Penrose Tilings, Explained

After reading this you will know how two rhombi cover the plane without ever repeating, how recursive subdivision by the golden ratio builds a Penrose tiling, and how to check the deflation counts against real numbers.

What a Penrose tiling is

A Penrose tiling covers the whole infinite plane with copies of just two rhombus shapes. The surprising part is that no matter how you slide the finished pattern, it never lands exactly on itself. It has no repeating unit cell. Ordinary tilings of squares, triangles or hexagons do repeat: shift a square grid by one square and it looks identical. A Penrose tiling refuses that. It is aperiodic.

The two tiles are rhombi with equal side length. One is fat, with corner angles of 72^{\circ} and 108^{\circ}. The other is thin, with angles of 36^{\circ} and 144^{\circ}. Every one of those angles is a multiple of 36^{\circ}, which is one tenth of a full turn. That is the source of the tiling's five-fold and ten-fold flavour. You cannot build a periodic tiling with true five-fold symmetry, yet these tiles carry it everywhere.

Here is the hook. In 1982 Dan Shechtman measured an aluminium-manganese alloy and saw a diffraction pattern with sharp ten-fold spots. Sharp spots mean long-range order. Ten-fold symmetry was supposed to be impossible in a crystal. The resolution was that the atoms sat on a three-dimensional analogue of a Penrose tiling. The mathematics came first, the matter followed, and Shechtman won the 2011 Nobel Prize in Chemistry.

When this model is worth your time

Reach for a Penrose tiling when you want the cleanest example of order without periodicity. It shows that "no repeating cell" and "no structure" are completely different statements. The tiling is rigidly determined and yet non-periodic.

It is also the friendliest doorway into quasicrystals. If you want to understand why a solid can produce sharp diffraction peaks at forbidden angles, the tiling gives you a picture you can draw by hand.

The golden ratio and the two triangles

The engine behind the tiling is not the rhombi themselves but two triangles you get by cutting them in half. These are the Robinson triangles. Both are isosceles.

Thick (acute) Robinson triangle

Angles 36^{\circ}, 72^{\circ}, 72^{\circ}. Half of a fat rhombus.

Thin (obtuse) Robinson triangle

Angles 36^{\circ}, 36^{\circ}, 108^{\circ}. Half of a thin rhombus.

Everything hinges on the golden ratio:

Here \varphi is the positive solution of \varphi^2 = \varphi + 1. In the thick triangle the ratio of the long side to the short side is exactly \varphi. When you draw a point that splits a side in the ratio 1 : \varphi and connect it, each triangle breaks into smaller Robinson triangles of the same two kinds. That step is called deflation, and it is the whole algorithm.

The subdivision rule, step by step

One deflation replaces every triangle with smaller ones, using cuts placed by the golden ratio. The counts are fixed:

• Each thick triangle splits into two thick plus one thin.
• Each thin triangle splits into one thick plus one thin.

Write T_n for the number of thick triangles and H_n for the number of thin ones after n deflations. The rule above gives a pair of linear recurrences:

The symbol T_n counts thick triangles and H_n counts thin triangles at depth n. Each new thick triangle comes from two per old thick plus one per old thin, and each new thin from one per old thick plus one per old thin. The matrix of this map is \begin{bmatrix}2 & 1\\ 1 & 1\end{bmatrix}, whose larger eigenvalue is \varphi^2 \approx 2.618. So the tile count multiplies by about 2.618 every step, and the ratio T_n / H_n settles toward \varphi.

Reading the pattern you get

Once the triangles pair back into rhombi, look for the recurring vertex figures. Where five fat rhombi meet point to point you get a sun. Where five thin rhombi share a point you get a star. These, along with named figures like the "ace", "deuce", "jack", "queen" and "king", are the only ways the tiles can legally meet. Their local five-fold symmetry is what your eye reads as order.

Yet the global pattern is aperiodic. A clean way to feel this is the ratio you just computed. In a periodic tiling the ratio of two tile types is always a simple rational, because the unit cell fixes it. Here the limiting ratio is \varphi, an irrational number. A single repeating cell can never produce an irrational tile ratio, so no such cell exists.

Another local property has a global consequence. Every finite patch you can find anywhere in the tiling appears infinitely often, and it recurs at a bounded spacing. This is repetitivity. So the tiling looks the same everywhere at the level of patches, while never repeating exactly. Local sameness and global difference sit side by side.

Common mistakes

Three errors show up again and again.

• Confusing aperiodic with random. The tiling has zero freedom once the seed and rules are fixed. Every tile position is forced. It is deterministic and non-periodic, not disordered.
• Cutting by 1/2 instead of by \varphi. Halving a Robinson triangle does not give Robinson triangles. The whole scheme collapses unless the cut sits at the golden proportion, so a long piece of length \varphi pairs with a short piece of length 1.
• Ignoring matching rules when assembling by hand. The two rhombi alone can tile the plane periodically if you allow any orientation. The aperiodicity only appears when you enforce edge-matching (the arrows or notches). Deflation builds a legal tiling automatically, which is why the generator uses it.

Related tools

Penrose tilings sit at the crossing of symmetry, recursion and the golden ratio, so several other toys pair naturally with this one. For non-flat tilings that also escape the ordinary grid, try the Hyperbolic Tiling Explorer. To meet the golden ratio in a growing biological pattern, open Phyllotaxis, where the golden angle packs seeds into Fibonacci spirals. The self-similar shrinking of deflation echoes the classic fractal built by chance in the Chaos Game, and it shows up again in the colored triangles of Pascal's Triangle mod n. For randomized tiling patterns without matching rules, see Truchet Tiles, and for a different kind of grid-solving that snaps possibilities into one pattern, look at Wave Function Collapse.

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