Lunch maths - fractals

My daughter and I were eating our lunch together one day at the end of last week, when suddenly I noticed that A. had stopped eating and was looking intensely at her food.
"Is everything all right, dear?" I asked.
"Mama, this is amazing," she replied.
"Well, I'm glad you like it" I said, pleased.
"No, well yes it IS delicious, but look, there's a fractal on my pickled cabbage!"

So what are fractals, you may be asking. This is how my daughter described them in her research project:

A fractal is a mathematical object of great complexity defined by simple algorithms (a collection of instructions or steps used to solve a problem or carry out a task.) It’s a semi geometrical object of which the basic structure repeats itself on different scales, partially or irregularly.

She continues to explain a little history:

Fractals first appeared because of the need to find a geometry which could describe natural objects.

The mathematics behind fractals was first studied during the 17th century by Gottfried Leibniz, who considered the recursive self-similarity. He was followed by Karl Weierstrass in 1872, who gave an example of a function which was everywhere continuous but nowhere differentiable; Helge von Koch, who in 1904 gave a more accurate geometric definition of a resembling function (the Koch curve or snowflake, a mathematical curve and one of the first fractal curves to have been described); in 1915 Waclaw Sierpinski constructed his triangle (one of the basic examples of self-similar sets, that is, a pattern that can be reproducible at any magnification or reduction) and one year later his carpet; Paul Pierre Lévy took the idea of self-similar curves further in 1938 with the Lévy C curve; and Georg Cantor explained the Cantor sets (constructed in the unit interval [0, 1] by deleting successive middle thirds of intervals) which are also now recognized as fractals.

In the 1960s, Benoît Mandelbrot, a mathematician born in Poland to a Jewish family who later went to live in France, started investigating self-similarity and studied fractals thoroughly. He illustrated the mathematical definition of fractal with striking computer-constructed visualizations. The term fractal comes from the Latin word “fractus,” which means “torn,” “fractured,” “irregular.”

Even today, in 2013, there is still no exact and generally accepted mathematical definition for the fractal concept.

Here are some of my favourite fractals, taken from Google images:

Here is a  link to a maths web which simplifies pretty much the whole concept, and if you have a look here, you will find 17 more fractals which appear in nature. Interesting stuff.