The T-Square Fractal

Jonathan Tarr

April 03, 2016

The T-Square Fractal

The point of this project was to study fractals that can be expressed as iterative function systems, such as the famous Sierpinski Gasket. These are, as the name implies, a way to iteratively, or step-by-step, remove a (generally) geometric object from a copy of the same, larger, geometric object.

One such example of this kind of fractal is the T-Square. Like most two dimensional fractals of this kind, the T-Square has an infinite length within a finite area, similar to the Koch Curve. Geometrically, the shape is constructed by taking a square and placing a square one-half the length of the starting square on each corner, centered on each respective corner. Repeat this ad infinitum and you will arrive at the T-Square Fractal. When creating this as an iterative function system, the code below (Figure 1) is used to generate the fractal, where r and s are scalars, Theta and Phi are rotations, and e and f are translations along the x-axis and y-axis.

Figure 1: The IFS code for generating a T-Square Fractal. Courtesy of Yale.edu.

For my specific T-Square Fractal, I chose to get experimental and exhibit a property of IFS’s that is intriguing, you can begin with any geometric shape and you will arrive at the aforementioned fractal. Namely, I used circles to generate my T-Square Fractal. Below, in Figure 2.1-2.3 is the code I used in a program named OpenSCAD in order to render my IFS. After rendering and exporting the file, I modified the file in Makerware for a Makerbot 2X 3-D printer, Figure 3.1-3.2, and printed it. Figure 4 is the printed model ready for display. Unfortunately, due to over hangs, there was a lot of excess material printed that needed cleaned out as it was used to support the structure so it did not collapse.

Figure 2.1: My OpenSCAD code for generating a T-Square Fractal, curtesy of Evelyn Sander.

Figure 3.1: The Makerware rendering of my OpenSCAD code after sizing it up to a ~4 hour print.

Figure 3.2: The Makerware rendering of my OpenSCAD code after sizing it up to a ~4 hour print.

Figure 4: My T-Square made up of Iterated Circles printed using a Makerbot 2X 3-D printer.