Self-Similarity

Some examples

Self-Similarity in the Koch Curve

Fractals usually possess what is called self-similarity across scales.

That is, as one zooms in or out the geometry/ image has a similar (sometimes exact)

appearance.

Self-Similarity in the Mandelbrot Set

Self-similarity in Nature

Natural: A picture of a fernHow do plants do it?

Mathematical: plant growth as simulated by a simple iterative procedure.

Is this a large piece of rugged terrain photographed from an aeroplane, or the side of a mountain, or a patch of dirt on the scale of a few meters, or a magnification of the surface of a rough rock?

Whichever it is, it could also easily be imagined to be any one of the others. So one could start at the large scale view from the air and apply successive zooms down to a microscopic scale, the surface maintains self similarity across those scales.