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What are Fractals?

Snakes film by Cristobal Vila Fractals are fractional shapes.  The term was originally coined by Benoît Mandelbrot in 1975 although the idea of fractals goes back further in time.  Romanesco Broccoli M C Escher did a lot of fractal like works and there is a beautiful animation by Cristóbal Vila based on Escher's last work called Snakes at Etérea Fractals are geometric shapes which are produced by recursive iteration and feedback.  Fractals are self similar on every scale.  Trees, coastlines, clouds, lightening and Romanesco broccoli are all fractal in essence.
Fractals are beautiful.  The best way to describe the geometry of a fractal is with the Menger Sponge.  If you take a cube and slice it, like a layered cake, into three slices then you do the same on each face you are left with a cube made up of 27 cubes that are each 1/27 of the volume of the whole cube.  Now if you remove the middle cubes from each face and the centre cube you are left with a cube with a hole through it in three directions.  You have removed seven small cubes leaving 20 in the block.
Menger Sponge 4 cubes
If the volume of the original cube is 1 unit then the volume of the remaining cube is 20/27 or approximately 3/4 units.  Repeat the process for each of the 20 cubes left in the second block.  The result is that the volume of the third block is 20/27 of the second block or 400/729 or roughly 1/2 unit.  The fourth is about 4/10 and so on.  The volume is tending towards zero whilst the surface area is tending towards infinity.  It is said to have a Hausdorff dimension of approximately 2.7
The fractal aspect of the Menger Sponge is that it is a repeat of itself at all scales.  This is the most defining feature of fractals in general.

The 4 Menger Sponges illustration above was kindly provided by SolKoll via the Wikimedia Commons. Daniel White's page at skytopia has a lot of interesting images of fractals and the child inside the Menger Sponge was gleaned from Gayla Chandler's site with thanks to her and Paul Bourke for the original pic before Gayla put her granddaughter throwing a soft Menger in it.

Child in Menger Sponge

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