he wasn’t actually an alcoholic. “It has been said that Winston used alcohol as a prop to his persona, rather like the cigars and pet bulldog, and that he rarely got monkey-arsed, or reached the falling-down, slurred-words state,” author Robert Sellers writes in An A-Z of Hellraisers: A Comprehensive Compendium of Outrageous Insobriety. “Total inebriation was something he abhorred, which says much for what must have been a steel constitution.”

The Christmas Book Flood: Iceland’s Literature-Loving Holiday Tradition

In Iceland, the most popular Christmas gifts aren't the latest iProducts or kitchen gadgets. They're books. Each year, Iceland celebrates what’s known as “Jólabókaflóðið:” the annual Yule Book Flood.

The holiday season is the Black Friday of the Icelandic publishing world—but it’s not just about one day. According to Reader’s Digest, at the beginning of November, each household in Iceland gets a copy of the Bokatidindi, the Iceland Publishers Association’s catalog of all the books that will be published that year, giving residents a chance to pick out holiday books for their friends and family. September to November marks Icelandic publishers’ biggest season, and many sell the majority of their yearly stock leading up to Christmas. Even grocery stores become major booksellers during the Book Flood season.

The Jólabókaflóðið (pronounced YO-la-bok-a-flothe) tradition dates back to post-World War II economic policies. Iceland separated from Denmark in 1918, and didn’t become a fully autonomous republic until 1944. During the Great Depression, the country created a rigid, intricate system of import restrictions, and its protectionist policies continued after the war. High inflation and strict rations on imported goods made it difficult for Icelanders to get their hands on many products. The one imported product that was relatively easy to get? Paper. As a result, books became the nation’s default gift purchase, and they still are, more than half a century later.

The "flood" in Christmas Book Flood has more to do with the deluge of books hitting bookstores than it does a flood of books flowing onto individual bookshelves. To take advantage of the tradition, most hardback books published in Iceland come out in the months leading up to Christmas, when Icelanders will be purchasing them for friends and family. (Cheaper paperbacks often come out a few months later, since people are more apt to buy those for themselves rather than their loved ones, according to The Reykjavik Grapevine’s Hildur Knútsdóttir.)

While family traditions vary from household to household, most Icelanders unwrap a book on December 24, according to Reader’s Digest. Some people get a book for every member of their family, while others do a swap exchange where everyone brings one title and everyone gets to pick one from the pile. After the exchange, many people cozy up with their new volume and get reading, preferably in bed, with chocolate.

As Icelandic writer Alda Sigmundsdóttir explained in a blog post in 2008, people in Iceland “will typically describe the pinnacle of enjoyment as lying in bed eating konfekt [filled chocolates] and reading one of the books they received under the tree. Later, at the slew of Christmas parties that inevitably follow, the Christmas books will be a prominent topic of conversation, and post-Yule the newspapers are filled with evaluations of which books had the best and worst titles, best and worst covers, etc.” Sounds like a pretty good tradition to us.

It’s not surprising that Iceland places such high importance on giving and receiving books. The country reads and publishes more books per capita than any other nation in the world, and one in 10 Icelanders have published a book themselves. (There’s an Icelandic adage, “ad ganga med bok I maganum,” that means “everyone gives birth to a book.” Well, technically it means “everyone has a book in their stomach,” but same idea.)

But the glut of books that flood the Icelandic market during the latter months of the year may not be as completely joyful as it sounds, some critics warn—at least not when it comes to the stability of the publishing market. Iceland is a nation of just 338,000 people, and there are more books than there are people to buy them. Some publishers, faced with a lack of space to store the unsold books, have had to resort to destroying unpurchased stock at the end of the holiday season. But marketing books outside of Yuletime is a relatively budding practice, one that Icelandic presses are still adapting to. It’s hard to beat the prospect of curling up after Christmas dinner with a freshly opened book and a bunch of chocolates, after all.