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YouTube // Veritasium
YouTube // Veritasium

Testing 'Six Degrees of Separation'

YouTube // Veritasium
YouTube // Veritasium

Are all the people on earth really connected via six steps of social connection? If you imagine that each person has at least 44 unique friends, and each friend does, and so on for six degrees...you get to more than 7 billion people, which happens to be roughly the population of our planet. It's a cool theory, but does it hold up to testing?

A variant of the "six degrees" theory (called the "small world problem") was one of many tested by Stanley Milgram (paper here [PDF]) in the 1960s. Prior to Milgram's work, mathematician Paul Erdős worked out much of the math involved. More recent tests (many of which use modern social networking sites) turn up surprising evidence about the interconnectedness of humans around the globe.

In this video, Derek from Veritasium summarizes the science behind "Six Degrees of Separation." Check it out, and watch for a special guest star right around the 7-minute mark:

Strongly related: Kevin Bacon on 'Six Degrees of Kevin Bacon.'

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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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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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