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# The Secret of Floppy Paper

In this video, mathematician Tadashi Tokieda shows us a simple demo with surprising implications. He takes two squares of ordinary paper, clips them together, and shows how they either stick together or flop apart, depending on how they're held (basically upside down and right side up). The weird thing is that the pieces of paper change their behavior based on where you hold them. If you hold them on one edge, they flop. On the next edge, they stick. Next edge, flop again. Why?

Most forms of paper are anisotropic, meaning the material behaves differently when oriented in different directions. A shorthand term might be to say the paper has a grain. This makes sense, because wood has a grain—a direction in which the fibers run—so the paper does too, unless it's been intentionally broken up.

Knowing that materials can be anisotropic, what does that tell us? How does this apply to making paper books? And how can we use this knowledge to amaze children? Watch this video and learn. (Bonus points: You'll also learn about corrugation in this video, and why corrugated cardboard is so strong.)

More Numberphile videos featuring Tokieda: Can You See These Freaky Dot Patterns? and The Spinning Tube Trick.

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Math Symbols Might Look Complicated, But They Were Invented to Make Life Easier
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Numbers can be intimidating, especially for those of us who never quite mastered multiplication or tackled high-school trig. But the squiggly, straight, and angular symbols used in math have surprisingly basic origins.

For example, Robert Recorde, the 16th century Welsh mathematician who invented the “equal” sign, simply grew tired of constantly writing out the words “equal to.” To save time (and perhaps ease his writers’ cramp), he drew two parallel horizontal line segments, which he considered to be a pictorial representation of equality. Meanwhile, plenty of other symbols used in math are just Greek or Latin letters (instead of being some kind of secret code designed to torture students).

These symbols—and more—were all invented or adopted by academics who wanted to avoid redundancy or take a shortcut while tackling a math problem. Learn more about their history by watching TED-Ed’s video below.

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