How did self-similarity come into mathematics? As far as I understand the interest in self-similarity was born outside of mathematics. The textbooks I came across give a few objects as examples (tree, broccoli, river, etc) yet it's clear that the examples serve purely didactic purpose. So how exactly did mathematicians become interested in self-similarity?
 A: Consider a regular pentagon with side length $s$ and diagonal length $d$. The ratio of the pentagonal diagonal length to side length is a constant we'll call $\phi=\frac{d}{s}$. The ancient Greek Pythagoreans originally thought that any number could be written as a fraction of two positive integers, and so naturally the assumed that the ratio $\phi$ must for some positive integers, say $p$ and $q$, be equal to:
$$\phi=\frac{p}{q}.$$
Start then with a pentagon having diagonal length $d=p$ and side length $s=q$. Now, logically there are two possible cases: either the integers $p$ and $q$ are both multiples of some common integer $a$, that is,
$$\begin{cases}\text{either }p=ma\text{ and }q=na,\\
\text{or }p\text{ and }q\text{ have no common factors other than }1.\end{cases}$$
Draw line segments connecting all five diagonals inside the pentagon. The five points where the diagonals intersect create a smaller pentagon nested inside the original pentagon. You can show just by using basic high school geometry that the length of this smaller pentagons diagonal is $d^\prime=p-q$, and it's side length is $s^\prime=2q-p$. Since the ratio diagonal to side is constant,
$$\frac{p}{q}=\frac{p-q}{2q-p}.$$
The result is that the side length of the smaller pentagon is an integer greater than or equal to one, and the side length of the larger pentagon is an integer multiple of the shorter side length. Here's where self-similarity comes into play. If you iterate the procedure of constructing ever smaller pentagons nested inside the starting pentagon ad infinitum (and the self-similarity of the construction implies there's no reason why we shouldn't be able to iterate it out to infinity), the argument above implies that the side length of each pentagon should always be an integer greater than or equal to one. But, if you divide a number by an integer greater than one enough times then you can eventually get a result as small as you desire (this is the Archimedean Principle in a nutshell). So we have generated a contradiction by infinite descent. Therefore, the number $\phi$ must be irrational. This number is more popularly known as the Golden Ratio.

Self-similarity arguments like these are how the Pythagoreans first discovered the existence of irrational numbers. It is unknown whether they discovered the irrationality of $\sqrt{2}$ or $\phi$ first, but regardless, once they had found these two irrational numbers, their view of mathematics was changed forever. You might say that geometric self-similarity was our first window into mathematical infinity, hence its importance.
A: Mathematics ultimately stems from the human impulse to explain, abstract, and analyze our world. Self similar objects appear everywhere in the world (your examples and many, many more: a quick google search for fractals in nature will include many), and we humans have been interested them since ancient times (look up "Apollonian Gasket" for a great example). As far as their mathematical properties, it is probably the paradoxical idea of "fractal dimension" that is most useful in application, but they are beautiful structures to study on an abstract level.
