|
Lesson Sierpinski triangle In Lesson 1, the Cantor function cuts up an input rod into three equal pieces and throws way the middle piece. Figure 1 shows the remaining rod pieces after each generation of applying the Cantor function up to the third. Here, the rod is assumed a line segment, which has length but no thickness to speak of, although we need to draw it thick enough to be seen. Line is a one-dimensional object in the Euclidean everyday world.
Figure 1. Cantor set
We now extend the idea of Canter function to a two-dimensional object in plane and, in particular, a triangle. This then is the Sierpinski triangle (or gasket) function, named after a Polish mathematician Vaclav Sierpinski (1882 - 1969). It divides up a triangle into four equal triangles and throws away the middle one, as the function diagram of figure 2 shows.
Figure 2. Sierpinski triangle
Given an input triangle, the Sierpinski triangle function generates 3 half-size triangles for the first generation. Applying it again to each of the half-size triangles, we get 9 quarter-size triangles for the second generation. For instance, from the initial equilateral (60º - 60º - 60º) triangle of side 12cm, we obtain 3 equilateral triangles of side 6cm of the first generation, 9 triangles of side 3cm of the second generation, and so on. With Prog#5a it becomes apparent that most of the input triangle gets thrown away, leaving only small patches of it, after several iterations of the Sierpinski triangle function.
[prev] [ 0 | 1 | 2 | 3 | 4 | 5 ] [next]
|
