The fractal (“FRACTional dimensioAL”) shape is perhaps the most wonderous manifestation of functional beauty arising from simple recursion. It is here that mathematics becomes intimate with nature to the point where often the eye
To see a world in a grain of sand
And a heaven in a wildflower
Hold infinity in the palkm of your hand
And eternity in an hour
– William Blake
What is a fractal? It is a fragmented geometric shape that can be split into parts in which one can see a reduced-size self-similar copy of the whole. The name fractal is due to Benoît Mandelbrot and comes Latin “fractus” meaning “broken”. Fractals have the following amazing properties: 1) they have fine structure at arbitrarily small scales, 2) they are too irregular to be easily described in traditional Euclidean geometric language, 3) they are self-similar (at least approximately or stochastically), they have a Hausdorff Dimension and 5) they have a simple and recursive definition.
If you want to create your own fractals I recommend the 3D strange attractor rendering software Chaoscope created and maintained by Nicolas Desprez. If you have an image like a cluster of galaxies or a cloud or a coastline or a fern that you want to calculate the fractal dimension of then you should try Fractalyse developed by the research team “City, mobility, territory” of the research centre ThéMA led by Pierre Frankhauser and Cécile Tannier. Fractalyse was initially developed to measure fractal dimension of built up areas of cities.
Natural Fractals
