Case Study: Fractals

Using recursion is ideal for displaying fractals, because fractals are inherently recursive. A fractal is a geometrical figure, but unlike triangles, circles, and rectangles, fractals can be divided into parts, each of which is a reduced-size copy of the whole. There are many interesting examples of fractals. This section introduces a simple fractal, the Sierpinski triangle, named after a famous Polish mathematician.

A Sierpinski triangle is created as follows:

1. Begin with an equilateral triangle, which is considered to be a Sierpinski fractal of order (or level) 0, as shown in Figure below (a).


2. Connect the midpoints of the sides of the triangle of order 0 to create a Sierpinski triangle of order 1 (Figure below (b)).


3. Leave the center triangle intact. Connect the midpoints of the sides of the three other triangles to create a Sierpinski triangle of order 2 (Figure below (c)).


4. You can repeat the same process recursively to create a Sierpinski triangle of order 3, 4, . . . , and so on (Figure below (d)).

A Sierpinski triangle is displayed in a SierpinskiTrianglePane. The order property in the inner class SierpinskiTrianglePane specifies the order for the Sierpinski triangle. The Point2D Class, represents a point with x- and y-coordinates. Invoking p1.midpoint(p2) returns a new Point2D object that is the midpoint between p1 and p2 (lines 83–85).