Sierpinski Triangle

I created this Sierpinski triangle in Geometer's Sketchpad by iterating the following algorithm hundreds of times:  Find the midpoint of each side of the triangle.  Connect the midpoints to form a new triangle.  Remove the inner triangle.  I ended up with this self-similar design.  This means that if you zoom in on any of the smaller triangles, it will look exactly like the entire figure!

Sierpinski 1: I created this Sierpinski triangle in Geometer’s Sketchpad by iterating the following algorithm hundreds of times:

  1. Find the midpoint of each side of the triangle.
  2. Connect the midpoints to form a new triangle.
  3. Remove the inner triangle.

I ended up with this self-similar design. This means that if you zoom in on any of the smaller triangles, it will look exactly like the entire figure!

Relaxed Sierpinski:  Some students found that if you relaxed rule #1 so that the inner triangles did not have to be on midpoints, you got some interesting new fractals.  Here is mine.  It's interesting how simply removing regularity makes the fracal somewhat sinister.  Notice that it is still self-similar however!

Relaxed Sierpinski: Some students found that if you relaxed rule #1 so that the inner triangles did not have to be on midpoints, you got some interesting new fractals. Here is mine. It’s interesting how simply removing regularity makes the fracal somewhat sinister. Notice that it is still self-similar however!

Cellular Sierpenski:  This is the same design as the shape above made in a completely different way.  I made this using a 1D cellular automata program with rule 90.  The program builds each successive line using a very simple set of rules on the line above. This is after about 100 iterations.

Cellular Sierpenski: This is the same design as the shape above made in a completely different way. I made this using a 1D cellular automata program with rule 90. The program builds each successive line using a very simple set of rules on the line above. This is after about 100 iterations.

Cellular Sierpenski 2:  This is the same rule after about 40,000 iterations.  Even though I've zoomed out, the shape still looks the same.  It now has over 17,000,000 squares!

Cellular Sierpenski 2: This is the same rule after about 40,000 iterations. Even though I’ve zoomed out, the shape still looks the same. It now has over 17,000,000 squares!

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