Sierpinski Triangle in 3 Dimensions
Mae Markowski
April 5, 2016
The following post is by Mae Markowski as a part of George Mason University
Math 493, Mathematics Through 3D Printing.
We will examine the 3-dimensional Sierpinski Triangle, Sierpinski tetrahedron, or Sierpinski pyramid, as it sometimes goes by. To begin, we will first examine the Sierpinski triangle in 2 dimensions, the multiple ways in which it is created, and then see the simplest way to generate the figure in 3-dimensions. Lastly we will take a look at the final 3D-printed model and
the code which led us there.
The Sierpinski triangle is what is known in general as an iterated function system. These are fractal, or self-similar in nature, meaning that you can zoom in as far as you want and you would keep seeing the same thing again and again. It received its namesake after Polish mathematician Waclaw Sierpinski, but has been around since well before Sierpinski. The triangle is formed in several ways. Easiest to grasp is the method of removing triangles.For instance, start with an equilateral triangle, cut it into 4 equal triangles, remove the center triangle, and repeat. In this way for n iterations we have
3n cells in our figure. (The cells being the smallest of the triangular units.) Below is an image after n = 3 iterations. Here, it is more clear that we will in the infinite limit remove an area of value 1, but still have infinitely many points in our set. Thus, this is a Cantor set.
More interestingly, the same triangle can be drawn via the Chaos Game, a game that one would guess is random, but actually always leads us to this self-similar pattern. Addtionally, this set also has the relationship that it is modulus 2 of Pascal’s Triangle. Because the third creation method is slightly more complicated, we will focus on the first method as we move to 3D. In 3 dimensions, we now no longer have planar cells in our shape. Each cell is now a pyramid. So in our first iteration, when we remove one pyramid from the middle, we are left with 4 cells leftover. Similarly, for n = 2, we have now 16 units. In general, after n iterations, there will be 4n cells left in the object, the extra dimension reflected in the increase of our base number being raised to the n power.
The code used to generate the Sierpinski pyramid in OpenSCAD constructs equilateral triangles, and then appropriately assigns vertices of the ones remaining after n iterations. The pyramids for n = 2; 3; 4; 5 are shown below
Because at 5 iterations we are already at 45 units in our pyramid, I opted to print out the cases of n = 1; 2; 3; 4, which are shown below. Also worth noting is that these objects were printed with supports. Because the n = 4 has so many cells that it has to print, I would think supports are necessary; however, you always run the risk of not being able to get them off. Thus, the pictures below still have the supports intact.
These were printed using the Makerbot 5th Generation. If interested in having your own, there are many other models up on Thingiverse. If interested in more of the mathematical models, the following links can further your reading:
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