Aperiodic tilings are arrangements of shapes that cover a surface without repeating patterns. Unlike regular tilings, which can be extended infinitely in a predictable way, aperiodic tilings create unique designs that never repeat, no matter how far they are extended. One famous example of aperiodic tiling is the Penrose tiling, discovered by mathematician Roger Penrose. These tilings use a limited set of shapes, such as rhombuses, to create intricate patterns that exhibit both order and randomness, showcasing the beauty of mathematical structures in art and nature.