Iterated System Bamboo Shoots

thingiverse

Iterated Function Bamboo Shoots Taylor Menetski, 9/26/2020 This object was developed for George Mason University Math 401, Mathematics Through 3D Printing. This object is a simple representation of an iterated function system, where the object is itself a set, consisting of a base object and the results of repeated use of a specific group of functions on that object. Its dimensions are 4cm by 4cm by 4cm for the base and four iterations, and if more iterations are rendered it will get taller. I still suggest leaving slightly more space on the printer, just in case, as one edge is deliberately misaligned with the edge of the base. It should not require additional supports for printing. It most closely resembles a stacking iteration of the Sierpinski Gasket (link to explanation: http://larryriddle.agnesscott.org/ifs/siertri/siertri.htm ), where three defined affine transformations are applied on a shape. In the original Gasket, performed on a triangle, each set of three ever-smaller triangles were aligned to the corners of the previous iteration. These systems form many common fractals ( https://en.wikipedia.org/wiki/Fractal#Common_techniques_for_generating_fractals ). My version consists of a cylinder with a slanted rectangle cut out of it, that spirals and shrinks within each iteration, with a slightly-larger scaling on the z-axis to create a more visually interesting object. While not completely true-to-life, the iterations resemble vegetative propagation in bamboo, where new shoots begin growing out of a cut plant surface. The code uses a recursive module to generate the layers of the set, using a defined module, slice(), for each instance of the shape. The scaling for each of the three transformations compared to the previous iteration are 0.35x, 0.4x, and 0.45x from smallest to largest, though all z-axis scaling is 0.7x to force additional height out of something that might otherwise be almost flat. Each transformation also includes a translation, moving the center of the cylinder from its origin in the xy-plane (that I'll call [0,0] for reference, but know that it moves for each iteration) to [0.5r,0], [0,0.6r], and [-0.5r,-0.1r] from smallest to largest, where r is the radius of the cylinder from the previous iteration. Finally, each new cylinder is rotated to align the high end of the cut to the outer edge of the previous iteration. These transformations were color-coded for easier visualization.