Fractals are an interesting and beautiful piece of mathematics. There are many ways to create fractal shapes, but one common way is through an iterated function system (IFS). Loosely, an iterated function system on the plane is a series of contractive functions which map points on the plane. If we iterate just one contraction, this is not very interesting. In fact, Banach's fixed point theorem asserts that this will result in a single fixed point as we take the limit in the number of iterations of this contraction. However if we consider several contraction mappings with different fixed points, what we are left with is an interesting IFS that fractalizes the plane.

This print started off as a solid arrow with no holes. The top portion of the arrow is simply a four sided polygon. The base of the arrow was slightly more complicated, and involved taking the difference of a triangle with three circles to create a sort of hyperbolic triangle. This formed two separate shapes which were then intersected to create the base of the final object as seen in the images above.

To create the final object, we needed to create some sort of iterative system. Instead of iterating the base object itself, we created a new base object, namely a pentagon, to be iterated. Starting with a pentagon in the center of the base object, we scaled the pentagon by 0.62 and translated it 35mm in the vertical direction. Iterating this process yielded the top portion of the fractal arrow. The left portion was made by scaling the initial hexagon by 0.5 and translating it 9mm to the left and 7mm down. The right portion was made by scaling the initial hexagon by 0.5 and translating it 9mm to the right and 7mm down. In the end, the iterated hexagons were subtracted from the initial shape creating the object you see above. If you notice, the initial hexagon cuts the base arrow into three more arrows creating a self-similarity of the object. Each further iteration of the hexagon has the same affect. Each of these three arrows constitute one-half of the side length of the initial shape. This leads to a box counting dimension of log(3)/log(2) or approximately 1.585. Due to the self similarity of the shape, this holds true in the limit. This makes sense since intuitively this shape lives between one and two dimensions as the number of iterations tends to infinity. This print was done on an Ultimaker S5 with PLA filament, and it took roughly one hour to print.

UltiMaker

Ultimaker S5

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