In my last post I described my initial experiments with my new EggBot pen plotter, drawing on clear plastic “DIY” Christmas ornaments. My initial goal with those experiments was to minimize the distortion that often occurs when translating flat, planar designs to the sphere. The next set of experiments I did went in the exact opposite direction: utilize intentional distortion on the sphere, so that shadows in the plane are undistorted. This was motivated by the cover of my friend Henry Segerman’s book, Visualizing Mathematics with 3D Printing, where he shows a 3D printed spherical design whose shadow is a regular grid of squares.
My first experiment was to try and replicate Henry’s design as closely as possible. Here’s the result:
Pretty close! The little diagonal tick marks in the corners of each square were simply because I didn’t have the settings on the pen plotter dialed in exactly right yet, but I was clearly on the right track. Next, I tried a phyllotaxis (sunflower-like) design. As described here, such designs are characterized by two factors: each “seed” should be placed around a central axis according to the “golden ratio”, and the spacing between seeds should fill the resulting surface as densely as possible. In my last post I showed what kind of design you get if you apply these two assumptions to the sphere. For my next design, I applied these assumptions to the plane, and projected back to the sphere to get a distorted design whose shadows would be correct:
Next up: the Earth. For as long as people have been drawing maps of the Earth, Cartographers have struggled with the fact that it is impossible to do so without distortion. To deal with this, they have come up with many schemes to make maps of specific regions that minimize this deficit as much as possible. One common method when drawing maps of the arctic regions is to project out from either the center of the Earth or the opposite pole. However, if you literally do this with a transparent globe, the map will look reversed. To compensate, I plotted a mirror-image globe to make shadows that look correct. This was similar to a 3D printed globe I designed several years ago, featured in the Brilliant Geometry exhibition by Henry Segerman, Saul Schleimer, and Sabetta Matsumoto. (There’s a fun walk-through of this exhibit posted here.)
Finally, I tried something a little different. I wanted to create a design whose shadow was the standard Mercator projection of the Earth, the map that people are most familiar with. The Mercator projection is not a spherical projection at all, so creating an image on the sphere that projects to such a map creates a globe that is completely unrecognizable!
