Crystallography & Symmetry
patterns derived from their groups
Concept

Rotoinversion — where it appears

A rotation followed by an inversion through a point on its axis, a single operation although neither half of it need be a symmetry. Its linear part fixes no direction at all, which is why a class containing one can never act on space without holding a point still.

Named by 10 essays across 5 fields — each of them below, with the objects they name alongside it.

The crystal classes 4, 4̅, 3̅, 6̅. 4, 4̅, 3̅, 6̅: the orbit of a general direction under each group, giving 4, 4, 6, 6 poles, with general positions and symmetry elements. Filled marks are poles above the plane of the page and open ones below it.

What a trace decides

Ten kinds of operation, and two integers tell them apart. The determinant and the trace name a symmetry operation completely — which is why the classification can be run on integer matrices in a lattice basis and never once ask what angle anything turns through.

point-groups · Crystal classes P4_1: a screw of 90°. One operation of P4_1, reduced to Chasles' three numbers: an axis, an angle of 90°, and a pitch of 1.25 along it. The points are the orbit of one position under repeated application, which climbs because the pitch is not zero — and it is not zero for any choice of origin, which is what makes this a screw rather than a rotation. It needs 4 mirrors, and their product was checked against the operation before this was drawn.

Every motion of space is a screw

A rigid motion of space that preserves handedness turns about some axis and slides along that same axis, and there is nothing else it can do. Rotations and translations are the two ends of that one description, the axis and the pitch are computed rather than recognised, and the operations a space group is made of stop being a list of kinds.

operations · What symmetry is P2₁/c from 27 marks. The marks of P2₁/c's plan, counted by kind, and what they rebuild to. Each mark is reduced to what a reader can see and handed to a closure with the matrices withheld: an axis gives its direction, its position and how far one turn advances along it; a plane gives its normal, its position and its slide. The lattice supplies the candidate matrices, the closure supplies the rest, and what comes back is the group — 4 operations against 4, with nothing missing and nothing extra.

The plan contains the group

A space-group diagram has always been treated here as a picture of the group. It is more than that: hand back the marks alone — no matrices, no operations, not even the centring — and the group comes out exactly, forty-five times out of forty-five.

space-groups · Space groups Every crystal class is a rotation group, read one of three ways. The 32 crystal classes sorted by their rotations. Each row is one of the 11 proper classes; beside it is the class obtained by adjoining the inversion, which doubles the order, and the classes obtained by negating the half of the group outside a subgroup of index two, which keeps it. The columns hold 11, 11 and 10 classes, and every class appears exactly once. 3 rows have nothing in the last column, because 1, 3, 23 have no subgroup of index two to leave alone. At most 2 classes share a row, which happens where a proper class has halves of two different kinds.

Eleven, eleven and ten

Twenty-one of the thirty-two crystal classes contain a mirror, a centre or a rotoinversion, and not one of them is a new group. Each is a group of rotations with the inversion added, or a group of rotations with half of itself negated — and which half is left alone is the whole of the choice.

restriction · Finite groups Three rotations of order twelve, and one twelve-fold. Three integer 4 × 4 matrices of order twelve, one from each rational class of that order on a four-dimensional lattice. A four-dimensional rotation turns two perpendicular planes, and each plane is drawn as a dial: a point, its images, and the path the operation takes round them. Φ₁₂ turns its planes by one twelfth and five twelfths of a turn, so a point runs round twelve positions. Φ₃ ⊕ Φ₄ turns one plane by a third and the other by a quarter, and Φ₄ ⊕ Φ₆ by a quarter and a sixth; each has order twelve because twelve is the least common multiple, and neither has a plane with twelve positions in it.

Three rotations of order twelve

A four-dimensional lattice can be carried onto itself by three different integer matrices of order twelve, and only one of them turns anything by a twelfth of a turn. The totient that is usually said to decide which orders a dimension permits decides something else: which angles. The orders are decided by a sum over blocks, the two lists part company at six dimensions, and space already has an operation whose order is not its angle.

restriction · Restriction The eighty layers by what rolling does to their two faces. The eighty layer groups sorted by how the two tubes a sheet makes, one with each face outside, are related. Seventeen one-sided layers make two different tubes in every direction. Forty-six make the same tube in every direction, because they have a mirror or glide in the sheet or a centre of inversion. Sixteen relate the two tubes only along two, four or six special directions, and there as mirror images. One, the layer with a four-fold rotoinversion and nothing else that turns it over, exchanges its faces and still makes two different tubes in every direction.

