Order is not periodicity
Two properties of a structure were treated as the same property for most of the twentieth century. They are not the same property, and separating them took an experiment nobody believed.
Periodicity is having a repeat: a translation that maps the structure onto itself.
Long-range order is having structure that is determined at arbitrary distances — knowing a patch here constrains what is over there.
Every known ordered material had both. The inference from one to the other became automatic, and it was the automatic inference rather than any theorem that failed in 1982.
Why sharpness meant periodicity
The reasoning that made the inference automatic was not foolish, and it is worth reconstructing.
A diffraction pattern is a Fourier transform of the structure. A periodic structure has a Fourier transform concentrated at discrete points — the reciprocal lattice — so a periodic crystal gives sharp spots. A disordered structure has a smeared transform and gives broad haloes, which is what glass does.
Sharp spots therefore mean something discrete in the transform, and for every structure anybody had examined, the discreteness came from a lattice. Given that, sharp spots were read as evidence of periodicity, and the reading was correct essentially every time it was used.
The gap in the argument is narrow and real. A discrete Fourier transform requires long-range order; it does not require a lattice. A structure whose transform is supported on a discrete set that is not a lattice would diffract sharply and have no repeat, and nobody had a reason to think such a structure could exist in matter.
The measurement
On 8 April 1982, Dan Shechtman, on sabbatical at the American National Bureau of Standards, put a rapidly cooled aluminium–manganese alloy into an electron microscope and recorded an electron diffraction pattern with sharp spots in a tenfold arrangement.
His laboratory notebook for the day reads, in Hebrew, “10 fold???”.
The standard explanation for an impossible-looking symmetry was twinning: several ordinary crystallites in different orientations, each perfectly periodic, superimposing their patterns to give an apparent symmetry none of them has. Twinning was common, well understood, and it had explained every previous case.
Shechtman ruled it out. Twinned crystallites can be separated by a sufficiently fine beam, and the tenfold pattern persisted from every region of the sample he could isolate. He also found the pattern’s symmetry was icosahedral in three dimensions — with sixfold, fivefold, threefold and twofold axes in the icosahedral arrangement — which is a much harder thing to fake by twinning than a single tenfold axis.
The reception
Publication took two years. Shechtman’s immediate colleagues did not believe him; the head of his research group suggested he re-read the textbook on diffraction. His first submission was rejected. The paper that eventually appeared, in Physical Review Letters in November 1984, was co-authored with Ilan Blech, Denis Gratias and John Cahn, and it was submitted only after Blech had produced a structural model that made the observation intelligible.
Linus Pauling, then the most decorated chemist alive, maintained the twinning explanation publicly for the rest of his life, arguing that the patterns came from large-cell cubic crystals twinned in a particular way. His formulation — “there are no quasicrystals, only quasi-scientists” — is the most quoted sentence in the episode.
Pauling was wrong, and the reason he was wrong is instructive rather than embarrassing. His alternative was the correct default. Twinning explains apparent forbidden symmetry in the overwhelming majority of cases where it is observed, and the prior probability that a new form of matter had been found was, reasonably, very low. What settled it was accumulating evidence — better samples, thermodynamically stable quasicrystals grown slowly rather than quenched, and eventually single grains large enough to see with the naked eye.
Shechtman received the Nobel Prize in Chemistry in 2011, alone.
What the theorem actually said
It is worth restating precisely, because the popular account of this episode is usually wrong in an important way.
The crystallographic restriction says: a pattern with a lattice may have rotations of order one, two, three, four or six, and no others. Its hypothesis is a lattice — a discrete group of translations with a shortest non-zero vector.
A quasicrystal has no lattice. Every step of the proof fails, not because the proof is weak but because its hypothesis is absent. No theorem was violated, weakened or amended.
What was amended was an unstated assumption: that a material giving sharp diffraction must be periodic. That assumption had never been proved, because it had never needed to be — it was true of everything anybody had looked at, and its failure required a material nobody had made.
This is a recognisable shape of scientific error, and it is worth naming. Not a wrong theorem. A correct theorem, applied to a case outside its hypotheses, with the mismatch invisible because no case outside the hypotheses had ever turned up.
