Fractals, part I
PROPERTIES OF FRACTALS
Although there is no precise definition for a fractal, the geometric
structures that people refer to when they use the word fractal generally have
the following properties.
- Detail at all levels of magnification. If you "zoom in", you continue
to see structure of all levels of detail.
- Self-similarity. Components of the fractal can be shifted and scaled
so that they are identical to the entire fractal itself. Sometimes we
allow probabalistic self-similarity, and sometimes we allow
quasi-self-similarity (as in the Mandelbrot set)
- Iteration - The process for constructing or defining the fractal involves
some infinite iterative process.
- Non-integer dimension. There are various ways of measuring the dimension
of a geometrical structure. Line segments are one-dimensional and squares
are two-dimensional. Fractals may have dimensions that are not whole numbers.
EXAMPLES OF FRACTALS
Note how the first three fractal properties are all apparent in each of these
examples.
- Cantor set (Picture of the Cantor Set)
- Koch's curve (Picture of Koch's snowflake)
- Sierpinski's gasket (Picture of Sierpinksi gasket
- Julia sets and the Mandelbrot set Interactive pictures of Mandelbrot and Julia sets)
- Fractals defined by iterative function systems (Picture of Barnsley's Fern)
THE "LENGTH" OF THE CANTOR SET
After all of the intervals have been removed and we're left with the Cantor
set,what remains looks like just dust. If the original segment weighed 1
pound, how much would the Cantor set weigh? In other words, what is the
total measure or length of the Cantor set? We can calculate this by
two different methods:
adding up the length of all of the intervals that were removed at each step, or
calculating the length of what remains at each step. In either case,
we finish by taking the limit as the number of steps goes to infinity.
Let's use the latter method.
After the first step, what remains is 2 intervals of length 1/3, for a total
length of 2/3.
After the second step, there are 4 intervals of length 1/9, for a total
length of 4/9.
After the third step, there are 8 intervals of length 1/27, for a total
length of 8/27.
In general, after the nth step, the length remaining is (2/3)n.
Taking the limit as n approaches infinity, we see that the length is 0.
For homework, you'll be asked to do a similar type of calculation for Koch's
curve and the Sierpinski gasket.
The fact that the length is 0 indicates that the Cantor set may not
be a dimension 1 set. It's more than a point, though, so it seems
that its dimension should be greater than 0. We'll see next how
to calculate its dimension.