Dragon Curve

thingiverse

This is a 3D version of the 7th iteration of the Heighway dragon curve. Multiple copies of the model can be used to demonstrate that the dragon curve covers the plane completely. Print Settings Printer Brand: SeeMeCNC Printer: Rostock MAX Rafts: Doesn't Matter Supports: No Notes: The model has a tendency to pop off the bed. Take proper precautions. Post-Printing Assemble multiple copies of this model into a larger but similar version - or just keep tiling. Since the dragon is self-similar, one can easily assemble multiple copies of this model into a larger but similar version - or just keep tiling. How I Designed This To create the 2D dragon curve in a printable format that can be tiled together, I determined I could create a polyline of the curve in Rhinoceros and from there quickly sweep the curve to create the printable object. To automate the process, rather than entering all the points by hand, I utilized Rhinoceros' Python integration. (N.L., MAT 363, December 2015) Custom Section More Details: Mathematics Behind the Design The Python implementation utilizes a specific understanding of the construction of the Heighway dragon curve. Effectively, to construct this curve we start with a straight line segment and proceed by duplicating our curve and rotating it 90 degrees around the end of our curve. To implement this, I just need to list the first two points in my curve and proceed with the algorithm. It was necessary to use the rotation matrix for R2. For computational efficiency, instead of using matrix multiplication, my program directly converted a point (x; y) into (-y; x). Note that this matrix rotates around the origin. Thus, our algorithm will need to translate our curve so it ends at the origin. From here, it is simple to sweep the curve with a rectangle to produce a 3D object that can be printed. Unfortunately, the object is not yet fit for tiling because corners would overlap. To allow the object to demonstrate a tiling it was necessary to curve the corners using the FilletCorners command. This will bring the corners in enough to prevent overlap, so long as we choose a suitable width rectangle to sweep the curve with for the given iteration. (N.L., MAT 363, December 2015)