Fractal fun.

As a Mathemagician (I didn't pick the name) I have been lucky enough to work with Prof. Marcus du Sautoy on a number of occasions. One of my favourite projects that I have worked on was when he asked me to do some illustrations for his most recent book, The Number Mysteries. Specifically, in Chapter 2, The Story of Illusive Shape, he wanted some pictures of fractals. So for those of you who have read the book you'll see some familiar figures below and for those of you who haven't then hopefully these will pique your interest and you will go and find out more.

Figure 1. Illustrations of how to measure the dimension of a fractal by covering it in boxes of different lengths.

Figure 2. The Koch snowflake is a very famous fractal. Normally, it is constructed using an equilateral triangle. Here, I alter the angle to make it isosceles. What do you think that might do to the dimension?

 Figure 3. The Koch snowflake is constructed by replacing the middle third of a triangle with a triangle a third of the size. Normally, all the triangles are taken to point outwards. However, above I randomise this process.

Figure 4. If you put three of the randomised Koch snowflakes together you get something that looks like a medieval map of Britain

If in the future I run out of things to say I may put up the codes so that you can also learn how to create such structures.