The idea that simple mathematical rules govern complex behaviour in nature is the basis of chaos theory and fractals.  

There is a nice applet on the cut-the-knot website that generates a Sierpinski triangle from a random process.  This is fun to do with students by hand, although it does take quite a while to get a good pattern!  This is connected to the Pascal's cube task!  

Changing the number of vertices and distance plotted from the vertex creates other patterns such as these, where n is the number of vertices and r is the distance plotted from the current point to the new vertex (below).  I haven't found a way to make this into a lesson yet, maybe you have a suggestion? 

The Koch snowflake is also fun to create and leads to an interesting discussion about perimeter and area.  I have used this with year 7 students to some success, drawing the shapes on triangle dotty paper to help find the area/perimeter.  I feel this could be developed for work with geometric series (C2). 

This leads nicely onto the section on chaos theory.