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.

In the 1970s Roger Penrose found a pair of diamonds, one wide and one narrow, that cover a floor without gaps or end, but never fall into a repeating pattern.

Some places have five-pointed stars, and others a star surrounded by a ring. These two diamonds create many different arrangements together, but overall, wide diamonds outnumber the narrow ones by the golden ratio.

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.