A never-repeating pattern: This pattern never settles into a repeating rhythm. Part of Fivefold, our family of never-repeating patterns built on the number five.

Roger Penrose found two shapes, a kite and a dart, that can cover a floor forever without the pattern ever repeating. John Conway noticed that if each tile carries two small arcs, the arcs flow across the seams into smooth curves. The tiles disappear, and loops and flowers with fivefold symmetry appear in their place.

This pattern shows just one of these arcs: its loops make five-petaled flowers and five-pointed stars, each colored by its size.

Order without repetition seemed impossible in nature until 1982, when Dan Shechtman found it in a real metal alloy: a quasicrystal. The discovery earned him the 2011 Nobel Prize in Chemistry.

Never-repeating tilings are important puzzles in computation, too: no computer program can always tell whether a given set of tiles will cover an endless floor; a geometric version of the halting problem.

Even simple rules can result in extraordinary complexity: Some tiles can fit together to cover a floor forever, but only if the pattern never repeats. This pattern is one beautiful example.