Fold-and-Cut Hat and Spectre Tiles

In the late 1990s, Erik Demaine, Martin Demaine, and Anna Lubiw proved that any pattern made from straight line segments in the plane—connected or not—can be cut out of a piece of paper by first making some strategic folds and then making a single cut. This is the now-famous Fold-and-Cut Theorem. (See Erik Demaine’s website,…