Fractals

Koch Curve

In the Koch Curve, we start with a line. Then, every iteration we draw an equilateral triangle on the middle third of each existing line. This after the first iteration, we get this figure.

After the second iteration, it looks like this.

After the third iteration, it looks like this.

Each iteration gets a more and more detailed picture.

If we iterate the Koch Curve ad infinitum, we get a very curious fact: The area inside the Koch Curve is finite. However, its perimeter is infinite. This may seem counterintuitive, but the area must be finite because we can put a square around it with a fixed side length. Thus the Koch curve has an area less than the square. However, on each iteration, we increase the number of lines, so the lengths of the new lines is always equal to 2/3 of the original length of the line. Thus the perimeter goes to infinity.

Sierpinski Triangle

In the Sierpinski Triangle, we start with the three vertices of an equilateral triangle, which are numbered 1, 2, and 3. Then we take the center of the triangle. Then we use a random number generator to choose the numbered vertices 1, 2, or 3. Then we move ½ the distance from the place where the point already is (initially at the center) and draw a new point at the new location halfway to the new vertex. If we iterate this infinitely many times, we get something amazing. It does not matter what the random numbers are. We always get the same picture. This is what we get with 100 random numbers.

Here, I colored the vertices rather than numbered them, but the outcome is the same. After 100 tests, it is hard to see what will happen. Here is after 1000.

A figure begins to emerge. After 10 000, it is obvious what will happen.

After 100 000, it changes little, but the lines become less blurry.

Interestingly, we reach the Sierpinski Triangle if we color the odd numbers in Pascal's Triangle. The even numbers take up the white spaces.

Sierpinski Carpet

There is no real reason that we must use a triangle for use in the Sierpinski Triangle. In the Sierpinski Carpet, we use a 3 by 3 square with no point in the middle. Then a random point from the eight is selected. However, unlike in the Sierpinski Triangle, in the Sierpinski Carpet, we move only 1/3 the distance to the point chosen rather than ½. The resulting figure, after 100 000 random points is

Mandelbrot Set

This is probably the most famous of all fractals. In the Mandelbrot Set, we start with the complex plane. Then we perform an infinite iteration: we perform the transformation , where c is a constant, and each iteration, x of the new iteration is the result of the previous iteration. c is a point on the complex plane that we are evaluating. If after infinitely many iterations the transformation has not gone to infinity, we say that this point is inside the Mandelbrot Set. If after infinitely many iterations the transformation has gone to infinity, we say that this point is not part of the Mandelbrot Set. We color the points inside the Mandelbrot Set in one color. In my picture they are black. We then color the divergent points in various colors based on how fast those points reach very large values. Below is my diagram of the Mandelbrot Set.