Snowflakes
It is said that no two snowflakes are exactly alike. That may be true, but all snowflakes do have similar structures. All snowflakes have hexagonal structures (six sides or six arms). This fact allows us to create our own fractal snowflakes quite easily. We just start with a hexagon, create an IFS, and iterate to form a pattern. We will create two different snowflakes here.
Our first snowflake will be simple. We take a hexagon:
and divide it into seven smaller hexagons:
We are thus applying a Sierpinski-esque transformation to our hexagon. Sibsequent iterations yield:
This creates a beautiful, if simplistic, snowflake.
If we alter this IFS just slightly, we can create an entirely different snowflake. We take our hexagon as before:
and we divide it into seven smaller hexagons. this time, however, we will make the center hexagon larger and the other six smaller, like so:
If we continue to iterate this, we get our new snowflake:
Again, even after only four iterations, it is very hard to remember that this is simply a bunch of hexagons in a regular pattern that is repeated over and over.
Both of these snowflakes can be created in Mathematica using code found here.
Fractals are often used in art to simulate nature. Computer-animated films, like Toy Story, use fractals to generate leaves, snowflakes, grass, and even clouds. Even explosions can be generated using three-dimensional fractal images. And many of the IFS used to create these images are not much more complex than the IFS we use here.