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The Math Behind Fractals

How to make a fractal...

A fractal is a never-ending pattern. Fractals are infinitely complex patterns that are self-similar across different scales. They are created by repeating a simple process over and over in an ongoing feedback loop. Driven by recursion, fractals are images of dynamic systems – the pictures of Chaos.
In mathematics, a fractal is a subset of a Euclidean space for which the Hausdorff dimension strictly exceeds the topological dimension. Wikipedia
Mathematician Benoit Mandelbrot coined the term "fractal" in 1975 to describe a shape that appears similar at all levels of magnification. Surprisingly, fractals occur everywhere in nature - so did nature help mankind devise mathematics?

The following infomation is taken from the educational  Mathigon website.

 "The name “fractals” is derived from the fact that fractals don’t have a whole number dimension – they have a fractional dimension. Initially this may seem impossible – what do you mean by a dimension like 2.5 – but it becomes clear when we compare fractals with other shapes."  Below is a step by step, how to create two famous fractals: theSierpinski Gasket and the von Koch Snowflake.

Sierpinski Gasket von Koch Snowflake
To create the Sierpinski Gasket, start with a triangle and repeatedly cut out the centre of every segment. Notice how, after a while, every smaller triangle looks exactly the same as the whole. To create the von Koch Snowflake you also start with a triangle and repeatedly add a smaller triangle to every segment of its edge. After a while, the edge looks exactly the same at small and large scales.