Beautiful math

The beauty can be very subjective. I may find chocolate glazed donuts beautiful, and for someone Boston cream donuts are simply irresistible. But the both of them are tasty, sweat and round. So, the both of us will agree that donuts are beautiful. We find beauty in small and big things, in material and spiritual things, in our day lives and in our night dreams. And, there are universal beauties, such as our World (a beautiful place to live), the creatures that live in it (among them - we humans) and mathematics.

I am not sure when we had met the beauty of mathematics, for the first time. Is it when we solved our first problem, or when we learned to count, or even when we chose the bigger toy to play with. However, it might happen that we become aware of this beauty much more later.

What makes us to see the beauty in mathematics? At the beginning, for sure, it is her simplicity and elegance. Lake a magic, the sentences "There are 5 apples on the table, and you take 3 apples. How many apples are on the table now?" turn into $$5 - 3 = 2.$$ Later, it is her power to turn a variety of concepts into one meaningful and solid idea, like

The sum of the angles in a triangle is equal to \(180^o\).

Then, there are plenty of helpful formulas, like

$$x_{1,2}=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$$ are solutions of the quadratic equation \(ax^2+bx+c=0\), \(a \ne 0\).

And many useful visualizations, like

Trigonometric circle
(image credit: globalspec.com) 

And many astonishing interpretations, like

The first derivative \(f'(x)\), if it exists, is the slope of the tangent line of \(f(x)\) at \(x\) i.e.

$$f'(x)=\lim_{\Delta x \rightarrow 0} \frac{\Delta y}{\Delta x} = \lim_{\Delta x \rightarrow 0} \frac{f(x+\Delta x)-f(x)}{\Delta x}.$$

Which can be illustrated as

The slope of the secant line PQ is the difference quotient \(\frac{\Delta y}{\Delta x}\),
and the slope of the tangent line at P is the rate of change
  of the sequence of slopes of the secant lines at P as \(\Delta x\) 
approaches to zero. (image credit: themathpage.com)

And many, many, many other beauties.

And then, when you think that you have learned mathematics enough to be able to apply it, you will enter a new room filled with surprising and unexplored beauties. You will realize the true meaning of the saying

"Mathematics is the queen and the slave of the sciences at the same time."

You will see that this new room has infinitely many doors, opened or closed, and you should not be surprised when some of the doors will return you at the very beginnings of some of the mathematical concepts, searching for the truth and only the truth. Then you are going to find the most delightful beauty of mathematics. The fact is that, the more you do mathematics, the more you realize how wonderful she is.

But you may have your own opinion about the beauty of mathematics?