Summary

This is a 3D version of the 7th iteration of the Heighway dragon curve. Multiple copies of the model may be used to demonstrate that the dragon curve tiles the plane.
(N.L., MAT 363, December 2015)

Print Settings

Printer Brand:

SeeMeCNC

                                            <p class="detail-setting printer"><strong>Printer: </strong>                     <div><p>Rostock MAX</p></div></p>                                                                                <p class="detail-setting rafts"><strong>Rafts: </strong>                     <div><p>Doesn't Matter</p></div></p>                                                                                <p class="detail-setting supports"><strong>Supports: </strong>                     <div><p>No</p></div></p>                                                        <br>                        <p class="detail-setting notes"><strong>Notes: </strong>                    </p><div><p>Model has a tendency to pop off the bed. Take proper steps.</p></div>

Post-Printing

Put em together!

            <div><p>Since the dragon is self-similar, one may assemble multiple copies of this model into a larger but similar version - or just keep tiling. </p></div>

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. I intended to be able to produce any iteration (up to computational limits) of the dragon curve so I needed some way to automate the process, rather than entering all the points by hand. To create such an automation, I utilized Rhinoceros' python integration.
(N.L., MAT 363, December 2015)

Custom Section

More Details: Mathematics Behind the Design

            <div><p>The python implementation utilizes a specific understanding of the construction of the<br>Heighway dragon curve. Effectively, to construct this curve we start with a straight line<br>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 eciency, instead of using matrix multiplication my program directly<br>converted a point (x; y) into (-y; x). Note that this matrix rotates around the origin. Thus,<br>our algorithm will need to translate our curve so it ends at the origin. From here, it is<br>simple to sweep the curve with a rectangle to produce a 3-dimensional object that can be<br>printed. Unfortunately, the object is not yet fit for tiling because corners would overlap.<br>To allow the object to demonstrate a tiling it was necessary to curve the corners using the<br>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.<br>(N.L., MAT 363, December 2015)</p></div>

Dragon Curve