Icosahedral symmetry

The icosahedral group is the largest finite rotation group in three dimensions apart from the reflections that go with it, and it is the one arrangement of axes that crystallography spent a century certain could not occur in matter. Then it did.

Both halves of that story are computations rather than assertions, and this essay is about doing them: counting what the group contains, showing why a lattice cannot hold it, and saying what the description of a real quasicrystal costs instead.

Sixty rotations, counted #

The group is found here from the solid rather than from a table. An icosahedron’s twelve vertices are the cyclic permutations of (0, ±1, ±φ); a rotation of it is fixed by where it sends one vertex and one of that vertex’s five neighbours; so the candidates are twelve times five, and every one of them turns out to carry the whole vertex set onto itself.

Sixty rotations. Sorting them by order gives six axes of order five, ten of order three and fifteen of order two — and the sorting is checked by counting the rotations twice: an axis of order k carries k−1 non-identity rotations, and the totals must come to fifty-nine.

That check caught a real error. The obvious way to find a rotation’s order is from its trace, since the trace of a rotation by θ is 1 + 2cos θ. But a fivefold axis carries rotations by a fifth and by two fifths of a turn, and those have different traces — so testing the trace against 2π/k for small k recognised only half the fivefold rotations and left twelve of the fifty-nine unaccounted for. Multiplying each matrix by itself until it returns to the identity is slower and cannot make that mistake.

Adding the inversion doubles the group to a hundred and twenty operations, which is the full symmetry of the icosahedron including reflections. The rotations alone are what the counting above produces, and they are what determines whether a lattice can carry the group.

Why no lattice can hold it #

The argument is the plane’s crystallographic restriction, unchanged except for the number of dimensions.

A rotation that maps a lattice onto itself is an integer matrix in the lattice basis. An integer matrix has an integer trace. The trace of a rotation by 2π/n in three dimensions is 1 + 2cos(2π/n), so 2cos(2π/n) must be an integer, which happens only for n = 1, 2, 3, 4 and 6. Five is not on the list, in three dimensions exactly as in two.

So a crystal with a fivefold axis is impossible, and a crystal with six of them is impossible six times over. That was not a conjecture or a rule of thumb: it is a theorem about integer matrices, it has no exceptions, and it is still true.

What Shechtman’s alloy did was not break it. The diffraction pattern had tenfold symmetry and sharp peaks, and both are compatible with the theorem — because the theorem is about lattices, and the material has no lattice. The response was not to weaken the restriction but to widen the definition of a crystal, which the International Union of Crystallography did in 1992 by defining one as a material with an essentially discrete diffraction pattern.

Six axes, six indices #

The practical consequence is in how such a material is described, and it is where the higher-dimensional picture stops being an elegance and becomes the working method.

The six fivefold axes of an icosahedron point at six of its twelve vertices — one of each opposite pair — and no three of them are coplanar. Any lattice whose reciprocal points lie along those directions therefore needs six independent basis vectors, so a diffraction pattern from an icosahedral quasicrystal is indexed by six integers rather than three.

This is exactly what the Fibonacci chain does in miniature. The chain lives on a line and its peaks need two indices, because it was cut from a two-dimensional lattice. An icosahedral quasicrystal lives in space and its peaks need six, because it is a slice of a six-dimensional one. The extra indices are not a bookkeeping device; they are the coordinates of the lattice the material is a shadow of.

The six-dimensional lattice in question is the hypercubic one, and the icosahedral group acts on it by permuting and negating the six axes — so in those coordinates every operation of the group is an integer matrix, and the restriction is satisfied rather than violated. Five-fold symmetry is legal in six dimensions for the same reason it is legal in four, which the essay on higher dimensions works through.

Tenfold, and what a diffraction pattern shows #

The photograph that started the argument did not show six fivefold axes. It showed a tenfold pattern down one of them, and the distinction is worth drawing because it is where Friedel’s law enters.

A fivefold axis in the material gives a tenfold pattern in the diffraction, because the intensity acquires a centre of symmetry whether or not the structure has one — so a fivefold rotation and the half turn it composes with produce ten. An experimenter seeing tenfold symmetry is therefore seeing a fivefold axis plus the measurement’s own contribution, and the first responses to Shechtman’s photographs included the suggestion that five ordinary crystals twinned in a fivefold arrangement would give the same thing.

That objection was serious and it was answered by other means: dark-field imaging showed the diffraction came from a single grain rather than from five, and moving the beam across the sample did not change the pattern the way a twinned aggregate would. The diffraction photograph alone could not settle it, and the site’s usual habit — a claim needs a test it could fail — applies as much to a famous experiment as to a figure.

