Crystallography & Symmetry
patterns derived from their groups
Order without repetition

Cut and project

Take a periodic lattice, cut a strip through it at an irrational angle, and keep the shadow of what falls inside. The result never repeats, has exactly two spacings, and is a quasicrystal — built from something perfectly periodic that is simply not where anybody was looking.
15 min read 6 figures Order without repetitionSymmetry is decidableThe lattice forbids

A pattern that never repeats sounds like a pattern that has to be described point by point. It does not. The construction below produces one from three ingredients — a lattice, a line, and a strip — and everything about the result is determined by the angle of the line.

Cut and projectA square lattice, a strip along a line of the given slope, and the shadow on that line of every lattice point inside the strip. The shadow has two gap lengths; whether their order repeats depends entirely on whether the slope is rational.the shadow2 gap lengths: 0.526 and 0.851no period in the windowlong ÷ short count = 1.6250LSLLSLSLLSLSLLSLLSLSLLSLSLLSLLSLSLmeasured from the projected points, not assumedslope 0.6180
Fig. 1 A square lattice, a line of irrational slope, and the strip swept along it by the unit cell. Every lattice point inside the strip casts a shadow on the line. The shadow has exactly two gap lengths, they occur in the golden ratio, and their order never repeats.

The construction is due to de Bruijn, from 1981, and its arrival changed what a quasicrystal is. Before it, an aperiodic tiling was a curiosity produced by a substitution rule. After it, an aperiodic pattern was a slice of something periodic, which is a description an experiment can index.

The three ingredients

A lattice. Here the square lattice Z2\mathbb{Z}^2, the simplest there is. Nothing exotic happens in the lattice; all the content is in how it is cut.

A line, through the origin, at slope ss. This is the “physical space” — the space the resulting pattern lives in.

A strip, the region swept by the unit square as it slides along the line. Its width is not a free parameter: it is exactly the projection of the unit cell onto the direction perpendicular to the line, and choosing it that way is what makes the result a point set with a positive minimum spacing rather than a dense mess or a set with gaps of unbounded size.

Then: keep every lattice point inside the strip, project each one perpendicularly onto the line, and read off the sequence of gaps.

What the slope decides

Everything. The construction is the same for every slope, and the character of the result flips entirely on whether ss is rational.

A rational slope gives a periodic sequence. If s=p/qs = p/q in lowest terms, the lattice contains the vector (q,p)(q, p), which lies exactly along the line — so translating the whole picture by it maps the lattice to itself, the strip to itself, and the shadow to itself. The shadow repeats with a period the figure finds by searching the gap sequence for a repeat.

An irrational slope gives an aperiodic one. No lattice vector lies along the line, so no translation maps the shadow onto itself. The sequence of gaps goes on forever without repeating, and the figure searches for a period exactly as it does in the rational case and reports finding none.

Those are opposite claims and both are asserted, which is the point of running the figure at more than one slope. The slope is supplied as an exact fraction or not supplied at all, so “is this slope rational” is decided rather than guessed — a floating-point test for rationality would be meaningless.

Cut and projectA square lattice, a strip along a line of the given slope, and the shadow on that line of every lattice point inside the strip. The shadow has two gap lengths; whether their order repeats depends entirely on whether the slope is rational.the shadow2 gap lengths: 0.447 and 0.894period 3long ÷ short count = 2.0000LSLLSLLSLLSLLSLLSLLSLLSLLSLLSLLSLLmeasured from the projected points, not assumedslope 0.5000
Fig. 2 The same construction at slope one half. The lattice vector (2,1)(2, 1) lies along the line, so the shadow repeats — the same two gap lengths, in a sequence with a period the figure finds and prints.
Cut and projectA square lattice, a strip along a line of the given slope, and the shadow on that line of every lattice point inside the strip. The shadow has two gap lengths; whether their order repeats depends entirely on whether the slope is rational.the shadow2 gap lengths: 0.555 and 0.832period 5long ÷ short count = 1.4444SLSLLSLSLLSLSLLSLSLLSLSLLSLSLLSLSLmeasured from the projected points, not assumedslope 0.6667
Fig. 3 And at slope two thirds, where the period is longer. Rational slopes give periodic shadows with periods that grow as the fraction’s denominator grows, and the limit of that process is the aperiodic case rather than a break from it.

