*"Through a Fractal on a Breaking Wall.."*

## What is chaos theory?

The beautiful images scattered throughout this website are made using fractal geometry which is a branch of chaos theory. Chaos is a relatively new area of mathematics and can be random or deterministic. Chaotic systems can actually be very structured.

Henri Poincare -known as the father of chaos - proved in 1887, that there was no solution to the 3 body problem, i.e. a way of determining how 3 planetary objects are attracted to each other. This discovery of a 'chaotic' solar system laid the foundations for chaos theory.

Also in the 1880s, George Cantor considered what would happen if he started with a line, removed the middle third, then the middle third of the remaining segments, and continued removing repeating segments. The resulting figure is known as 'Cantor dust'. This is possibly the first known fractal, it has the key characteristics of fractals: namely a repeating pattern on a constantly smaller scale.

In the 1960s, Edward Lorenz, a meteorologist at MIT, was working on a computer simulation to predict weather patterns. One day he rounded up some figures he had been using but found the computer emitted a wildly different weather prediction then when using the earlier, slightly more accurate, numbers. This is known as 'The Butterfly Effect' where minute changes can have massively disproportionate effects.

**Early chaos**Henri Poincare -known as the father of chaos - proved in 1887, that there was no solution to the 3 body problem, i.e. a way of determining how 3 planetary objects are attracted to each other. This discovery of a 'chaotic' solar system laid the foundations for chaos theory.

Also in the 1880s, George Cantor considered what would happen if he started with a line, removed the middle third, then the middle third of the remaining segments, and continued removing repeating segments. The resulting figure is known as 'Cantor dust'. This is possibly the first known fractal, it has the key characteristics of fractals: namely a repeating pattern on a constantly smaller scale.

**Modern chaos**In the 1960s, Edward Lorenz, a meteorologist at MIT, was working on a computer simulation to predict weather patterns. One day he rounded up some figures he had been using but found the computer emitted a wildly different weather prediction then when using the earlier, slightly more accurate, numbers. This is known as 'The Butterfly Effect' where minute changes can have massively disproportionate effects.

## Uses of Chaos Theory

Weather patterns

Measuring coastlines

Economic forecasting

Behaviour patterns

Population movements

Astronomy

Business cycles

Ecology

Engineering

Fluid mechanics

Medicine

Fractal geometry

Measuring coastlines

Economic forecasting

Behaviour patterns

Population movements

Astronomy

Business cycles

Ecology

Engineering

Fluid mechanics

Medicine

Fractal geometry

## Fractal Geometry

**Characteristics of fractals**

Fractal geometry can be said to be the graphical representation of chaos theory.Fractals are formed by iterative equations that have some form of recursion.

Fractal geometry can be said to be the graphical representation of chaos theory.

**Key characteristics of Fractals:**

**They are self-similar**

**They have a simple and recurring definition**

**They have a Hausdorff dimension greater than their topological dimension**

Iterated functions in the complex plane have been studied before in the late 19th and early 20th centuries. However, without the assistance of modern technology, mathematicians were unable to visualize the beauty of many of the objects that they had created. Some of the most stunning fractals are the Julia sets, devised by the French mathematician Gaston Julia (1893-1978).

**Benoit Mandelbrot**

In the 1960s,

**Mandelbrot wrote the paper**

*How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension*, investigating objects whose Hausdorff-Besicovitch dimension is greater than its topological dimension. He actually invented the term 'fractal' and was able to graph these objects which led to these beautiful computer visualisations including, the ethereal

**Mandelbrot Man**.

However you don't need to be a mathematician or use sophisticated computer software to appreciate fractals, since they are essentially repeating patterns on a diminishing scale factor, they can actually be observed in many natural phenomena...

## Fractals found in nature

Coastlines

Ferns

Trees

Snowflakes

Rivers

Mountains

Clouds

Ferns

Trees

Snowflakes

Rivers

Mountains

Clouds

## How long is the coastline of Britain?

click on image to find the coastline

A question posed by Benoit Mandelbrot - the father of fractal geometry - who suggested that the smaller the measuring instrument, the greater the accuracy, hence the greater the ability to measure every crag, every inlet, every nook and cranny along the British coastline. His conclusion? Click on the image left to find the answer.

## Make your own paper fractal - Sierpinski's Triangle

## Take an A4 piece of paper and fold it lengthways

## Cut halfway along the width at mid-point, and fold this flap over

## Fold this flap inwards

## Repeat this process along the next 2 steps and fold the flaps inwards

## Repeat again for all 4 steps - don't forget to fold the flaps over first

## Repeat the process for the next 8 steps

## Cut halfway along the 8 steps and fold the flaps over..

## You should have 16 steps, fold all the flaps inwards

## Carefully open up - congratulations, you now have a 4th iteration fractal!

## Need more chaos in your life?

a fractal tree made using Logo