Forbidden bracelet

thingiverse

This is an illustration of the so-called "forbidden" tiling of the plane by regular pentagons and decagons. Although the vertex angles of two regular pentagons and one regular decagon add to 360 degrees, one is not able to tessellate the entire plane with these two shapes; you can start, but you get stuck. This frustrum is an illustration of how you might start to tile the plane, get stuck, and then wrap your partial tiling around a cone. This material is based upon work 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.