Two gap lengths, always

The most striking feature of the shadow is one that does not depend on the slope at all: there are exactly two distinct gaps, whatever the line’s angle.

The reason is a counting argument. A point of the shadow comes from a lattice point in the strip, and the next point along comes from one of only two possible lattice neighbours — a step of (1,0)(1, 0) or a step of (0,1)(0, 1), since any larger step would leave the strip. Those two steps project to two lengths, and there is nothing else to project.

That is asserted on every call: the gaps are collected, the distinct values counted, and the figure fails if there are not exactly two. It is the assertion that would fire if the strip were the wrong width — too wide and a third kind of step fits inside, too narrow and some gaps become sums of two others.

The proportions of the two gaps are where the slope reappears. For the golden slope the long gaps and short gaps occur in the ratio φ\varphi, measured on the sequence and printed on the figure. And the sequence of long and short — written LSLLSLSLLSLLS… — is the Fibonacci word, the same string produced by repeatedly substituting L → LS and S → L.

Two constructions, one object

That coincidence is not one. The cut-and-project shadow at the golden slope and the Fibonacci substitution sequence are the same sequence, and the equality is the one-dimensional shadow of the equality between de Bruijn’s construction and Penrose’s.

The two routes could hardly look less alike. One is a substitution rule applied to a string; the other is a geometric slice through a two-dimensional lattice. What connects them is that the substitution matrix (1110)\begin{pmatrix}1 & 1\\ 1 & 0\end{pmatrix} has eigenvalues φ\varphi and −1/φ-1/\varphi, and its eigenvectors are exactly the direction of the cut and the direction perpendicular to it. The substitution is what the lattice’s own linear map looks like after the projection.

This is the strongest argument for taking the higher-dimensional description seriously. It is not merely a way of generating aperiodic patterns; it is a way of generating the same aperiodic patterns that the substitution rules generate, from a construction that has a lattice in it and can therefore be indexed.

A Penrose tiling, 5 inflationsTwo rhombs, subdivided into smaller copies of themselves over and over. The result covers the plane, has five-fold symmetry about its centre, and never repeats — there is no translation that maps it to itself.890 tilesthick ÷ thin = 1.6176golden ratio = 1.6180generated by substitution, never by placing tilesdepth 5
Fig. 4 The two-dimensional case, generated by substitution. The same tiling comes out of a five-dimensional cut-and-project with a pentagonal window, and the equivalence of the two routes is the reason the higher-dimensional description is worth its awkwardness.
InflationOne tile subdivided into smaller copies of the same two shapes, repeatedly. The rule is local and deterministic, and the pattern it builds has long-range order without any repeating cell.one tile1 tile1 inflation2 tiles2 inflations5 tiles3 inflations13 tilesthe substitution rule applied to a single tile
Fig. 5 The substitution route, on Penrose’s tiles. Each tile becomes several, and the growth factor is the golden ratio — which is the eigenvalue of the same matrix whose eigenvector is the slope of the cut.

Reading the shadow as a tiling

The one-dimensional shadow is a set of points, and turning it into a tiling is a change of language rather than of content: put a tile between each pair of neighbouring points, and the two gap lengths become two tile lengths.

Done that way, the Fibonacci chain is a tiling of the line by two intervals whose lengths are in the ratio φ\varphi, arranged in a sequence that never repeats. It is the one-dimensional Penrose tiling in every respect worth having — same construction, same substitution, same diffraction behaviour, and small enough to write out.

The diffraction is the property that matters physically and it is the one worth stating carefully. The Fourier transform of the Fibonacci chain is a set of sharp peaks, like a crystal’s, and not a smooth spectrum like a random arrangement’s. But the peaks are indexed by two integers rather than one: their positions are m+n/φm + n/\varphi for whole numbers mm and nn, so they form a dense set on the line with only finitely many strong ones in any window.

That is the general signature of a quasicrystal, in its smallest case. Sharp peaks mean long-range order. Needing more indices than there are dimensions means no lattice. Both at once was the thing believed impossible until 1982, and this construction shows the two are compatible in the simplest setting there is.

