The self-similarity is not the only simple relation because, again, it means that in the case of objects in space; the reductions go from the whole to the parts and are the same in all directions. The structure is isotropic. In the case of graphs of functions, for example of pressure as a function of temperature, or volume as a function of temperature, the volume and temperature play very different roles.
This is also true for mountain shapes.
For mountain shapes, the gravity has very strong direction, therefore the up and horizontal are the same. Under these conditions it seems natural to introduce transformation in which the horizontal reduction and the vertical reduction are different as you go from the whole to the parts, and those shapes are called self-affining. The word affinity was introduced by Euler. It's a terrible word, but Euler will not be overruled, so it's called affinity. Self-affinity actually is more general than affinity in some ways, but the main idea is still the same. So how can one construct a self-affining variability? Well, the procedure is very straightforward. The simplest case is you imagine a function with values for zero at time zero to one at time one. First approximation is a line. Second approximation, break the line, replace the line by a zigzag made, for example, of three intervals of arbitrary length. After that you reduce the zigzag in different ratios horizontally and vertically to fit each of the three intervals, and you replace each zigzag, each zig, zag and zig by a whole zigzag reduced. And so on and so on. The procedure is that of repetition, of recurrence; the same algorithm appears again and again. When such procedures were used by mathematicians around 1900 they were certainly not viewed as being in any way significant. It was just a way of writing a simple description. They never thought of computers, of course. But once computers came to be, that's the kind of thing computers adore. You just write your instruction, and say do it again and again. Which is why there is an uncanny profound attraction, if you will, between the computer with its particular usefulness for recursive constructions on the one hand, and the constructions of fractals which were introduced around 1900, of which there're about half a dozen. I call them pre-fractal, because they came well before fractal geometry and they were done for entirely different purposes.