What the shells look like #

There is a second, much older place where icosahedral symmetry appears in matter, and it is worth separating from quasicrystals because it is not the same phenomenon.

Clusters of atoms frequently adopt icosahedral arrangements: thirteen atoms as an icosahedron with a centre, fifty-five as a second shell, and so on through the sequence of Mackay icosahedra. These are perfectly ordinary objects — finite clusters, with no requirement to tile space — and their symmetry is unconstrained because the restriction only applies to arrangements that repeat.

The comparison worth carrying is with the largest point group a three-dimensional lattice can actually carry, which is the cube’s forty-eight. The icosahedral group with its reflections has a hundred and twenty — two and a half times as many — and not one of the extra operations is available. So the forbidden group is not marginally too symmetric for a lattice; the excess is not made of exotic operations either, but of the six fivefold axes and everything they generate.

Virus capsids are the other standard example, and their icosahedral symmetry is a solution to a packing problem rather than a crystallographic one: a shell built from many copies of one protein has few ways to close, and the icosahedral arrangements are the efficient ones. Nothing about them bears on the restriction, and conflating them with quasicrystals is a common enough confusion to be worth naming.

A finite cluster may have any symmetry it likes. The restriction is a constraint on infinite repeating arrangements, and it says nothing about a single icosahedron, a shell of them, or a molecule. What made 1982 an argument was that the alloy’s diffraction pattern was sharp, which is the signature of long-range order rather than of a cluster.

The group is simple, and that has consequences #

The sixty rotations are not merely sixty; they are a group whose internal structure is unlike that of every crystallographic point group, and naming the structure explains two things this essay has already used without justifying.

The sixty rotations are the alternating group on five letters. The icosahedron contains five inscribed cubes, each using eight of its twenty face centres, and every rotation permutes those five cubes. The permutation determines the rotation, and only the even permutations arise — sixty of them, matching the count.

That group is simple: it has no normal subgroup except the trivial one and itself. Every crystallographic point group, by contrast, is soluble, and its normal subgroups are what a descent of symmetry descends through.

So an icosahedral parent has nowhere gentle to fall. There is no subgroup of index two, so nothing analogous to the two-domain descents that dominate the ferroelastic transitions — a phase transition out of icosahedral symmetry drops to something much smaller in one step, and produces correspondingly many domains.

And it is the same simplicity that makes the general quintic unsolvable by radicals. Klein’s account of the quintic runs through this very group acting on this very solid; the icosahedron is where the two subjects meet, and the coincidence is not one. Both are asking what a group of order sixty with no normal subgroups permits.

The reflections change none of this. Adding the inversion gives a group of order one hundred and twenty which is not the symmetric group on five letters, but the direct product of these sixty rotations with the two-element group generated by the inversion — a distinction worth keeping, since the two groups of order one hundred and twenty are easily confused and only one of them is a symmetry of anything here.

The group that is largest, and the lattice that is largest #

Putting the two extremes side by side makes the size of the gap concrete.

The largest point group a three-dimensional lattice can carry is the cube’s, with forty-eight operations. The icosahedral group with its reflections has a hundred and twenty. So the forbidden group is not marginally too symmetric for a lattice — it is two and a half times the size of the largest one available, and the excess is not made of exotic operations but of the six fivefold axes and everything they generate.

That ratio explains why the restriction feels surprising rather than obvious. Nothing about an icosahedron looks incompatible with filling space; it is a perfectly ordinary solid, and children’s toys are made of them. What fails is not the solid but the repetition: copies of an icosahedron cannot be stacked to fill space without gaps, and the arrangement that comes closest is exactly the one quasicrystals adopt.

The group is simple, and one consequence #

The sixty rotations form a group that group theory knows well: it is isomorphic to the alternating group on five letters, the smallest non-abelian simple group there is. That is not a coincidence of size — the isomorphism is realised by the five inscribed cubes an icosahedron contains, which the rotations permute.

Simple means it has no normal subgroup except the trivial one and itself, and that has an immediate consequence for anything built on this symmetry.

A two-colouring of a pattern is a homomorphism onto the group of order two, and its kernel is a normal subgroup of index two. A simple group has none. So an icosahedral arrangement cannot be two-coloured so that every rotation either preserves or exchanges the colours — the same argument that rules out two-colourings of p3 in the plane, reaching a much larger group by exactly the same route.

The result is worth pausing on because it is a constraint on decoration derived from pure group theory with no geometry in it at all. Anybody colouring a sixty-sided arrangement in two colours is either breaking some of its symmetry or using a colouring that is not compatible with the rotations, and no amount of ingenuity changes that.

