Sierpinski's Triangle
| It has been said that the three great developments in twentieth century science are relativity, quantum mechanics, and chaos. That strikes me the same as saying that the three great developments in twentieth century engineering are the airplane, the computer, and the pop-top aluminum can. Chaos and fractals are not even twentieth century ideas: chaos was first observed by Poincare and fractals were familiar to Cantor a century ago, although neither man had the computer at his disposal to show the rest of the world the beauty he was seeing. |
| -Robert L. Devaney from "Introduction: Special Issue on Dynamical Systems," College Mathematics Journal, Vol. 22, no. 1, Jan. 1991 |
From the picture above it should be clear how to create Sierpinski's Triangle.
You start with the first triangle, take the midpoint of each
edge 
connect the midpoints to create a new (upside down triangle)
now remove the center triangle to arrive at:
.
Now your left with three triangles
continue this pattern with each of these
triangle to arrive at 
.
And now again with the remaining black triangles to arrive at
. Do this infinitely many times and the
resulting set of points is called Sierpinski's triangle.
Do the following:
1. Calculate the area of the Sierpinski's triangle (or gasket) if the original triangle is an equilateral triangle with side lengths 1.
2. Show that there are infinitely many points in the set.
3. Generalize this idea to create other gaskets.
4. Write a program that plots 5-10 thousand points in the following way:
a. First plots the vertices of an equilateral triangle call them S, T, and U.
b. Plot a random point call it A.
c. Plot the next 5000 points by randomly choosing one of the three vertices (say T) and picking the midpoint of the segment ST as your next point to plot
d. Call that new point A and repeat.
5. Does the plot look familiar?
6. Show that there can be only finitely many points plotted in the area inside the biggest "removed" triangle.
7. Generalize this algorithm to create other gaskets.