Tiles that never repeat
A pattern with a kind of symmetry that wallpaper cannot have.
What you are looking at
The dots arrange into a star-like pattern with five-way symmetry. It looks orderly, but if you slide it over, it never lines up with itself.
The idea
Wallpaper and floor tiles repeat. If you slide the pattern over by the right amount, it lands exactly on itself. Tiles that repeat can only have certain kinds of symmetry: you can turn them by a half, a third, a quarter, or a sixth of a full turn and the pattern still looks the same. A fifth of a turn is not allowed.
In the 1970s, Roger Penrose found two simple tile shapes that cover a floor completely, have five-way symmetry, and never repeat. The pattern has rules and order, but it never lines up with itself, so you cannot slide it into place.
For a long time that seemed like a game with shapes. Then, in 1982, Dan Shechtman saw the same five-way symmetry in a real metal alloy. That discovery won a Nobel Prize in 2011.
Where else you see it
- Decorative tile patterns in some old buildings come close to this.
- Quasicrystals have been used in some hard, slippery coatings.
Go deeper: the proper names and the math
The crystallographic restriction theorem says a pattern that repeats in two dimensions can only have 2-, 3-, 4-, or 6-fold rotation symmetry. Penrose tilings avoid the restriction by not repeating (they are aperiodic).
They are built using the golden ratio: the ratio of the two tile counts approaches φ, and the tiling can be generated by repeated "inflation" (cutting each tile into smaller tiles). Another way to make one is to take a slice through a five-dimensional lattice. Their diffraction pattern is sharp with ten-fold symmetry, which is how quasicrystals were recognized.