The most of an icosahedron a crystal can keep

Before the lattice has a say builds the sixty rotations of an icosahedron from its own vertices and states the fact the restriction forbids: no three-dimensional lattice admits a five-fold rotation, so no crystal has this group.

That is true and it leaves the interesting question unasked. Icosahedral molecules crystallise. C₆₀ does, in a face-centred cubic arrangement; so do virus capsids, in the crystals that structural biology solves them from; so do the boron icosahedra in every boron-rich solid there is. None of them stops being icosahedral on entering a crystal. The crystal simply cannot fix all of it.

So: how much of it can a crystal fix? The site symmetry of a position in a space group is a crystallographic point group, and it has to be a subgroup of whatever sits there. The answer is a subgroup enumeration.

Fifty-nine subgroups, and forty-six of them are available #

The icosahedral rotation group has order sixty and is the alternating group on five letters. Closing every element, then every pair, then repeating until nothing new appears, gives fifty-nine subgroups: the trivial one, fifteen of order two, ten of order three, six of order five, five of order four, ten of order six, six of order ten, five of order twelve, and the whole group.

Lagrange is the check that the enumeration is complete in the only sense a search can be: every order found divides sixty, and the orders that appear are 1, 2, 3, 4, 5, 6, 10, 12 and 60.

Now apply the restriction — not to a matrix, but to a whole group. A subgroup can be a crystal’s site symmetry only if every rotation order in it is one a three-dimensional lattice permits, and that list is 1, 2, 3, 4, 6, computed from integer matrices rather than quoted. Thirteen subgroups fail: the six of order five, the six of order ten, and the whole group. Forty-six survive.

The largest is 23, at index five #

The survivors’ orders are 1, 2, 3, 4, 6 and 12, and the largest is the tetrahedral rotation group — 23 in Hermann–Mauguin, of order twelve. There are five of them inside the icosahedral group, and they are conjugate, so the index is five.

That index has a name older than group theory. The five tetrahedral subgroups correspond to the five cubes inscribed in a dodecahedron — Kepler drew them in the Harmonices Mundi of 1619 — and the icosahedral group permutes those five cubes, which is why it is the alternating group on five letters in the first place.

So the sharpest form of the answer is this: a crystal may fix a twelfth of an icosahedral molecule’s symmetry and no more. Put a C₆₀ molecule at a site of symmetry m3̅ — the tetrahedral class with a centre, order 24 — and 24 of its 120 operations are operations of the crystal. The other 96 are symmetries of the molecule that the crystal does not know about, and every one of them is a way the molecule could be reoriented without the crystal noticing.

What the crystal does with the rest #

The unfixed symmetry does not disappear. It goes into one of three places, and all three are things real fullerene crystals do.

Orientational disorder. The five orientations related by cosets of the site group are equivalent as far as the crystal is concerned, so a molecule may sit in any of them. If it chooses at random from site to site the crystal is orientationally disordered, and the average structure has more symmetry than any of its molecules — which is exactly the symmetry of an average. Solid C₆₀ above 260 K is the standard example: the molecules rotate freely and the diffraction pattern is of a sphere of charge.

An ordering transition. Cool it and the orientations order, the site symmetry drops, and the cell may enlarge — C₆₀ goes from Fm3̅m to Pa3̅ at 260 K, taking the site symmetry from m3̅m to 3̅, which is a descent of symmetry with the daughter’s domain count given by the index.

Merohedral twinning. When the ordering breaks a symmetry the lattice still has, the crystal can order one way in one region and another way elsewhere, with the lattice continuous across the boundary. That is twinning by merohedry, and its variants are counted by the same coset arithmetic as the orientations.

All three are consequences of one number — the index of the site group inside the molecule’s group — and none of them is a fact about fullerenes in particular.

Why the restriction bites here at all #

It is worth being precise about what forbids what, because the argument is often stated in a way that proves too much.

The restriction is about the lattice, not about the contents. A five-fold rotation cannot be a symmetry of a lattice, because an integer matrix of order five would need trace 2 cos 72° = 0.618, which is not an integer. Nothing in that argument says a molecule cannot be five-fold. The molecule is not required to map the lattice to itself; only the crystal’s operations are.

So the constraint on a site is a compatibility, not a prohibition. The site symmetry group is the subgroup of the space group fixing that point, so it is crystallographic by construction; the molecule’s group is whatever it is; and what the crystal can impose is their intersection, which must be a subgroup of both. The enumeration above computes the possible intersections and finds their maximum.