A diffraction pattern with tenfold symmetrySharp spots, arranged with a symmetry that no periodic crystal can have. When this was measured in 1982 the immediate reaction was that the sample must be a twinned crystal, because the alternative was that a theorem with a one-line proof had a case nobody had considered.181 reflections10-fold symmetryforbidden to any latticeinteger combinations of ten star vectorsaperiodic
Fig. 6 The same phenomenon in two dimensions, as an experiment sees it: sharp reflections in an arrangement no lattice can produce, needing more indices than the space has dimensions. The construction on this page is what supplies the extra indices.

The window, and what it selects

In the two-dimensional picture the strip’s cross-section is an interval, and the general name for that cross-section is the window or acceptance domain. It lives in the space perpendicular to the physical one, and it is the part of the construction with genuine freedom in it.

The choice of window changes the pattern. Shifting it produces a different aperiodic sequence with the same statistics — locally indistinguishable from the first, in the sense that every finite patch of one occurs somewhere in the other, and globally different. Changing its shape changes the pattern more substantially.

For Penrose tilings the construction runs from five dimensions rather than two: project Z5\mathbb{Z}^5 onto a two-dimensional plane invariant under a five-fold rotation, with a window that is a pentagon. Different pentagon positions give the uncountably many distinct Penrose tilings, all locally indistinguishable, and the fact that a five-fold rotation of Z5\mathbb{Z}^5 exists at all is the totient condition permitting what the plane forbids.

The window is also where the physics enters. A real quasicrystal’s structure is described by a higher-dimensional lattice and an acceptance domain, and refining a quasicrystal structure means refining the shape of that domain — which is a genuinely strange thing to be measuring, and is what quasicrystallography actually does.

Why the strip has to be exactly that wide

The width of the strip looks like a parameter and is not, and the two ways of getting it wrong are instructive because they fail differently.

Too narrow, and the shadow acquires gaps that are sums of the two basic ones. Lattice points that should have been kept fall outside, so a step that should have been short-then-long becomes one long step, and the gap census finds three values or more. The assertion in the figure fires on the count.

Too wide, and points that should have been excluded come in. Two lattice points project to positions closer together than the minimum spacing, so the shadow acquires a third, shorter gap — and in the limit of a very wide strip the shadow becomes dense, which is what the projection of a lattice does with no window at all.

The correct width is the projection of the unit cell onto the perpendicular direction, and the reason is a tiling argument. As the window slides along the perpendicular direction, each lattice point enters it and leaves it exactly once, and the intervals during which the points are inside tile the perpendicular axis without gap or overlap. That is the same partition condition a fundamental domain has to satisfy, one dimension down, and it is what guarantees the shadow has exactly two spacings.

So the window is not a tuning knob. It is determined by the lattice and the direction of the cut, and the only genuine freedom is where along the perpendicular axis it sits — which is the freedom that produces uncountably many distinct patterns with identical statistics.

Where the exactness stops

Two boundaries, and the first is inherent to the construction rather than to this implementation.

The projection uses irrational numbers. The whole point is a slope with no rational description, so every coordinate in the shadow is a floating-point number, every gap is a floating-point difference, and every comparison between two gaps is made with a tolerance. The assertions here are correspondingly framed: the gap count is exact because the two values are separated by far more than any rounding, and the golden-ratio claim is a measurement with its distance from φ\varphi printed.

Periodicity is decided within a window. The figure searches for a repeat in the gap sequence it drew, and finding none is evidence rather than proof — a period longer than the sequence would be missed. For the rational cases the period found is short and the evidence is conclusive; for the irrational case the honest statement is “no period within the window drawn”, which is what the figure prints.

The proof that no period exists at all is the argument given above, about no lattice vector lying along an irrational line, and it lives in the prose because it is a claim about an infinite object.

The surprising part

The aperiodic case is not a break from the periodic ones. It is their limit, and the sequence of rational slopes approaching an irrational one produces periodic shadows whose periods grow without bound.

That reframes what aperiodicity is. A pattern with period 10610^{6} is periodic and utterly indistinguishable, in any finite observation, from one that never repeats. The mathematical distinction between the two is absolute and the physical distinction is not — which is why the question “is this material a quasicrystal or a very large-celled crystal” was a genuine controversy for a decade after 1982 and could not have been settled by any single measurement.

