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.

See What It Was Like to Live in a Secret NYC Library Apartment


Ever wanted to live in a library? For the dozens of custodians who once helped take care of New York Public Library branches, that dream was a reality. Recently, Sarah Laskow of Atlas Obscura stepped into one of these now-vacant apartments in upper Manhattan and explored it in all of its creepy, dilapidated glory (think falling plaster and unsafe floors—there's a reason the space isn't usually open to the public). Since the branches no longer require live-in custodians to shovel the coal that once kept the furnaces humming, the apartments have all been closed down, and are slowly being converted into new public uses. In 2016, one custodian's apartment in Washington Heights was converted into a teen center and programming space. The secret apartment at the Fort Washington library will also eventually be converted—which means that Laskow's trip helped document a space that may soon be only a memory. You can see more inside the space, and learn more about the history of these apartments, in the video below.

The British Codebreaker Who Convinced the U.S. to Join WWI

National Archives
National Archives

On January 16, 1917, Nigel de Grey, a cryptologist working for the British military, intercepted a coded German telegram sent via standard diplomatic channels. This alone was nothing special. The British cryptanalytic office where de Grey worked, called Room 40, had cracked a handful of Germany's ciphers and intercepted their messages daily. Today, however, was different: The jumble of numbers revealed a political bombshell.

We intend to begin on the 1st of February unrestricted submarine warfare. We shall endeavor in spite of this to keep the United States of America neutral. In the event of this not succeeding, we make Mexico a proposal of alliance on the following basis: make war together, make peace together, generous financial support and an understanding on our part that Mexico is to reconquer the lost territory in Texas, New Mexico and Arizona.

The coded message, sent by the German Foreign Minister Arthur Zimmermann, was destined for the German Minister in Mexico City. (Along the way, it had passed through Germany's ambassador in Washington, D.C., Count Johann Heinrich von Bernstorff.) At the time, America was not involved in the Great War. In fact, President Woodrow Wilson had just secured a second term by riding the slogans "He Kept Us Out of War" and "American First." The decrypted message plainly indicated that Germany was hoping Wilson would stick to his campaign trail talking points.

But the message also showed that Germany was afraid. An escalation of submarine warfare could provoke the United States, compelling it to abandon its isolationist policies and enter the war. If that happened, Germany hoped to distract the U.S. by forcing American troops to focus on an enemy closer to home: Mexico.

Upon realizing the message's significance, de Grey immediately sprinted to the office of his superior, William Reginald "Blinker" Hall.

"Do you want America in the war, Sir?" he shouted.

Hall gave de Grey, who was gleaming with sweat, an incredulous look. "Yes, why?"

"I've got a telegram that will bring them in if you give it to them."

According to an exhibit at the National Cryptologic Museum in Fort Meade, Maryland, "[The British] realized that they held a cryptanalytic 'trump card' that virtually guaranteed America's entry into WWI on the Allied side." Historian David Kahn put it thusly: The "codebreakers held history in the palm of their hands."

When a select few in the U.S. federal government were finally notified of the secret message, many doubted its authenticity, believing it was just a deceitful ploy by the British to win American support. The British assuaged those doubts by acquiring a fresh copy of the coded telegram and handing it over to the Americans. On February 23, a U.S. diplomat saw the message be decrypted with his own eyes—again with the help of codebreaker Nigel de Grey—and independently verified Germany's intentions. The diplomat immediately contacted President Wilson.

When Wilson saw it, he was shocked and insulted. "Good Lord! Good Lord!" he shouted. About one week later, he leaked the message to the press. Americans were similarly outraged.

(As for Mexico, the country knew it was getting a raw deal and never took Germany's bait. Reclaiming the American southwest—what was formerly Mexican territory before the Mexican-American War in the 1840s—was a recipe for disaster. Besides, Germany would have never been able to help anyway: They were blocked by the British Navy.)

By early April, the secret code had compelled the U.S. to join Britain and its allies. Today, the work of de Grey and the other Room 40 codebreakers is widely considered one of the most consequential events in cryptologic history.


Hungry for more details about the Zimmermann telegram? Mental Floss’s coverage of the World War I Centennial has you covered here and here.