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
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.
|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.