Activity 1-20: The Mandelbrot Set

1. www.carom-maths.co.uk Activity 1-20: The Mandelbrot Set

2. You may have heard of the butterfly effect;the claim that a butterfly flapping its wings on one side of the world can trigger a tornado on the other. A tiny change to initial conditionssometimes creates a huge change in the final outcome.

3. This is a topic from Chaos Theory, a young branch of mathematics, and the Mandelbrot Set is an example of how beautiful this theory can be.

4. To understand how the Mandelbrot Set comes about, you first need to know about complex numbers. Make sure you understand the following: We call the square root of -1, i. A complex number can be written as a + bi, where a and b are both real numbers. A complex number can be displayed on the Argand diagram, with real part plotted horizontally, and the imaginary part plotted vertically. Complex numbers can be added, subtracted, multiplied and divided, each time giving a complex number.

5. For an object of such depth and beauty, the Mandelbrot Set is created by a remarkably simple rule. We pick a complex number c on the Argand diagram; is it in the Mandelbrot set or not? To decide,we start with the complex number 0 + 0i, the origin of the Argand diagram. Square this, and add c. Now square this new number, and add c. Now repeat this lots of times.

6. One of two things can happen: 1. The iteration remains bounded, in which case we say cis IN the Mandelbrot Set.

7. 2. The iteration diverges to infinity, in which case we say c is NOT in the Mandelbrot Set.

8. Task: using the grid (link given below), and the Excel spreadsheet (link given below), draw a rough sketch of the Mandelbrot Set. Grid Link Spreadsheet Link http://www.s253053503.websitehome.co.uk/carom/carom-files/carom-1-20.xls http://www.s253053503.websitehome.co.uk/carom/carom-files/carom-1-20-grid.pdf Stays bounded...

9. You should end up with something like this: A rough first approximation!

10. Computers now allow us to draw the Mandelbrot Set with much greater accuracy. In fact, the Mandelbrot Set could only sensibly be explored in the computer age.

11. One of the extraordinary things about the Mandelbrot Set is its edge. There seems to be copies of parts of the Set on the edge, and the further we zoom in on the edge, the smaller and smaller the copies of the Set we seem to encounter. The area enclosed by the Mandelbrot Set is finite, but its perimeter is infinite.

12. Zooming in on the edge of the Mandelbrot Set.

13. The Mandelbrot Set is what we call a fractal. Fractals are typically self-similar patterns, where self-similar means they are ‘the same from near as from far’. Wikipedia. Von Koch Curve Link http://en.wikipedia.org/wiki/File:Von_Koch_curve.gif

14. Approximate fractals are found in nature... Frost Romanesco broccoli

15. Click below for deep zoom video aimed at the edge of the MS. Mandelbrot Set Deep Zoom Link http://www.youtube.com/watch?v=0jGaio87u3A

16. With thanks to: Wikipedia, for another excellent article. Orson Wang. Carom is written by Jonny Griffiths, hello@jonny-griffiths.net