What replaces the lattice
If not periodicity, then what produces sharp diffraction?
The modern answer is a cut-and-project construction, and it is the same idea that de Bruijn used in 1981 to analyse Penrose tilings.
Take a periodic lattice in a higher-dimensional space — six dimensions for an icosahedral quasicrystal, five for a Penrose tiling. Cut it with a three-dimensional plane at an irrational angle, and project the lattice points near that plane down onto it. The result is aperiodic, because the irrational angle means the slice never repeats. And it diffracts sharply, because the higher-dimensional object is periodic and the projection of a discrete Fourier transform is still discrete.
So a quasicrystal is a projection of something periodic, and its forbidden symmetry is inherited from a symmetry that is perfectly legal upstairs. In six dimensions, icosahedral symmetry is compatible with a lattice. In three it is not, and the projection carries the symmetry down while leaving the periodicity behind.
That construction also explains the diffraction pattern’s structure. The reflections are indexed not by three integers but by six, and their positions are integer combinations of six basis vectors projected into three dimensions — which is why they appear dense, with sharp peaks at every scale rather than on a grid.
The tenfold pattern, computed
The figure at the top of this page is not a photograph and not a redrawing of one. It is computed, and the computation is short enough to describe.
Take ten unit vectors at intervals — the star that five-fold symmetry generates, counting each direction and its reverse. Form every sum of a bounded number of them. The resulting set of points is where a decagonal quasicrystal scatters, and it has three properties worth noticing.
It is dense: between any two points there are others, at every scale. A periodic crystal’s reciprocal lattice is discrete in the ordinary sense, with a minimum spacing; this is not.
It is discrete in the sense that matters: the intensities fall off, so only finitely many reflections exceed any given threshold, and what an experiment records is a finite pattern of sharp spots.
And it has exact tenfold symmetry, because the generating star does.
The third property is what makes the pattern impossible for a periodic crystal, and the first is what makes it unmistakable once seen. A twinned periodic crystal produces a superposition of lattices, which is a finite union of grids and still has a minimum spacing. A quasicrystal’s pattern does not, and the difference is measurable given a good enough sample.
Repetitivity, which is what order means here
There is a local way to say the same thing, without leaving three dimensions.
A structure is repetitive if every finite configuration that occurs in it occurs infinitely often, and within a bounded distance of every point. A Penrose tiling is repetitive. So is a quasicrystal.
Penrose tilings are repetitive too, which is the property that makes a structure determinate. Any finite window sees only a bounded number of distinct scenes, each recurring at a bounded spacing, so the structure is constrained everywhere by what it is anywhere.
That is enough for sharp diffraction and it is strictly weaker than periodicity. Periodicity says the same scene recurs at exactly the same displacement everywhere; repetitivity says only that it recurs nearby. The gap between those two statements is the entire subject of this essay.
Three kinds of solid, and where the boundary sits
Placing quasicrystals correctly requires distinguishing three things that a casual account tends to blur.
A glass has short-range order and no long-range order. Its atoms sit at sensible distances from their neighbours and the arrangement loses coherence within a few atomic spacings. Its diffraction pattern is broad haloes, and no sharpening of the experiment turns them into spots.
A crystal in the old sense has long-range order supplied by a lattice. Sharp spots on a reciprocal lattice, indexed by three integers.
A quasicrystal has long-range order supplied by something else. Sharp spots, but their positions are not the points of a lattice — they are dense in reciprocal space, with intensity concentrated at particular ones, and indexing them takes more than three integers.
The distinction between the second and third is measurable rather than interpretive. Count the integers needed to index every observed reflection: three for a periodic crystal, six for an icosahedral quasicrystal, five for a decagonal one. That number is the practical definition, and it is the reason the 1992 redefinition could be written in terms of diffraction without becoming vague.
The definition changed
In 1992 the International Union of Crystallography’s Commission on Aperiodic Crystals replaced the definition of “crystal”.
