Chapter 12-3 Sierpinski's Triangle

There are many ways to generate fractals, but they are at heart geometrically-growing figures.

The Sierpinski triangle is sometimes called the Sierpinski gasket because of its final shape. It is a simple fractal that can be understood by generating geometric sequences to characterize its features.

Generate each new level of the fractal by clicking on the midpoints of each side and then clicking on the "Step" button. What are the implications, for your work here, of the geometric increase in the number of midpoints? Shade the inner or the outer triangles of each level (or both).

You can also use contraction mapping to generate a fractal. Try this with the rectangle and the bird to see what this process means for the final figure.

Questions
1. Describe, in visual terms, a recursive formula for Sierpinski's Triangle.

2. Starting from the original Sierpinski triangle, collect data by counting;
a) the number of new triangles produced at each level, and
b) the number of the new ones that will be subdivided.
How many of each type are there at level 3? Level 4? (Hint: at level 1, connecting the midpoint makes 4 new triangles; 3 will be subdivided in the next level. At level 2, there are 12 new triangles; 9 will be subdivided at the next level).

3. How many triangles will be subdivided at the nth level?

4. Write a sequence and its rule that describes the length of the sides of each triangle at given level n. Be sure and refer back to the diagram as it builds the triangles in order to check your sequence.

5. If the length of each side starts at 1, find the area of the original triangle. What is the area after one iteration? Two iterations? n iterations?

6. Propose a basic algorithm (in words) that describes contraction mapping.