A Penrose tiling can be extended forever, yet no shift ever maps it onto itself. Still, it is full of order: five-pointed stars crop up everywhere, and in the limit there are exactly φ ≈ 1.618 times as many thick rhombs as thin ones. This order without a period is called quasiperiodic, and some alloys turn out to freeze into it.
How it works
The first method is de Bruijn's multigrid: n families of parallel lines, turned 360°/n against each other and shifted slightly. Every crossing of two lines becomes a rhomb with sides perpendicular to those lines, and the tile vertex for any point is found by counting how many lines of each family lie behind it. With n = 5 and a whole-number sum of shifts this gives exactly the Penrose tiling of thick (72°) and thin (36°) rhombs; with 8, the Ammann–Beenker tiling of squares and rhombs. It also shows that the Penrose tiling is the shadow of a five-dimensional cubic lattice.
The second method is deflation. Rhombs, like kites and darts, split into Robinson triangles with angles 36–72–72 and 36–36–108, and each triangle splits into copies φ times smaller. The generator starts from a five-fold “sun”. On kites and darts the arcs cut the edges in the golden ratio, the classic matching rule: if neighbours' arcs must continue each other, no periodic tiling can be built. On rhombs the arcs follow a two-colouring of the vertices, so they too join into unbroken curves.
A bit of history
In 1619, in Harmonices Mundi, Kepler fitted pentagons together with stars and suspected the pattern never repeats. Roger Penrose built an aperiodic set of six tiles in 1974 and soon cut it to two; Robert Ammann found the rhombs independently. The public met them in Martin Gardner's column in Scientific American in January 1977, and Nicolaas de Bruijn devised the multigrid in 1981. In 1982 Dan Shechtman saw five-fold symmetry, forbidden for crystals, in the diffraction pattern of a rapidly cooled aluminium–manganese alloy. He was long disbelieved, but won the 2011 Nobel Prize in Chemistry for quasicrystals.
What to tweak
- “Symmetry” at 7, 9 or 11: odd orders have more distinct rhombs; at 12, squares sit beside 30° and 60° rhombs.
- Slide “Shift” slowly with the multigrid and rhombs flip three at a time inside hexagons while the pattern stays put; physicists call these phason flips.
- Set “Construction” to “kites & darts (P2)”, “Arcs” to “golden” and “Colouring” to “one colour”, and the arcs join into long curves. Conway and Penrose proved that every closed curve encloses a region with five-fold symmetry.
- Add the “Relief” effect on top in “emboss” mode for the look of pressed ceramic tiles.
Archimedean tilings
Islamic star patterns
The aperiodic hat
Truchet tiles
Maze
Voronoi and Delaunay
Circle packing
Celtic knots
Weaves