A sheet with two faces rolls into two tubes

Roll a sheet whose two faces differ and every operation that turns it over dies, as expected: a tube's inside is not its outside. But the sheet can be rolled with either face out, and the operations that died are exactly the congruences between the two tubes that result. Over the eighty layer groups that settles, direction by direction, whether the two are one tube, a pair of mirror images or two different objects, and one layer exchanges its faces and still never rolls the same tube twice.

restriction · Finite groups Patches of an icosahedral quasicrystal grow with the dimension their symmetry fixes. The number of distinct patches about a vertex of the canonical icosahedral quasicrystal that keep a given symmetry, against the patch's radius in edges on logarithmic axes, counted per axis or per mirror plane: along a five-fold, a three-fold and a two-fold axis, and across a mirror plane. The axes' counts grow in proportion to the radius and the mirror's as its square, the slopes given are the last four points'. Along the bottom, the patches with an inversion centre: one up to a radius of 1.9 edges, and none from 2.15 on.

In space, a five-fold axis is a line

In a Penrose tiling the patches with a five-fold centre stop at four, however large they are drawn, because a five-fold rotation fixes only a point of the tiling's internal space. Slice a quasicrystal from six dimensions instead of four and the internal space has three dimensions, where a rotation fixes a whole line. The five-fold patches of the icosahedral quasicrystal then grow in number with their radius, from five to forty-six per axis, while the patches with a centre of inversion, which fixes only a point, vanish past two edges. Counting them takes internal space: a sample of vertices cannot.

aperiodic · Complexity Two cuts that give back the mirror a single cut takes away. A sheet whose only symmetry beyond translation is a centre of inversion: each flag has a copy turned half round and moved to the underside, drawn open, and the dots are the centres. Every single cut leaves a chiral edge, and the piece on the far side carries its mirror image. The upper ribbon is cut symmetrically about a row of centres, so each centre on its middle line carries the ribbon onto itself with its two edges exchanged, and the ribbon is achiral. The lower ribbon, cut at the same width about a line with no centre on it, keeps nothing but translations and is chiral.

A ribbon can be achiral where its edges are not

Cut an achiral sheet once and the edge is usually chiral, in almost every direction. Cut it twice, into a ribbon, and the ribbon may be achiral even when both its edges are chiral, because an operation can carry the ribbon onto itself by exchanging the edges — but only if the ribbon's middle line sits on a centre of inversion or a mirror parallel to the cut. Seven of the eight sheets with no achiral edge give achiral ribbons, and one achiral sheet gives nothing but chiral strips however it is cut.

space-groups · Chirality Which two kinds of operation make each of the 64 two-generator space groups. For each of the fifteen pairs of kinds of operation, rotation, screw, mirror, glide and rotoinversion (the inversion included), the number of the 64 space-group types generated by two of their own operations that some pair of those kinds generates. Every one of the 960 cases is decided: by a generating pair found, by the pair's linear parts failing to generate the point group, or by a proof in a finite quotient of the group. A screw and a glide make 32 types; two mirrors and a rotation with a mirror make none.

Two mirrors never make a crystal

Sixty-four space groups are generated by two of their own operations. Which two kinds can do it is a finite question, and it has been settled for all sixty-four and all fifteen pairs of kinds. A screw and a glide make thirty-two of them. Two mirrors make none, and neither does a rotation with a mirror, for reasons that need no computation: two operations that undo themselves generate a dihedral group, and an axis and a plane always share a point or a direction. Two plain rotations make six groups, every one of them cubic and without a mirror.

space-groups · Screws and glides Which three kinds of operation make each of the 93 three-generator space groups. For each of the 35 triples of kinds of operation (rotation, screw, mirror, glide, rotoinversion with the inversion), how many of the 93 space-group types that need three of their own operations some triple of those kinds generates. Of the 3255 cases 978 are generating triples found, 2275 are refuted (by the point group or in a finite quotient) and 2 remain undecided. Three mirrors make none; a screw and two glides make the most, 53.

Three mirrors never make a crystal either

Ninety-three space groups need three of their own operations to generate them, and which three kinds can do it is a finite question with 3,255 cases. All but two are now decided. Three mirrors make none of the ninety-three, as a theorem about reflection groups requires. A rotation with two mirrors makes exactly three, all cubic. Three plain rotations make thirteen, every one without a mirror. One group can be made thirty ways and three can be made only one.

space-groups · Screws and glides

Named alongside it

The objects these essays reach for when they reach for this one.

Inversion centreScrew axisGlide planeSpace groupChiralityCrystallographic restrictionGenerating setImproper operationIndex two subgroupLayer groupMirrorQuasicrystal

All concepts