Fractals

A fractal is a never-ending pattern that repeats itself at different scales and is created by a simple process that repeats continuously.

Fractals in Nature

Romanesco Broccoli is an example of the ‘ultimate’ fractal vegetable. Its pattern represents the golden spiral. fractal_10

The golden spiral is a logarithmic spiral where every turn is farther from the origin by a factor of the golden ratio (phi).

golden spiral

Salt flats also contain remarkably consistent but random patterns created by the encrusted salt.

fractal_11a

Ammonites have been extinct for 65 million years and were predatory, squid-like creatures that lived inside coil-shaped shells. The walls between the chambers inside their spiral shells were complex fractal curves. Similarly to the romanesco broccoli, the shells of ammonites also grow as a logarithmic spiral.

fractal_1a

Mountains are formed from tectonic forces that push the crust upwards, joined with erosion, which breaks some of the crust down, resulting in a fractal pattern.

IDL TIFF file

Fractal patterns are also present in many plants, as they generate their branching shapes and leaf patterns through simple recursive formulas.

fractal_6a

When water crystallises it forms repeating patterns in snowflakes or on frosty surfaces. fractal_12a

These patterns inspired the first described fractal curves – the Koch snowflake – in a 1904 paper by Swedish mathematician Helge von Koch.

The first four iterations of the Koch snowflake

The first four iterations of the Koch snowflake

Finally, the path that lightning takes is formed step by step as it moves towards the ground and closely resembles a fractal pattern.

fractal_8a

Fractals in Mathematics

Mathematical fractals are formed by calculating a simple equation thousands of times, feeding the answer back to the start. The mathematical beauty of fractals is that infinite complexity – meaning we can zoom into them forever – is formed from relatively simple equations.

A very famous example of a mathematical fractal is the Mandelbrot Set, which was discovered by Benoit Mandelbrot in 1980.

The Mandelbrot set is a collection of numbers that are generated from the recurrence equation:

NumberedEquation1

Firstly we specify an initial value of z and c (a constant). We are looking for starting values of z, for which the sequence of numbers generated by the equation remains bounded. For example, if we start with z = -1 and c = 0, then the values will always be 1 or -1, so the sequence is bounded. Hence, -1 is included in our set of solutions.

When extending this to use complex numbers, the results become very interesting; we get the Mandelbrot Set:

Mandel_zoom_00_mandelbrot_set

In this picture, the black indicates numbers in the set, blue are numbers not in the set, and white is the boundary.

For more information, I suggest clicking here or here.

Additionally, Fractals are closely related to Chaos Theory (as they are complex systems that have definite properties) which is a subset of and area in mathematics called Dynamical Systems. These allow us to determine the general behaviour of solutions to systems of equations without actually solving the equations.

Let me know what you think of fractals! x

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