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Boring or Not?
Mathematics is one of the only areas of knowledge that can objectively be described as "true," because its theorems are derived from pure logic. And yet, at the same time, those theorems are often extremely strange and counterintuitive.
Some people find math boring. As these examples show, it's anything but.

Random Patterns
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Random Patterns
Weirdly, random data isn't actually all that random. In a given list of numbers representing anything from stock prices to city populations to the heights of buildings to the lengths of rivers, about 30 percent of the numbers will begin with the digit 1. Less of them will begin with 2, even less with 3, and so on, until only one number in twenty will begin with a 9. The bigger the data set, and the more orders of magnitude it spans, the more strongly this pattern emerges.
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Prime Spirals
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Prime Spirals
Because prime numbers are indivisible (except by 1 and themselves), and because all other numbers can be written as multiples of them, they are often regarded as the "atoms" of the math world. Despite their importance, the distribution of prime numbers among the integers is still a mystery. There is no pattern dictating which numbers will be prime or how far apart successive primes will be.
The seeming randomness of the primes makes the pattern found in "Ulam spirals" very strange indeed.
In 1963, the mathematician Stanislaw Ulam noticed an odd pattern while doodling in his notebook during a presentation: When integers are written in a spiral, prime numbers always seem to fall along diagonal lines. This in itself wasn't so surprising, because all prime numbers except for the number 2 are odd, and diagonal lines in integer spirals are alternately odd and even. Much more startling was the tendency of prime numbers to lie on some diagonals more than others — and this happens regardless of whether you start with 1 in the middle, or any other number.
Even when you zoom out to a much larger scale, as in the plot of hundreds of numbers below, you can see clear diagonal lines of primes (black dots), with some lines stronger than others. There are mathematical conjectures as to why this prime pattern emerges, but nothing has been proven.
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Sphere Eversion
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Sphere Eversion
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In an important field of mathematics called topology, two objects are considered to be equivalent, or "homeomorphic," if one can be morphed into the other by simply twisting and stretching its surface; they are different if you have to cut or crease the surface of one to reshape it into the form of the other.
Consider, for example, a torus — the dougnutshape object shown in the intro slide. If you turn it upright, widen one side and indent the top of that side, you then have a cylindrical object with a handle. Thus, a classic math joke is to say that topologists can't tell their doughnuts from their coffee cups.
On the other hand, Moebius bands — loops with a single twist in them — are not homeomorphic with twistfree loops (cylinders), because you can't take the twist out of a Moebius band without cutting it, flipping over one of the edges, and reattaching.
Topologists long wondered: Is a sphere homeomorphic with the insideout version of itself? In other words, can you turn a sphere inside out? At first it seems impossible, because you aren't allowed to poke a hole in the sphere and pull out the inside. But in fact, "sphere eversion," as it's called, is possible. Watch the video above to see how it's done.
Incredibly, the topologist Bernard Morin, a key developer of the complex method of sphere eversion shown here, was blind.
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Wall Math
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