Forbidden bracelet

thingiverse

This is an example of what's known as a "forbidden" tiling of the plane by regular pentagons and decagons. Despite the fact that the vertex angles of two regular pentagons and one regular decagon add up to 360 degrees, it's impossible to completely cover the entire plane with these two shapes; you can start, but you'll inevitably get stuck. This illustration shows how you might begin to tile the plane, only to become frustrated and then wrap your partial tiling around a cone. This work is based on research supported by the National Science Foundation under Grant No. DMS-1439786 and the Alfred P. Sloan Foundation award G-2019-11406 while the author was in residence at the Institute for Computational and Experimental Research in Mathematics in Providence, RI, during the Illustrating Mathematics program.