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

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.

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History
Scientists Reveal Long-Hidden Text in Alexander Hamilton Letter
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iStock

Age, deterioration, and water damage are just a few of the reasons historians can be short on information that was once readily available on paper. Sometimes, it’s simply a case of missing pages. Other times, researchers can see “lost” text right under their noses.

One example: a letter written by Alexander Hamilton to his future wife, Elizabeth Schuyler, on September 6, 1780. On the surface, it looked very much like a rant about a Revolutionary War skirmish in Camden, South Carolina. But Hamilton scholars were excited by the 14 lines of writing in the first paragraph that had been crossed out. If they could be read, they might reveal some new dimension to one of the better-known Founding Fathers.

Using the practice of multispectral imaging—sometimes called hyperspectral imaging—conservationists at the Library of Congress were recently able to shine a new light on what someone had attempted to scrub out. In multispectral imaging, different wavelengths of light are “bounced” off the paper to reveal (or hide) different ink pigments. By examining a document through these different wavelengths, investigators can tune in to faded or obscured handwriting and make it visible to the naked eye.

A hyperspectral image of Alexander Hamilton's handwriting
Hyperspectral imaging of Hamilton's handwriting, from being obscured (top) to isolated and revealed (bottom).
Library of Congress

The text revealed a more emotional and romantic side to Hamilton, who had used the lines to woo Elizabeth. Technicians uncovered most of what he had written, with words in brackets still obscured and inferred:

Do you know my sensations when I see the
sweet characters from your hand? Yes you do,
by comparing [them] with your [own]
for my Betsey [loves] me and is [acquainted]
with all the joys of fondness. [Would] you
[exchange] them my dear for any other worthy
blessings? Is there any thing you would put
in competition[,] with one glowing [kiss] of
[unreadable], anticipate the delights we [unreadable]
in the unrestrained intercourses of wedded love,
and bet your heart joins mine in [fervent]
[wishes] to heaven that [all obstacles] and [interruptions]
May [be] speedily [removed].

Hamilton and Elizabeth Schuyler married on December 14, 1780. So why did Hamilton try and hide such romantic words during or after their courtship? He probably didn’t. Historians believe that his son, John Church Hamilton, crossed them out before publishing the letter as a part of a book of his father’s correspondence. He may have considered the passage a little too sexy for mass consumption.

[h/t Library of Congress]

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History
7 of History’s Most Unusual Riots
Scott Barbour/Getty Images
Scott Barbour/Getty Images

Some sociologists theorize that most rioters only join a crowd because the crowd is big enough to justify joining. But there’s always that one person who sparks the violence, and sometimes the reason for doing so can seem pretty baffling. Maybe a work of art scandalizes its audience, like the famous premiere of Igor Stravinsky’s The Rite of Spring. Or maybe it’s simply a notable act of disrespect, like history’s first recorded mooning (in Jerusalem in the first century CE). From balloonists to brown dogs to daylight saving time, here are seven weird reasons things just got out of hand.

1. THE MELBOURNE DART RIOT

The Darts Invitational Challenge, an international tournament held in Melbourne, attracted international gawking in January 2015 during the finals match between Michael "Mighty Mike" van Gerwen and Simon "The Wizard" Whitlock. The dart players weren’t making a scene, though: Rather, hundreds of spectators, many of them drunk and in costume, began throwing plastic chairs as they watched (pictured above). The reasons for the fight remain unclear; footage and photos show police trying to control adults dressed as Oompa-Loompas, numerous superheroes, and, in one instance, in a ghillie suit (heavy camouflage meant to resemble foliage).

2. THE LEICESTER BALLOON RIOT

In 1864, balloonists were the great daredevils of their time, and a major draw for eager audiences. That summer, Henry Coxwell, a famous professional aeronaut, was set to make an appearance for 50,000 paying ticketholders in Leicester, England. Unfortunately, a rumor spread that he hadn’t brought his biggest and best balloon to the event. After heckling from the crowd, Coxwell deflated his balloon, and attendees rushed it, ripping it to shreds, setting it on fire, and threatening to visit the same fate on Coxwell. Rioters even paraded the remains of the balloon through the streets of town, which briefly brought residents a new nickname: Balloonatics.

3. THE TORONTO CLOWN AND FIREFIGHTER RIOT

Toronto was still a pretty rough place in the 1850s, but not so rough that the circus wouldn’t come to town. As it turns out, circus entertainers were also a tough lot back then, so when a group of off-duty clowns spent an evening at a brothel popular with the city’s firefighters on July 12, 1855, tensions came to a head. Accounts differ as to who started the fight, but after one firefighter knocked the hat off a clown things escalated into a full-on rabble intent on chasing the circus out of town. Only the mayor calling in the militia put an end to the uproar, an incident Torontonians credit with kicking off much-needed local police reforms.

4. THE BELGIAN NIGHT AT THE OPERA RIOT

A painting by Charles Soubre of the Belgian Revolution
Charles Soubre, Wikimedia Commons // Public Domain

Not many nations can claim their independence started with an aria, but for 19th-century Belgians sick of living under Dutch rule, an opera was just the right fuse for a revolution. To honor the birthday of King William I of the Netherlands, a theater in Brussels put on La Muette de Portici, about an uprising in Naples against Spanish rule. One song, "Amour Sacre de la Patrie" ("Sacred Love of the Fatherland"), aroused nationalistic passions so much that after the opera ended, the crowd began destroying factories and occupying government buildings. That was August 25, 1830; Belgium declared independence on October 4.

5. THE NEW YORK DOCTORS' RIOT

Hamilton fans, take note: Everyone’s favorite Founding Father once tried to quiet a mob bent on burning corpses. For centuries, anatomists and medical students relied on gruesome means to learn about the human body. Cadavers for dissection class often came from grave robbers, since the corpses of executed criminals were the only legal source—and they were in limited supply. In New York in 1788, rumors abounded that medical students were digging up paupers’ graves and black cemeteries. When one mob came after the doctors responsible, Alexander Hamilton tried, and failed, to restore the peace. The crowd swelled to about 5000 before militiamen intervened, leading to up to about 20 deaths.

6. THE BROWN DOG RIOTS

Photo of an anti-vivisection demonstration in Trafalgar Square, London, to protest the removal from Battersea Park of the Brown Dog statue
The Anti-Vivisection Review, Wikimedia Commons // Public Domain

Riots against the dissection of dead human bodies were not rare in the United States at one time. But on December 10, 1907, a thousand Britons marched in support of vivisection, or surgery on live animals. At the center of the controversy was a small terrier allegedly vivisected without anesthetic in 1903 during a class at London’s University College. Animal rights activists erected a statue to the dog in 1906, which enraged area medical students, and protesters tried to destroy the statue using crowbars and hammers. For the 1907 march, 400 mounted police were deployed to contain marchers. The statue became such a flashpoint (and an expense to local authorities) that in 1910, it was removed and melted down.

7. THE EEL-PULLING RIOT

Palingtrekken (eel-pulling) was once a popular contest in Amsterdam, in which a writhing eel was suspended over a canal and hopefuls on boats would leap to snatch it as they passed beneath (usually landing in the water instead). However, “eel-pulling” was also illegal—the government deemed it a “cruel popular entertainment”—and in July 1886, police intervened at a particularly large gathering in the city’s Jordaan district. Civilians threw stones and bricks at police, and when some nearby socialist protestors joined them, a riot broke out that lasted for several days. The army finally intervened and opened fire on the protestors. All in all, 26 people died and 136 were wounded, but somehow, the eel itself at the center of the riots was allegedly saved and auctioned off in 1913.

A version of this story originally ran in 2015.

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