The consequence is asymmetric in an odd way. A molecule with less symmetry than its site is impossible — the crystal would be claiming a symmetry the contents do not have, and the structure would be wrong. A molecule with more symmetry than its site is ordinary and is the normal case. The site symmetry is a lower bound on the molecule’s, never an upper one.

Reading the enumeration: which subgroups a crystal actually uses #

The forty-six crystallographic subgroups are not forty-six different possibilities in practice, because conjugate subgroups give the same site symmetry in different orientations. Sorted by conjugacy the list is short: 1, 2, 3, 222, 32 and 23 — six kinds, with 1, 15, 10, 5, 10 and 5 copies apiece.

That list is worth reading against the thirty-two crystal classes. Every one of the six is a class, so any of them can be a site symmetry in some space group; and the ones missing from the list are missing for a reason that is not the restriction. There is no 4 in it, and no 6, because the icosahedral group has no rotation of order four or six at all — its axes are of orders 5, 3 and 2 and nothing else. So a molecule with icosahedral symmetry can never sit at a four-fold site with its symmetry aligned, even though four-fold sites are perfectly crystallographic.

Two constraints, meeting. The lattice forbids five; the molecule has no four or six. What survives is their intersection, and the intersection is smaller than either constraint alone would suggest.

Where the exactness stops #

The subgroup enumeration is exact and complete. Closure to a fixed point reaches every subgroup, since a subgroup is generated by its own elements and each is added at some round; Lagrange checks the result.

The list of permitted orders is computed, not quoted. It comes from asking which integer matrices of finite order exist in three dimensions, which is the same call the restriction essays use.

What is not decided here is which site a molecule takes. Nothing in this arithmetic says a C₆₀ molecule sits at a site of symmetry m3̅ rather than at one of symmetry 3̅ or 1. That is decided by packing and by energy, and this site computes neither. What the arithmetic supplies is the list of possibilities and the index attached to each — which is exactly what a structure refinement needs before it can choose between them.

And the classification of the molecule’s own group is a finite-group question. A molecular point group is a finite subgroup of O(3), enumerated by the axis equation, and the site-symmetry side of the argument stays with the crystal.

The same argument for the other exceptional groups #

Icosahedral symmetry is the interesting case because it is the one a crystal cannot have at all. Running the same enumeration on the other finite rotation groups is quicker and worth stating for contrast.

The tetrahedral group of order twelve is entirely crystallographic — every one of its subgroups has rotation orders in {1, 2, 3}, so a crystal can fix all of it, and 23 is one of the thirty-two classes.

The octahedral group of order twenty-four is too, and it is 432. Both appear in the thirty-two, and both appear as site symmetries in real space groups.

The cyclic and dihedral groups split by order. C₅, C₇, C₈ and their dihedral partners are as forbidden as the icosahedral group and for the same reason, so a molecule with a seven-fold axis — and they exist — has the same story as C₆₀ with different numbers.

So the icosahedral case is not special in its logic. It is special in that the group is large and the crystallographic part of it is small, which is what makes the leftover symmetry so consequential.

Why twelve is the ceiling, from the group alone #

The largest crystallographic subgroup has order twelve, and the restriction is only half the reason. The other half is a fact about the icosahedral group itself, and it says twelve would be the ceiling even if the restriction permitted more.

The icosahedral rotation group is the alternating group on five letters, and it is simple — it has no normal subgroup but the identity and itself. That has an immediate consequence for indices: a subgroup of index k gives an action on k cosets, hence a homomorphism into the permutations of k things, and simplicity makes that homomorphism injective. So the group of order sixty must embed in the permutations of k objects, which needs k! at least sixty.

Two, three and four are ruled out at once: 2! = 2, 3! = 6 and 4! = 24 are all smaller than sixty. Five is the smallest index a proper subgroup can have, and a subgroup of index five has order twelve.

So there is no subgroup of order fifteen, twenty or thirty — not because the restriction forbids them, but because the group has none — and the largest proper subgroup of the icosahedral rotations is of order twelve whatever else is true. The restriction then has nothing left to forbid, since the tetrahedral group of order twelve is entirely crystallographic.

That is a tidier answer than the enumeration alone gives. The search finds forty-six crystallographic subgroups and reports the largest; the argument says the largest could not have been larger, and says so from the abstract group without any matrices.

The other way to keep all of it #

Everything here asks what a lattice can fix, and the whole of the constraint is in that word. There is a class of solids that keeps the entire icosahedral group, and it does so by giving up the lattice instead.

An icosahedral quasicrystal has the full sixty rotations — a hundred and twenty with the inversion — as symmetries of its diffraction pattern, exactly and not approximately. Nothing is fixed to a subgroup, no orientational disorder is required, and the five cubes have no role.

