Inflation, and where the golden ratio comes from

There is no way to lay out a Penrose tiling tile by tile. Following the matching rules locally can lead to a position where no legal tile fits, and no local test predicts the dead end in advance.

The construction that works runs the other way: start with one tile, cut it into smaller ones by a fixed rule, rescale, and repeat. That is inflation, and it produces a legal tiling of any size with no possibility of failure.

The factor is the golden ratio, and this essay is about why it has to be.

The substitution rule #

The tiles are most conveniently handled as half-rhombs — the Robinson triangles, obtained by cutting each rhombus along its short diagonal. There are two, thick and thin, and each subdivides into smaller copies of both.

Write $F$ for the thick triangle and $T$ for the thin one. The rule is that a thick triangle becomes two thick and one thin, and a thin triangle becomes one thick and one thin.

That is all the rule is: two replacements, fixed once and applied everywhere. Everything else about the tiling follows from iterating them.

The bookkeeping is a matrix. If a patch contains $f$ thick tiles and $t$ thin ones, one inflation gives

$$\begin{pmatrix} f' \\ t' \end{pmatrix} = \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix} \begin{pmatrix} f \\ t \end{pmatrix}.$$

The substitution matrix is the whole of the tiling’s combinatorics, and reading it is a standard piece of linear algebra. Note what it does not contain: no angles, no lengths, no matching rules. Everything about how many of each tile appears is in four whole numbers.

Where the ratio comes from #

Iterating a matrix drives any starting vector toward the direction of the eigenvector belonging to the largest eigenvalue. So the ratio $f/t$ converges to the ratio of that eigenvector’s components, whatever the tiling started from.

The matrix above has characteristic polynomial $\lambda^2 - 3\lambda + 1 = 0$, with roots $\lambda = (3 \pm \sqrt5)/2$. The larger is $\varphi^2 \approx 2.618$, where $\varphi = (1+\sqrt5)/2$ is the golden ratio, and the corresponding eigenvector has components in the ratio $\varphi$.

So the tile counts grow by $\varphi^2$ per inflation, the linear scale grows by $\varphi$, and the ratio of fat to thin converges to $\varphi \approx 1.618$.

None of that was designed in. The subdivision rule is a statement about geometry — which smaller tiles fit inside a larger one — and the golden ratio is what the resulting arithmetic produces.

Watching it converge #

The convergence is fast, and the numbers are worth seeing because they are what the figures on this page actually report.

Starting from the five-fold seed — a wheel of ten thin triangles, so $f = 0$ and $t = 10$ — and iterating the matrix gives the thick-to-thin ratio at each depth:

The golden ratio to five decimal places is 1.61803. By depth seven the tiling agrees with it to every digit shown, and the agreement is not a fit — it is convergence toward an eigenvector, at a rate set by the ratio of the two eigenvalues.

The intermediate values are also informative: 1, 3/2, 8/5, 21/13, 55/34, 144/89. Those are ratios of alternate Fibonacci numbers, which is the same fact from a different direction — the substitution matrix is a Fibonacci recurrence in disguise, and the golden ratio is the limit of Fibonacci ratios.

Why an irrational ratio proves aperiodicity #

This is the cleanest argument for aperiodicity there is, and it takes two sentences.

Suppose a tiling by these two tiles were periodic. Then it has a unit cell, the cell contains some whole number of thick tiles and some whole number of thin ones, and the ratio over the whole tiling is that ratio of whole numbers — a rational number.

But the ratio in a Penrose tiling is $\varphi$, which is irrational. So no such tiling is periodic.

The argument is worth appreciating for what it does not use. It says nothing about matching rules, nothing about five-fold symmetry, and nothing about the shapes beyond how many of each there are. It is pure counting, and it applies to any substitution tiling whose matrix has an irrational eigenvalue ratio.

Why local rules cannot build it #

It is worth being precise about why inflation is necessary rather than merely convenient, because the reason says something about what aperiodic order is.

Assembling a tiling by matching rules is a local procedure: place a tile, check that its edges agree with its neighbours, place another. Locally legal, at every step. And yet such a procedure can reach a configuration where every remaining space is unfillable — a dead end — and there is no bound on how far ahead the trouble was decided.

That is not a defect of a particular rule set. It is a property of aperiodic tilings in general: legality of a patch is a local condition, but extendability to the whole plane is not, and no finite lookahead settles it.

Inflation sidesteps the whole issue by never making a local choice. Each tile’s subdivision is determined, so no decision is ever taken that could turn out to be wrong later. The price is that the tiling grows from the inside outward rather than being extended at an edge, which is a different thing from what a tiler with a bag of tiles is doing.

There is a converse worth noting. Every legal Penrose tiling can be obtained by inflation from somewhere, because deflation always succeeds — group the tiles into supertiles, and the grouping is forced. So the constructive method loses nothing.

Self-similarity, which is the replacement for translation #

Inflation is not only a construction method. It is a symmetry of the finished tiling, of a kind the classification of plane patterns does not include.

Take an infinite Penrose tiling. Rescale it by $\varphi$ about a suitable point and redraw. What appears is the same tiling — not a similar one, the same one, up to the local indistinguishability that all Penrose tilings share.

So the tiling is invariant under a scaling, where a periodic pattern is invariant under a translation. That is why the crystallographic restriction has nothing to say about it: the restriction constrains operations that preserve distance, and a scaling does not.

It also explains why the structure is determinate despite having no repeat. A pattern invariant under scaling is constrained at every length scale at once, and the constraint propagates outward from any patch. That is a different mechanism from periodicity and it delivers a comparable amount of order.

