Stereographic projection is a mapping technique that allows a spherical object to project a pattern onto a 2-D surface. Mathematical concepts and physical models of stereographic projection have been around since ancient Greece, when Ptolemy and others used it to visualize their maps of the stars. Later on, cartographers used the method to map the Earth.
In the era of 3-D printing, designers Henry Segermen and Saul Shleimer have made objects which sterographically project Escher-like patterns of light and shadows. Seeing these, mathematician Jason Cole was inspired by their work to create his own models of 3-D printable lampshades which can project these intricate patterns. Like Segerman and Shleimer, Cole started with a 2-D design and used it to reverse-engineer a 3-D physical object (in this case a lampshade).
Cole describes his process in a blog post, including a rundown of the mathematical formulas. (If you’re into that kind of thing.) Using patterns found through Google image search, Cole made the necessary adjustments to bend them into spheres that will project an accurate representation of the 2-D design when a light is placed at the proper point inside. With the open-source 3-D modeling software Blender, Cole was able to test his models and determine whether they could actually be 3-D printed. Spoiler: it worked!
Finally, he tweaked the original design to produce more fun projected results. In a gif, he modeled what patterns the lampshade would project if it were different sizes, and created more spheres using different patterns, including a famous street art design of a panda holding two pistols (frequently misattributed to Banksy). “This might also be the first time ever that a panda bear has been meshed with an FEA mesher,” Cole wrote.