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Mathematics is, without any doubt, one of the only areas of knowledge that can accurately be defined as "true," because its theorems are result of pure logic. And yet, at the same time, those theorems are often exceptionally extraordinary and counter-intuitive.

“The Most Beautiful Equation”
Stanford mathematician Keith Devlin the called Euler's formula "The Most Beautiful Equation." But why is Euler's formula so magnificent? First, the letter in the equation "e" signifies an irrational number (with endless digits) that begins 2.71828... Discovered in the situation of nonstop compounded interest, it directs the rate of exponential growth, from that of a tiny insect populations to the accumulation of curiosity to radioactive decay. In math, the number shows some very astonishing properties, for example, to use math expressions, being equal to the sum of the inverse of all factorials from 0 to infinity. Indeed, the constant "e" pervades math, acting apparently from nowhere in an immense number of important equations.

Next, "i" signifies the so-called "imaginary number": the square root of negative 1 also known as “iota”. It is thus called because, in actuality, there is no such number which can be multiplied by itself to yield a negative number (and so negative numbers have no actual square roots). But in mathematics, there are numerous circumstances where one is required to take the square root of a negative. The letter "i" is hence used as a kind of stand-in to mark places where this was done. Next thing in the equation is Pi and almost everybody knows about it, the ratio of a circle's circumference to its diameter. Pi is one of the best-loved and most remarkable numbers in mathematics. Like "e," it seems to rapidly rise in a huge number of mathematics and physics formulas.

Lastly, the constant "e" raised to the power of the iota "i" multiplied by pi is equal to -1. And, as seen in Euler's equation, addition of 1 to that gives 0. It appears nearly unbelievable that all these weird numbers — and even one that isn't genuine — would chain so easily. But, as a matter of fact, it's a proven fact.

Wall Math
Though they might be decorated with an endless variation of flourishes, mathematically speaking, there's just a limited number of separate geometric patterns. All Escher paintings, wallpapers, tile designs and certainly all two-dimensional, iterating arrangements of shapes can be recognized as fit in to one or another of the so-called "wallpaper groups." And there are only 17 wallpapers group.

Prime Spirals
Since prime numbers are indivisible (except by digit 1 and themselves), and because all other numbers can be inscribed as multiples of them, they are frequently viewed as the "atoms" of the math world. Regardless of their importance, the scattering of prime numbers amongst the integers is still a mystery. There is no outline dictating which numbers will be prime or how far apart consecutive primes will be.

The apparent randomness of the primes creates the pattern found in "Ulam spirals" very odd indeed. Stanislaw Ulam in 1963 observed an odd arrangement while sketching in his notebook during a presentation: When integers are written in a spiral, prime numbers permanently appear to fall along diagonal lines. This in itself wasn't so astonishing, because all prime numbers excluding the number 2 are odd, and crosswise lines in integer spirals are consecutively odd and even. Much more surprising was the leaning of prime numbers to lie on few diagonals more than others — and this occurs regardless of whether you start with 1 in the mid, or any other number.
Even when you zoom out to a quite bigger scale, as in the scheme of hundreds of numbers below, you can see perfect diagonal lines of primes (black dots), with some lines stronger than others. There are mathematical estimations as to why this prime outline arises, but nothing has been verified.

Random Patterns
Oddly, random data isn't really random at all. In a given list of numbers signifying anything from stock prices to town or a city populations to the elevations of buildings to the lengths of rivers, about 30 percent of the numbers will start with the digit 1. Less of them will start with 2, even less will start with 3, and so on, till only one number in twenty will start with a 9. The larger the data set, and the more orders of magnitude it extents, the more intensely this form appears.

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