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.