Friends, readers, random strangers! I hope you have all had a very merry Christmas (which of course is Sir Isaac Newton’s birthday, so those of us of a non-religious disposition may celebrate too) and enjoyed the beautiful snow we had last week. I journeyed from Edinburgh to London for the festive season and was flabbergasted at the beauty of the snow-covered fields with the camouflaged sheep in them. Poor McHaggis, my son, is not quite so well-camouflaged as I, what with his black face and wee feet, so unfortunately he was discovered and used by the cruel humans as one of their Christmas decorations.
As for myself, I have been staying amused by pondering the nature of the Maslov Index (don’t ask – I don’t understand it…) and by regaling my guests with tales of the infinity of infinities that exist in the mathematical realm. It is always good to test out ideas on the (adult) general public before letting them loose on school pupils!
Anyway, it being Christmas and all, I thought I should write a post on something more festive than usual. So I’m going to tell you about the idea of fractals and, in particular, how to make (and bake!) yourself a fractal snowflake.
A fractal is an shape with the property that any zoomed in part of it looks the same as the original shape. One example from real life is a coastline. A coastline looks pretty wiggly from a distance, but when you zoom in on a small part of the coastline it still looks wiggly! If you zoom in further, till you can see the actual pebbles, it still looks wiggly, and even on a molecular level it is still…wiggly. The difference between real life and mathematics is that in real life there is a limit to how far you can zoom in, since you can’t go any further than the atoms and quarks. But in the mathematical world, you can zoom in infinitely far.
Another real-life place to find fractals is snowflakes. Generally, if you zoom on on any of the ‘leaves’ of a snowflake, you’ll find that the zoomed-in bit looks the same as the whole leaf.
One way of making fractals is by a recursive method, i.e. you repeat the same set of instructions over and over again. We are going to make a fractal snowflake using a method described by Helge von Koch (whence it is called the Koch snowflake). First, draw an equilateral triangle.Then replace each side of the triangle with the following shape:
This gives us the following nice star shape:
Now repeat! Wikipedia has a nice video of the Koch snowflake evolving in a few time steps, as well as some more information about its mathematical properties. Sci-Fun also have a good article about the Koch snowflake, proving why it has an infinite perimeter but a finite area.
Try starting out with a random polygon and a rule for changing it, and see what happens! It’s easy to make beautiful, symmetric, intricate pictures that would look great on the front of a Christmas card. Then, if you get bored with that, you can try baking your own fractal cupcakes! Have fun!