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.

Amazon's Under-the-Radar Coupon Page Features Deals on Home Goods, Electronics, and Groceries

Stock Catalog, Flickr // CC BY 2.0
Stock Catalog, Flickr // CC BY 2.0

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Now that Prime Day is over, and with Black Friday and Cyber Monday still a few weeks away, online deals may seem harder to come by. And while it can be a hassle to scour the internet for promo codes, buy-one-get-one deals, and flash sales, Amazon actually has an extensive coupon page you might not know about that features deals to look through every day.

As pointed out by People, the coupon page breaks deals down by categories, like electronics, home & kitchen, and groceries (the coupons even work with SNAP benefits). Since most of the deals revolve around the essentials, it's easy to stock up on items like Cottonelle toilet paper, Tide Pods, Cascade dishwasher detergent, and a 50 pack of surgical masks whenever you're running low.

But the low prices don't just stop at necessities. If you’re looking for the best deal on headphones, all you have to do is go to the electronics coupon page and it will bring up a deal on these COWIN E7 PRO noise-canceling headphones, which are now $80, thanks to a $10 coupon you could have missed.

Alternatively, if you are looking for deals on specific brands, you can search for their coupons from the page. So if you've had your eye on the Homall S-Racer gaming chair, you’ll find there's currently a coupon that saves you 5 percent, thanks to a simple search.

To discover all the deals you have been missing out on, head over to the Amazon Coupons page.

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Why Does the Supreme Court Have Nine Justices?

Front row, left to right: Stephen G. Breyer, Clarence Thomas, (Chief Justice) John G. Roberts, Jr., Ruth Bader Ginsburg, Samuel A. Alito. Back row: Neil M. Gorsuch, Sonia Sotomayor, Elena Kagan, Brett M. Kavanaugh.
Front row, left to right: Stephen G. Breyer, Clarence Thomas, (Chief Justice) John G. Roberts, Jr., Ruth Bader Ginsburg, Samuel A. Alito. Back row: Neil M. Gorsuch, Sonia Sotomayor, Elena Kagan, Brett M. Kavanaugh.
Fred Schilling, Collection of the Supreme Court of the United States // Public Domain

Some facets of the U.S. government—like presidential terms and post offices—were written into the original Constitution after (often lengthy) deliberations by the Founding Fathers. The number of Supreme Court justices was not one of those things.

The document did establish a Supreme Court, and it stated that the president should appoint its judges; it also mentioned that a “Chief Justice shall preside” if the president gets impeached. Since it was left up to Congress to work out the rest of the details, they passed the Judiciary Act of 1789, which outlined an entire court system and declared that the Supreme Court should comprise one chief justice and five associate justices. As History.com explains, they landed on six because the justices would have to preside over federal circuit courts, one of which was located in each state. Traveling wasn’t quick or easy in the horse-and-carriage days, so Congress wanted to minimize each justice’s jurisdiction. They split the courts into three regions, and assigned two justices to each region.

According to Maeva Marcus, director of the Institute for Constitutional History at George Washington University Law School, the even number of justices was a non-issue. “They never even thought about it, because all the judges were Federalists and they didn’t foresee great disagreement,” she told History.com. “Plus, you didn’t always have all six justices appearing at the Supreme Court for health and travel reasons.”

Over the next 80 years, the number of Supreme Court justices would fluctuate for two reasons: the addition of federal circuit courts, and presidents’ partisan motives. John Adams and his Federalist Congress reduced the number to five with the Judiciary Act of 1801, which they hoped would prevent Democratic-Republican Thomas Jefferson from getting to fill a seat after he took office that year. By the following year, Jefferson’s Congress had passed another judicial act that returned the number of justices to six, and they upped it to seven after forming another circuit court in 1807.

The nation grew significantly during the early 19th century, and Congress finally added two new circuit courts—and with them, two new Supreme Court seats—during Andrew Jackson’s presidential tenure in 1837. Republican Abraham Lincoln then briefly increased the number of justices to 10 in order to add another abolitionist vote, but Congress shrunk it to seven in 1866 to keep Andrew Johnson from filling seats with Democrats. As soon as Republican Ulysses S. Grant succeeded Johnson, Congress set the number back to nine, where it’s remained ever since.

Sketched portraits of the U.S. Supreme Court justices through 1897.Popular and Applied Graphic Art Print Filing Series, Library of Congress Prints and Photographs Division // No Known Restrictions on Publication

In 1911, Congress did away with circuit courts altogether, so the number of Supreme Court justices stopped being contingent upon their expansion (though each justice does still oversee a region to help with occasional tasks). As for presidents shifting the number to serve their own goals, it’s now looked down upon as “packing the court.” When Franklin D. Roosevelt tried to increase it to 15 in the 1930s to push his New Deal through the Supreme Court, the Senate opposed the bill by a whopping 70 to 20 votes.

In short, the depth of the Supreme Court’s bench changed a lot in America’s early years not only because the country was expanding, but also because the federal government was still testing out its system of checks and balances. And though presidents do still appoint justices based on their own political party, we’ve gotten used to the idea that the Supreme Court is, at least ideologically, supposed to be unbiased. If Congress and the president kept up the habit of adding and subtracting justices at will, it would tarnish this ideal.

“If Congress increases the size of the Supreme Court for transparently partisan political reasons, it would cement the idea the justices are little more than politicians in robes, and that the court is little more than an additional—and very powerful—arm through which partisan political power can be exercised,” Steve Vladeck, a professor at the University of Texas School of Law, wrote for NBC News. “Indeed, that Congress has not revisited the size of the court in 150 years is a powerful testament to just how ingrained the norm of nine has become—and how concerned different political constituencies have been at different times about preserving the court’s power.”

[h/t History.com]