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Fractals

The places where you can find fractals include almost every part of the universe, from bacteria cultures to galaxies to your body. Fractals provide a simple solution to capture the enormous detail and irregularity of clouds and landscapes. Fractal geometry is an efficient way to draw realistic natural objects on a computer screen. Landscape designers start with basic shapes and iterate them over and over. Science Fiction films design imaginary landscapes likewise for backdrops. The wide range of applications includes those shown below:  Click here to find more details

Astronomy Biology / Chemistry Other
Galaxies
Rings of Saturn
Bacteria Cultures
Chemical Reactions
Human Anatomy
Molecules
Plants
Population Growth
Clouds
Coastlines and Borderlines
Data Compression
Diffusion
Economy
Fractal Art
Fractal Music
Landscapes
Newton's Method
Special Effects (Star Trek)
Weather

Fractal Cuts

What is a fractal - Simple Explanation

This applet allows you to generate simple fractals by moving the mouse - Click here

These 3 activities require repeated "fractal" cuts to generate a 3D shape

Fractal Cut Out Activity 1
Fractal Cut Out Activity 2
Fractal Cut Out Activity 3

 

Calculator Chaos

Chaos doesn't just relate to complicated systems and abstract theories - you can see it in numbers. Try iterating this function: 2x2 - 1, with a starting value for x between 0 and 1. If you're not familiar with the concept of iterative functions, it means that you take the answer you get from the function and you use it as your x-value and find another solution to the function, etc. You can do this on a calculator, or if you'd like, write a computer program or use a spreadsheet. I chose the value 0.75. If you looked at the graph of the function, it might look a little irregular, but not unusual. The interesting part is that if you chose a value like 0.74999, which is very close to 0.75, and graph it, you'll see that it is similar at first, but then becomes totally different. This ties in with how the initial state of a system can totally change what the final result will be - Click here

What happens if you begin each time with the seed 0 but change the rule so that you are iterating the function x2 + c for different values of c? - Click here
(This is a simplified version of Julia Sets see below)

Model Rabbit Population growth and explore the sensitivity of the population to small changes - Click here            Spreadsheet to explore further

Games of Life

This video gives a simple introduction to the rules of John Conway's Game of life - Click here

Nice written explanation with examples - Click here

This fantastic video illustrates the power of the simple rules of Conway's Game of Life - Click here

Run the classic game of life, learning about probabilities, chaos and simulation. This activity allows the user to run a randomly generated world or test out various patterns. This is a very powerful activity with a wide range of options - Click here

Predator - Prey Simulation - Click here

Visual Version which uses a simulation to generate population numbers - Click here

Fractals

This applet generates a Hilbret Curve fractal step by step - Click here

Sierpinski's Carpet is a simple fractal generated by colouring the centre of each square white - Click here

This document outlines the general principle of fractals and looks at some simple examples including the Sierpinski Triangle - Click here

For an online applet allowing you to generate 3 of these shapes - Click here

This worksheet explores the perimeter of the Koch snowflake shown on the left - Click here

 

Sierpinski Triangle

The Sierpinski Triangle is a simple example of a fractal and provides a rich avenue into the world of fractals.

The Triangle can be generated by what is known as the chaos game - Click here for more information or Click here for a PowerPoint introduction

This can be modelled using a spreadsheet - Click here for Version 1    Version 2

For an online Applet allowing you to weight the probabilities or change the number of vertices - Click here

The Sierpinski Triangle can amazingly be generated from Pascal's Triangle by colouring in all the odd numbers - Click here for a spreadsheet model

Explore what happens if you colour all multiples of 3?  or 5? or 7? ... - Triangle      Applet

These Think Maths worksheets have all the instructions and printable nets required to build 3D fractals, both a Menger Sponge and a Sierpinski Tetrahedron. They work fine on A4 plain paper, but when printed on A3 card, they can result in some truly impressive structures - Original Link - Alternative Link

The graph of all moves for Tower of Hanoi with 3 discs. Sierpinski's Triangle - Video

Impossible Fractals

A lovely and imaginative set of "Impossible Fractals" compiled by Cameron Browne

Click here

Fractal Attractors

Fractal Attractors look at the result of repeatedly performing a set of transformations and exploring whether the point or shape is attracted to a particular point.

This task can be accessed and investigated by younger pupils by using software such as Autograph but would need AS Further Maths content to fully understand the underlying mathematics.

Strange Attractor Task

 

Mandelbrot & Julia Sets

The Mandelbrot set is made up of points plotted on a complex plane to form a fractal: a striking shape or form in which each part is actually a miniature copy of the whole. The incredibly dazzling imagery hidden in the Mandelbrot Set was possible to view in the 1500s thanks to Rafael Bombelli's understanding of imaginary numbers -- but it wasn't until Benoit Mandelbrot and others started exploring fractals with the aid of computers that the secret universe was revealed.

This is a clear explanation of the derivation of the Mandelbrot set with commentary on Numberphile - Click here

Another Version from Mathologer - Click here

An alternative excellent explanation with images and videos - Click here
 

For a spreadsheet that looks at what happens to a selected point - Click here

For a crude rendering of the set done by hand - Click here

For an online applet - Click here

Explore Julia Sets                Zoom in on Mandlebrot Set

This document explains how the Mandelbrot Image to the left is generated mathematically by beginning with a 1 dimensional example and then extending to 2D using a little knowledge of complex numbers before explaining the difference between the Mandelbrot set (where the iteration changes and the seed remains 0) and Julia Sets (where the iteration remains constant and the seed changes), two examples of which are shown on the right - Click here

For an alternative explanation of the generation of the Mandelbrot set - Click here

Using the Mandelbrot set to generate an approximation for Pi (Numberphile) - Click here

Fractal Extras

Fractals and Fibonacci  - Click here

A thought on the Dimensions of Fractals takes us outside the 1, 2 or 3 Dimensional World we are familiar with - Click here

Generate your own spreadsheet to better understand the generation of the Mandelbrot set and the resulting image.
For an exemplar spreadsheet - Click here

Video Clips

Online: Fractal Zoom Online: Mandelbrot 1 Online: Mandelbrot 2
Doodling 1 Doodling 2  

Websites

Motivate Thinkquest  
Yale University's Introduction to Fractals classes.yale.edu/Fractals/  

Cultural Fractals

'I am a mathematician, and I would like to stand on your roof.' That is how Ron Eglash greeted many African families he met while researching the fractal patterns hed noticed in villages across the continent.

Ron Eglash is an ethno-mathematician: he studies the way math and cultures intersect. He has shown that many aspects of African design -- in architecture, art, even hair braiding -- are based on perfect fractal patterns.

TED Talk - Click here