From Snowflakes To Galaxies To Shrek…

Spot The Squiggle In Mathematical Models!!!

The Order and Chaos of Fractals

In apparent chaos we find the complex mathematical models of fractal geometry. Fractals repeat simple, basic geometric shapes – Triangles, Squares, Circles and of course starting or ending with….. The Squiggle. Fractal  patterns create infinite variations and forms. We see fractals at work in cloud formations, eroded land edges, spidery blood vessels, lightening forks, crystal growth, the formation of galaxies and more obviously in plants such as the broccoli.

Animated films (or cartoons) were revolutionised once the mathematics of fractals had been modelled on computers. Now undulating grass, moving clouds, fur, feathers, realistic clothing, trees, skin, hair….. could all be animated quickly without the need for vast numbers of people drawing image after image. A good example from the time fractal models made an impact can be seen in ‘Shrek’ (Dreamworks).

shrek

How are fractals generated?

2000px-KochFlake

A simple example is the Koch Snowflake.

A triangular squiggle is made in each straight line, in proportion to the length of the line. Then a smaller triangular squiggle scaled accordingly is made on every straight line.

This process is repeated again and again…infinitely.

 

 

Here is an example of recursive drawing that demonstrates how expansive simple fractals are:

kidney-tree

Finally, let’s look at some of the history fractal mathematical models:

Sierpinski Carpet – is a plane fractal first described by Wacław Sierpiński in 1916.

SierpinskiCarpet_730

Julia Set – 1918 – French mathematician Gaston Julia. The first to create a complex mathematical model on the generation of geometric patterns. His work was popularised by Mandelbrot.FMF_Julia8_b&w

Menger Sponge – 1926 – Karl Menger – This is essentially a 3D version of the Sierpinski Carpet.

sponges-l

Let’s take a trip inside a Menger Sponge!

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Mandelbrot Set – March 1980 –  Benoit B. Mandelbrot, a mathematician at the IBM Thomas J. Watson Research Centre.

The MOST COMPLEX mathematical object ever!

CLICK HERE to go a deep Mandelbrot zoom……

Mandel_zoom_00_mandelbrot_set

 

 

 

 

 

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