Sierpinski Fractals

Triangle
Pyramid
Carpet (Square)
Pentagon
Hexagon
Hexagon Prism
Interesting Links on the Web
Random Walk

Sierpinski Triangle

back to top

Here is the image of my Sierpinski Triangle written in Maple with 9500 iterations.  If you are interested, here is a link to download the code used.  (Here is the code for the above animation.)  I was able to write this with the help of the website of Harold T. Boas, a Professor in the Department of Mathematics at Texas A&M University.

Steps to create the Sierpinski Triangle:

  1. Start at any point within a larger triangle.  (I used the points (-1, 0), (1, 0), and (0, 1) as my larger triangle.)
  2. Roll a fair die.
  3. If a 1 or 2 is rolled, move to the midpoint of your current location and the bottom left point (in this case, it is (-1, 0)).
  4. If a 3 or 4 is rolled, move to the midpoint of your current location and the bottom right point (in this case, it is (1, 0)).
  5. If a 5 or 6 is rolled, move to the midpoint of your current location and the top center point (in this case, it is (0, 1)).
  6. Mark your new location with a dot.
  7. Repeat steps 2 through 6 n times.

Sierpinski Pyramid

back to top

Here is my Sierpinski Pyramid file.  It is constructed in a method similar to the Triangle, only it is 3-dimensional.  Instead of 3 points to go to, the Sierpinski Pyramid takes the midpoint between a random choice of 5 points (in this case it is (0, 1, 0), (-1, 0, 1), (1, 0, 1), (-1, 0, -1), and (1, 0, -1)).

Sierpinski Carpet

back to top

The Sierpinski Carpet gave me some trouble. I initially thought that it would work the same as the Triangle and Hexagon, BUT, you need to include the midpoints between the four corners also, then take the trippoint (instead of finding the midpoint, you divide by 3 instead of 2) of your random move with one of the 8 points. I am not entirely sure why you need to this, but as soon as I figure it out, I will post it on here.  I was able to do this with the help of this website.  This is the image that comes up after 15500 iterations:

Sierpinski Pentagon

back to top

Here is the file, and here is my animation after 9500 iterations:

Sierpinski Hexagon

back to top

Here is my Hexagon file, and it makes an image that looks like this with 9500 iterations:

Sierpinski Hexagon Prism

back to top

I wrote the Hexagon Prism within the same file as the regular Sierpinski Hexagon.  I do not think this is correct, so I need to work on it some more.  As you can see in the second picture, there are simply 8 Sierpinski Hexagons repeated, not really what the Sierpinski Pyramid looks like.  Here are two picture of what it looks like with 25000 iterations:

   

Interesting Sierpinski Links on the Web

back to top