What it does not have is a lattice, so its reflections need six integers rather than three and every count in this essay is inapplicable. There is no site symmetry, because there are no Wyckoff positions; no index, because there is no subgroup relation; and no coset disorder, because nothing is being compromised.

So the subject has exactly two answers and they are the two the restriction leaves. Keep the lattice and fix at most a twelfth of the molecule’s symmetry, with the remaining five-fold structure disposed of by rotation, ordering or twinning. Or keep the symmetry and lose the lattice. Solid fullerene takes the first and an aluminium–manganese quasicrystal takes the second, and the arithmetic on this page is a description of the first choice’s cost.

Who found it, and when #

Kepler’s five cubes are from 1619 and the connection to A₅ is Galois’s inheritance: the icosahedral group’s simplicity is why the quintic is unsolvable, and Klein’s Lectures on the Icosahedron of 1884 made that the centre of a whole book.

The crystallographic side is much later and was forced by measurements. Boron was the first case: the β-rhombohedral structure, solved through the 1950s and 1960s, is built of B₁₂ icosahedra in a rhombohedral cell, and the icosahedra sit at sites of symmetry 3̅m — order twelve with the centre, and a proper subgroup of the icosahedron’s own group in exactly the way described here.

Virus capsids raised it again in a form that mattered for method. Caspar and Klug’s quasi-equivalence theory of 1962 explains capsid architecture through icosahedral symmetry, and a capsid in a crystal has a site symmetry that is at best a small subgroup of it — so most of the icosahedral symmetry is non-crystallographic, exact for the particle and absent from the crystal, and structure determination exploits it as a constraint rather than as a symmetry. That is the practice where symmetry stacks the vectors describes from the Patterson side, and it is why the phrase non-crystallographic symmetry is a working term in protein crystallography and nowhere else.

C₆₀ closed the circle in 1990–91, when solid fullerene turned out to be face-centred cubic with freely rotating molecules — an icosahedral object in a cubic crystal, with the disorder being precisely the symmetry the site could not fix.

What the enumeration cost, and why it is worth saying #

The subgroup enumeration is a small computation with a large constant, and getting it wrong the first time is instructive about the shape of this kind of work.

Written directly on matrices — close a set, look each product up by formatting nine floating-point numbers into a string, repeat — it took twenty-six seconds. Written on a multiplication table of indices, computed once, it takes forty milliseconds. The mathematics is identical; the difference is that the first version formatted a few million numbers.

Twenty-six seconds is not a performance problem in a script. It is a limit on what a figure can show. A generator that costs half a minute is a generator nobody draws at more than one set of options, and a family of figures that exists at one point of its own parameter space is exactly what this site’s own standard forbids — every placement is meant to ask for the numbers its essay argues about. So the cost of an enumeration decides how much of it can appear in the collection, which is a reason to care about it that has nothing to do with waiting.

The same question for a molecule with a five-fold axis alone #

Icosahedral symmetry is the large case, and the small one comes up more often in ordinary chemistry.

A ferrocene molecule has a five-fold axis. So do the cyclopentadienyl rings in a great many organometallic compounds, and so does every porphyrin’s four-fold cousin’s five-membered relative. None of them can have its five-fold axis fixed by a crystal, and the consequences are the same three: disorder, a superstructure with the axis broken, or an ordered structure in which the molecule sits at a site of symmetry 1 and the five-fold symmetry is exact for the molecule and absent from the crystal.

The largest crystallographic subgroup of C₅ is trivial, which is the extreme case of this essay’s arithmetic: five is prime and none of its divisors above one is a permitted order, so a crystal can fix nothing of a five-fold axis. That is a sharper statement than the icosahedral one, where at least a twelfth survives.

It is also why five-fold molecular symmetry is so often reported as disorder in the crystallographic literature. The molecule is fine; the site cannot hold it; and the refinement sees a smeared ring.

Where the ladder goes next #

This rung answers the question for one exceptional group. Two rungs sit above it.

The lattice that can hold it. Six dimensions. The icosahedral group permutes its own six five-fold axes with signs, so in coordinates along those axes every operation is an integer matrix — and Z⁶ is a lattice it maps onto itself. That is six integers and the lattice that holds them, and it is where the forbidden symmetry becomes legal by paying in dimensions.

The quasicrystal. A real material with icosahedral diffraction symmetry is not a crystal with a large cell; it is an aperiodic structure whose symmetry is exact and whose lattice does not exist. What Shechtman measured is the experiment, and the six-dimensional construction is the description that made it respectable.