Snowflakes and Other Fractals - Peter Giblin
27th October 2001
We begin with the idea of similarity of figures: when a figure in the plane is enlarged so that all lengths double, then the area is multiplied by four = two squared; when all lengths treble then the area is multiplied by nine = three squared. We also go the other way: if all lengths are halved then areas are multiplied by ½ × ½ = ¼ for instance.
Several examples are given, one of them being the snowflake, which is illustrated in the applet: snowflake applet . Here we start with an equilateral triangle and begin by adding an equilateral triangle one-third the size (in terms of length) on to the middle of each side: press the 'iterate' button once to see this. Press 'iterate' several times to see the effect of repeating the procedure several times. In the presentation, a method was used to calculate the area inside the resulting figure at any stage. Also another, rather miraculous method is given in the notes for calculating the 'limiting area' which is obtained by repeating the procedure (iterating) infinitely often. The little number displayed, 60, means the number of degrees in the angle at the base of the added-on triangle, which in this case is equilateral so giving 60 degrees. If you 'reset', 'iterate' once and then change the angle by sliding the bar then you find the shape of the added-on triangle changes. The rule is that the two sticking-out sides and the two outer pieces of the edge of the equilateral triangle are all equal. See the text of the presentation for details.
