Other Fractals

Julia Set

    There are other fractals related to the ones already talked about. One of those fractals is the Julia Set. This set was created by Julia. Gaston Julia was a French mathematician. Julia was the one who introduced the concept of the iterative function system. An iterative function is a function from X to X which is obtained from having another function put onto itself so many times. He then used it to derive the Julia set in 1919. The concepts of the Julia set were used by Mandelbrot. Mandelbrot used the concepts of the Julia set to help him create the Mandelbrot set and also the concept of fractals.

Sierpinski Triangle

    Another fractal that has mathematical relationship is the Sierpinski Triangle fractal. This fractal was created in 1915 and is another well known fractal. The most common application of this fractal is with the Pascal’s triangle. The reason for this is because the Sierpinski Triangle fractal is made by starting off with one triangle. Then that triangle is split into 4 triangles. Then each of the four triangles is split into four more triangles which gives you 4^4 triangles. This process is then continued on forever. The way that this is similar to Pascal’s triangle is because of the pattern of the multiples of 2 on Pascal's triangle.

Sierpinski Carpet

    Sierpinski also has another famous fractal. The other fractal that Sierpinski has is called the Sierpinski Carpet fractal. This fractal was also discovered in 1915. This fractal is also a fractal that is known to others. This fractal is very similar to Sierpinski Triangle fractal. The difference is the carpet fractal is a square shape instead of a triangle. This fractal is also made from an iterative procedure like Sierpinski Triangle. The only difference is that for the carpet fractal you add 8 out of 9 squares. What I mean by this is that inside your square it will add 8 more little squares around it. This then will make the area of the Sierpinski Carpet \[(8/9)^n\]. This fractal is one that is made quite often when laying floor tile in a house.

Menger Sponge

    Another fractal that is very similar to the Sierpinski Carpet fractal is the Menger Sponge fractal. This fractal was discovered by Karl Menger in 1926. The reason that this fractal is similar to the Sierpinski Carpet fractal is because it is the 3D version of it. This fractal has several cavities in it. This fractal can also be made with other shapes besides squares.

Fibonacci Spiral

    There are so many different types of fractals out there. The last one that will be briefly mentioned is the Fibonacci spiral. This is a fractal because it is rectangles followed by rectangles. This is another fractal that is very well known because it is talked about in high school and college math classes. The way that it is talked about is when they talk about the golden ratio. It is also talked about when talking about the Fibonacci sequence.