# Beautiful Mathematics

While reading an article on the General Unified Theory, I came across a reference to what the author considered beautiful mathematical equations. It got me thinking: what would I consider beautiful math? I must confess right away, that it has been many years since I have done anything remotely advanced in the area of mathematics and not since my master thesis on the mathematical properties of artificial neural networks – drop me a line if you are unable to fall asleep, and I will send you a copy – have I really done any heavy math.

However, the seed was laid in my mind and during my run early this morning I began to consider if I could actually remember any mathematical equations, beautiful or not.

The first one that came to mind was Archimedes’ Recurrence Formula. Being a diver, I have always had a soft spot for old Archi 🙂

Let $a_n$ and $b_n$ be the perimeters of the circumscribed and inscribed n-gon and $a_{2n}$ and $b_{2n}$ the perimeters of the circumscribed and inscribed 2n-gon. Then

$\displaystyle{a_{2n} = \frac{2a_nb_n}{a_nb_n}, b_{2n} = \sqrt{a_{2n}b_n}, a_\infty = b_\infty}$

Of course the cool thing is that the successive application gives the Archimedes algorithm, which can be used to provide successive approximations to $\pi$.

The next equation that came to my mind was Eulers Formula or Eulers Identity, which states that

$\displaystyle{e^{ix} = \cos x + i \sin x}$

where $i$ is the imaginary unit. The beautiful equation arises when setting $x = \pi$,

$\displaystyle{e^{i\pi} + 1 = 0}$

The term Mandelbrot set is used to refer both to a general class of fractal sets and to a particular instance of such a set. In general, a Mandelbrot set marks the set of points in the complex plane such that the corresponding Julia set is connected and not computable.

“The” Mandelbrot set is the set obtained from the quadratic recurrence equation

$\displaystyle{z_{n+1} + z_n^2 + C, z_0 = C}$

where points $C$ in the complex plane for which the orbit of $z_n$ does not tend to infinity are in the set.

The image below displays the famous and well known Valley of the Sea Horses.

The next one that came to my mind was the Riemann Hypothesis.

The Riemann hypothesis is a deep mathematical conjecture which states that the non-trivial Riemann zeta function zeros, i.e., the values of $s$ other than $-2, -4, -6, \ldots$ such that $\zeta(s) = 0$ (where $\zeta(s)$ is the Riemann zeta function) all lie on the “critical line” $\sigma = \mathbb R [s] = 1/2$  (where $\mathbb R[s]$ denotes the real part of $s$).

The Riemann zeta-function $\zeta(s)$ is the function of a complex variable $s$ initially defined by the following infinite series:

$\displaystyle{\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}}$

The Riemann hypothesis can be stated as:

$\displaystyle{\zeta(\alpha + i\beta) = 0, \beta \not= 0 \Rightarrow \alpha = \frac{1}{2}}$

One could go on and on, but I will finish off with another of my favorites, namely the Gaussian Integral.

The Gaussian integral, also called the probability integral is the integral of the one-dimensional Gaussian function over $(-\infty, \infty)$

$\displaystyle{\int_{-\infty}^\infty e^{-x^2}dx = \sqrt{\pi}}$