The connection worth carrying is to continued fractions. The rational slopes that best approximate a given irrational one are its continued-fraction convergents, and for the golden ratio those are the ratios of consecutive Fibonacci numbers. So the periodic shadows that best approximate the Fibonacci chain have periods 1,2,3,5,8,13,…1, 2, 3, 5, 8, 13, \ldots — and the golden ratio, being the irrational hardest to approximate by rationals, is the slope whose shadow is furthest from periodic. The most aperiodic pattern comes from the most irrational number, and the two senses of “most” turn out to be the same sense.

What this makes of the restriction

The construction changes the status of the crystallographic restriction without contradicting a word of it, and the reframing is worth stating on its own.

The restriction says: a periodic pattern in the plane cannot have five-fold symmetry. Cut-and-project says: a five-fold pattern in the plane can be the shadow of a periodic pattern in five dimensions. Both are true, and together they say that five-fold symmetry is not forbidden, it is expensive — it costs three extra dimensions to store.

That is a better description of what quasicrystals are than “exceptions to a theorem”, and it has a practical consequence. A quasicrystal’s diffraction pattern cannot be indexed with three integers, and can be indexed with six, and the six are the ordinary Miller indices of the six-dimensional lattice. Every technique that depends on indexing — structure solution, absence rules, refinement — becomes available again once the extra indices are admitted.

Which is why the International Union of Crystallography’s 1992 redefinition of “crystal” reads the way it does. It requires an essentially discrete diffraction pattern and says nothing about periodicity, because periodicity turned out to be a special case of something better defined by what an experiment sees.

Who built it

Nicolaas de Bruijn published the cut-and-project description of Penrose tilings in 1981, in two papers that also gave the “pentagrid” construction — a second, equivalent route through five families of parallel lines. Both were written before anybody had measured a quasicrystal.

The generalisation was immediate once there was something to apply it to. When Shechtman’s result appeared in November 1984, papers describing icosahedral quasicrystals as six-dimensional projections were published within months, by Levine and Steinhardt, by Kramer and Neri, and by others independently. The mathematics was ready and waiting, which is not the usual order of events.

Meyer’s earlier work is worth naming as well. Yves Meyer studied what he called model sets in the early 1970s, for reasons in harmonic analysis with no connection to crystals at all, and they are exactly cut-and-project sets. The subject was constructed three times over — by a number theorist, by a combinatorialist and by physicists — before anybody knew there was a material.

Where the ladder goes next

The property being constructed is order without periodicity, and the pattern it produces in two dimensions is the Penrose tiling.

The higher-dimensional lattice being cut is where five-fold becomes legal, and the alternative route to the same tilings is substitution with its matching rules.

The measurement that made any of it urgent is what Shechtman measured.

What the pictures here cannot show. The shadow drawn on each figure is a few dozen points from an infinite sequence, and aperiodicity is a property of the infinite object. A picture of a finite stretch of a non-repeating sequence is indistinguishable from a picture of a finite stretch of a sequence whose period is longer than the stretch — which is exactly the difficulty the prose describes, appearing in the figure as well.

Where this goes next

181 reflections10-fold symmetryforbidden to any latticeinteger combinations of ten star vectorsaperiodic Order is not periodicity For most of a century the two words were used interchangeably, because every known ordered structure repeated. A diffraction pattern measured in 1982 forced them apart, and the definition of a crystal was rewritten. 4 dimensionsorder 5integer entries625 shadow points 0 0 0 -1 1 0 0 -1 0 1 0 -1 0 0 1 -1the lattice's shadow on the plane it rotates5-fold Where five-fold becomes legal A five-fold rotation with whole-number entries exists — in four dimensions, as a four-by-four matrix that can be written down. The plane forbids it because the plane is too small, and knowing which dimension is large enough changes what a quasicrystal is. 890 tilesthick ÷ thin = 1.6176golden ratio = 1.6180generated by substitution, never by placing tilesdepth 5 Penrose tilings Two rhombi, a rule about how their edges may meet, and a tiling that covers the plane completely and never repeats. The five-fold symmetry a lattice forbids, obtained by giving up the lattice.