One Tile to Rule Them All
Tile a floor with identical triangles, and you get a boring repeating pattern. But what if you want the pattern to never repeat, yet use just one tile? For a long time, mathematicians were sure it couldn't be done: you'd need at least two different shapes. The famous Penrose mosaic is a case in point: rhombi with acute and obtuse angles arrange into an infinitely varied but strictly ordered pattern.