Deflation, and reading the hierarchy backwards #

The inverse operation is deflation: group tiles into larger ones according to the inverse of the substitution rule, and rescale down.

Deflation is what makes the hierarchy visible, and it is why a finite patch determines the family but not the member. Take any Penrose tiling, deflate once, and a legal Penrose tiling of larger tiles appears. Deflate again and a still larger one appears. Every tile belongs to a supertile, which belongs to a supersupertile, and the nesting continues without limit.

That hierarchy is the tiling’s real structure, and it is why local information determines the family but never the individual. A finite patch determines its supertile, which determines its supersupertile, and so on — but only up to the point where the patch runs out, and the choices further up the hierarchy are unconstrained by anything visible.

The seed, and what depends on it #

One detail of the construction has consequences, and it is the choice of what to start from.

The figures here begin with a wheel of ten thin triangles meeting at a point — the five-fold seed — which is why the tilings shown have an exact five-fold centre. Most Penrose tilings do not. Start from a single triangle instead and the tiling that grows has no global rotational symmetry at all, while being locally indistinguishable from the symmetric one in every finite window.

So the five-fold centre visible in these pictures is an artefact of the seed rather than a property of Penrose tilings, and saying otherwise would be exactly the kind of over-claim this site tries to avoid. What is a property of every member of the family is the orientational statistics: every tile edge points in one of five directions up to reversal, in equal proportions, and that is what produces the tenfold diffraction pattern.

The seed also fixes the tile counts quoted above. A different seed gives different numbers at low depth and the same limit, because the limit belongs to the matrix rather than to the starting vector — which is the usual behaviour of an iterated linear map, and a reason to trust the ratio more than any particular count.

Substitution beyond Penrose #

The mechanism is general, and seeing it in a simpler case makes it clearer.

The Fibonacci chain is the one-dimensional version. Take two segment lengths, long and short, and substitute long → long short, short → long. Iterating gives L, LS, LSL, LSLLS, LSLLSLSL, and the ratio of long to short converges to $\varphi$ for exactly the reason above — the substitution matrix is $\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}$, whose dominant eigenvalue is $\varphi$.

That chain is aperiodic, has a sharp diffraction pattern, and is the standard toy model for a one-dimensional quasicrystal — and its sharpness is the whole content of the claim that order does not require periodicity. Everything conceptually difficult about Penrose tilings is present in it, in a form that can be written out on one line.

Other substitutions give other constants. The Ammann–Beenker tiling, with eight-fold symmetry, has silver ratio $1+\sqrt2$ in place of $\varphi$. The chair and table tilings use integer factors and are aperiodic for different reasons. What the family has in common is that a fixed local replacement rule, iterated, produces global structure that no finite rule could specify directly.

Quadratic irrationality, which is the condition #

There is a reason the inflation factors that appear are always numbers like $\varphi$ and $1+\sqrt2$ rather than $\pi$ or $e$, and it is worth stating because it connects back to the impossibility proof.

A substitution matrix has whole-number entries, so its eigenvalues are algebraic numbers of degree at most the matrix size — for a two-tile substitution, roots of a quadratic with integer coefficients. Those are exactly the quadratic irrationals.

That is the same class of number that arises from the descent construction, where the shrinking factor $2\cos 72° = 1/\varphi$ appears. Five-fold rotation generates an algebra whose fundamental unit is $\varphi$, and both the impossibility of periodicity and the existence of an inflation rule come out of that single algebraic fact.

Irrationality forbids periodicity. Quadratic irrationality permits self-similarity. A tiling built on a transcendental ratio would be aperiodic and would have no inflation rule, and would be correspondingly less interesting — it would lack the hierarchy that makes Penrose tilings tractable.

How the figures here are built #

The inflation figures on this site are generated by running the substitution, and the honest description of what they check is narrower than for the periodic figures.

The generator starts from a seed triangle, applies the subdivision rule to a chosen depth, and draws the result. Legality is guaranteed by construction rather than verified afterwards, since every tile is produced by a legal replacement. The tile counts are counted, and the ratio is reported as a measurement.

What is not available is the round trip that certifies every periodic figure here. That machinery works in coordinates along a lattice’s repeat vectors, and there are none. So an aperiodic figure on this site is constructed and measured, where a periodic one is generated and proved, and the difference is stated rather than glossed.

Where inflation came from #

The idea is older than Penrose tilings and arrived through several routes.

Substitution systems on sequences were studied by Axel Thue from 1906 and are the basis of a substantial part of combinatorics on words. The Fibonacci chain is essentially Thue’s construction with different letters.

For tilings, the hierarchical mechanism appears in Robert Berger’s 1964 aperiodic set and in Raphael Robinson’s 1971 simplification, both of which force aperiodicity by building an infinite hierarchy of nested squares. Penrose’s tiles have the same property in a much more elegant form, and Nicolaas de Bruijn’s 1981 analysis showed the tilings could equivalently be obtained by projecting a slice of a five-dimensional cubic lattice — the cut-and-project construction, which explains the golden ratio from an entirely different direction and connects the subject back to periodic lattices in higher dimensions.

Where the ladder goes next #

The physical consequence is order without periodicity: why an aperiodic structure diffracts sharply, what Shechtman measured, and how the definition of “crystal” was rewritten.

The construction these tilings escape is the periodic one, and the theorem they politely do not violate is the crystallographic restriction.

What the pictures here cannot show. Every figure is a finite patch at a finite inflation depth. Self-similarity is a property of the limit, and the tile ratio quoted converges rather than holding at any depth drawn — the pictures illustrate a process whose defining properties belong to its limit.