The simplicity has a second consequence, historically more important. A group with no normal subgroups has no non-trivial quotients, so it cannot be built up from smaller symmetries in stages — which is why icosahedral order in matter cannot arise as a small distortion of some more ordinary arrangement, and why the discovery of it in an alloy was such a discontinuity. There was nothing for it to be a slight modification of.

What “forbidden” was taken to mean #

The restriction was known and secure for a century before 1982, and the way it was held is part of why the discovery was contested rather than merely surprising.

A theorem about integer matrices does not say a material cannot scatter with fivefold symmetry. It says a periodic arrangement cannot have a fivefold rotation. Between those two statements sits an assumption so universal that it was rarely written down: that any solid producing sharp diffraction is periodic.

That assumption had excellent support. Sharp peaks were understood as the interference condition of a repeating lattice, and every material that had ever produced them had turned out to have one. It was not a definition anybody had chosen carelessly — it was an inference from the whole history of the subject, and it was wrong in exactly one direction.

The counterexample is a set of points with no repeat whose diffraction is nonetheless discrete. The Fibonacci chain is one, the Penrose tiling is another, and both were known as mathematics before Shechtman’s alloy: de Bruijn’s analysis of the tilings appeared in 1981, and the diffraction of quasiperiodic point sets had been studied by mathematicians without any suggestion that matter might do it.

So the resolution was available before the observation, in a different literature, and it took several years for the two to be connected. That gap is the most instructive part of the history and it is not unusual. The theorem was never in danger; what needed changing was an assumption about which objects the theorem applied to, and no amount of care with the mathematics would have surfaced it.

The 1992 redefinition of “crystal” is the formal end of that process. A crystal is now a material with an essentially discrete diffraction pattern — a definition by what an experiment sees rather than by what the structure is, which is a considerable philosophical shift and one made deliberately.

Two ways to have five-fold symmetry #

It is worth separating the two mechanisms by which a material can show a fivefold axis, because they are different physics with the same signature.

Quasiperiodic order. The atoms sit at the positions of a quasiperiodic point set — a projection from a higher-dimensional lattice — and the fivefold symmetry is exact and global. The diffraction is sharp, dense, and indexed by six integers. This is what an icosahedral quasicrystal is.

Twinning. Several ordinary crystals grow joined together in an orientation that produces a fivefold arrangement of grains. Each grain is perfectly periodic and none has a fivefold axis; the aggregate appears to. Fivefold twins are common, well known, and were the first explanation offered for Shechtman’s pattern.

The two are distinguishable and it takes work. A twinned aggregate has boundaries, and imaging finds them; moving the beam across the sample changes which grains contribute, so the pattern shifts. A quasicrystal has no boundaries to find and its pattern does not shift. The peak shapes differ too — twinning gives peaks at slightly inconsistent spacings, since the grains are periodic with a common lattice constant that cannot fit the fivefold arrangement exactly.

That last point is the one worth carrying away. A fivefold pattern from twinned periodic grains cannot be exactly fivefold, because the underlying periodicity forbids it; the deviation is small and it is systematic. A quasicrystal’s is exact. So the discrimination is in the end a measurement of how well the fivefold symmetry holds, which is a quantitative question and was answered quantitatively.

Where the exactness stops #

The rotations are computed in floating point. The icosahedron’s coordinates involve the golden ratio, so its symmetry cannot be settled by integer arithmetic in three dimensions — the whole point being that no integer basis exists. The comparisons here use a stated tolerance of a part in a million, and the group’s structure is discrete enough that nothing is close to the threshold.

The six-dimensional description is stated, not derived. This site’s machinery is two-dimensional and periodic. The claim that an icosahedral quasicrystal is a projection of a six-dimensional lattice is standard crystallography and it is not computed anywhere here; what is computed is the axis count, the order of the group, and the fact that four dimensions suffice for one fivefold rotation.

The restriction’s proof here is the trace argument. It is complete for rotations about a single axis and it is the argument the three-dimensional essay gives in full. Nothing on this page proves that no exotic lattice evades it; that follows from the same theorem, applied to every element of the candidate group at once.

Where the ladder goes next #

The measurement that made an impossible symmetry a physical observation is what Shechtman measured.

The smallest object with the same structure, where the whole story can be checked by reading a string, is the smallest quasicrystal.

The dimension count that makes five-fold legal, and what it costs, is where five-fold becomes legal; the theorem it is escaping is the crystallographic restriction.

What the pictures here cannot show. The axis diagram is a table, and the six-axis figure draws six directions in a plane because a page is flat — the real six axes are not coplanar, and that non-coplanarity is exactly the property the essay is about. No two-dimensional drawing can exhibit it, and the claim rests on the coordinates rather than on the picture.