The old definition was structural: a crystal is a solid with a periodic arrangement of atoms. The new one is operational: a crystal is any solid having an essentially discrete diffraction diagram. Periodicity is not mentioned; a periodic crystal becomes the special case in which the discrete diagram happens to come from a lattice.
Changing a definition in response to an observation is sometimes a sign that a field is protecting itself. Here it is the opposite. The old definition named a mechanism; the new one names what is measured, and the change made the category match the evidence rather than the other way round.
It also has a pleasing consequence for this site’s organising theme. The new definition is stated in terms of the diffraction pattern — which is to say, in terms of the second independent route rather than the first. A crystal is now defined by what it scatters.
An older near-miss
Shechtman’s was not the first observation of something that ought to have prompted the question, and the earlier cases are instructive about how a well-established framework absorbs anomalies.
Incommensurate structures had been known since the 1930s and studied seriously from the 1960s. These are materials whose diffraction patterns contain reflections that cannot be indexed with three integers — a periodic lattice plus an additional modulation whose wavelength is an irrational multiple of the lattice spacing. Sodium carbonate does it; so do many minerals.
The community had a framework for them: superspace, developed by Aloysio Janner and Ted Janssen in the 1970s. The idea is exactly cut-and-project — describe the material as a periodic structure in more than three dimensions and take a slice — and it was working well before 1982.
So the machinery that explains quasicrystals was in place, being used, and not connected to the question. What incommensurate structures had that quasicrystals do not is a lattice underneath the modulation, which kept them inside the old definition. The step to a material with no underlying lattice at all was small mathematically and enormous conceptually, and nobody took it until an experiment forced it.
That is the most interesting part of the episode. The obstacle was not a missing technique. It was that the relevant technique was filed under a heading that did not appear to apply.
What quasicrystals turned out to be
They are not a curiosity, and three developments are worth recording.
Stable quasicrystals exist. Shechtman’s original samples were rapidly quenched and metastable, which supported the view that they were an artefact. In 1987 An-Pang Tsai and colleagues produced thermodynamically stable icosahedral quasicrystals in aluminium–copper–iron, growable as large single grains with facets. That removed the last plausible version of the twinning objection.
They occur naturally. In 2009 Luca Bindi and Paul Steinhardt identified a natural quasicrystal — icosahedrite — in a mineral sample from the Koryak Mountains in far-eastern Russia. It turned out to be of meteoritic origin, formed in the early solar system, and it demonstrated that quasicrystalline order arises without laboratory conditions.
They have properties that follow from the structure. They are typically hard, brittle, poor conductors of heat and electricity despite being metallic alloys, and have unusually low surface friction. The electronic behaviour follows from the absence of a periodic potential — there is no Brillouin zone in the ordinary sense, and the usual band-structure account does not apply.
What this site can and cannot say about them
An honest boundary, because the machinery that certifies every periodic figure here has no purchase on this material.
The round trip works in coordinates along a lattice’s repeat vectors, where operations are integer matrices and translations are small rationals. A quasicrystal has no repeat vectors. There is nothing to use as coordinates, no finite holohedry to enumerate, and no exact comparison to make.
So the aperiodic figures on this site are constructed and measured: the tilings are built by inflation, which guarantees legality by construction; the tile ratios are counted; the diffraction patterns are computed from the structure by the same sum a crystallographer writes down. Where a number appears, it came from counting or from evaluating a sum, and not from a theorem about the infinite object.
That is a weaker claim than the periodic pages make, and stating it is the point. Implying that the integer machinery had decided something about an aperiodic structure would undermine every claim elsewhere on the site.
Where the ladder goes next
The route by which any of this is known is diffraction, and its most informative feature is what is missing rather than what is present.
The construction is inflation, and the tilings are Penrose’s.
And the theorem that survived all of this intact is the crystallographic restriction — worth revisiting once, to see that it says exactly what it always said.
What the pictures here cannot show. A diffraction pattern computed here is computed from a finite patch, so its peaks have finite width; the sharpness that is the whole point of this essay is a property of the infinite structure. And no figure can display long-range order directly — order is a statement about correlations at arbitrary distance, and a page is finite.