Crystallography & Symmetry
patterns derived from their groups
Concept

Glide reflection — where it appears

A reflection combined with a slide along its own line, which is a symmetry although neither half of it is. It fixes no point at all, which is why it is one of the two operations a pattern can have and still fold into a plain surface.

Named by 9 essays across 5 fields — each of them below, with the objects they name alongside it.

A glide. The motif in the first colour, its images under a single glide in the second, and the symmetry element marked where the operation itself says it lies.

The four motions of the plane

Slide, turn, flip, and the odd fourth thing that is a flip and a slide together but neither on its own. Every symmetry of every flat pattern that has ever been made is one of these.

operations · What symmetry is The seven frieze groups. Every way of repeating a motif along a strip. Seven, and no more: the only ingredients are a translation, a mirror across the strip, a mirror along it, a half-turn and a glide, and most combinations of those turn out to generate one another.

Seven friezes

The same classification argument on a strip instead of a plane, where it is short enough to check by hand. Seven ways to repeat a motif along a line, with names like hop, step and sidle.

classification · Friezes Every group decided by a window of radius 1. For each of the seventeen, the radius at which a round window on the pattern admits exactly the group's own operations and no others — with the numbers it admits at each smaller radius beside it. Two opposite failures are visible. Most groups under-report at small radii, because an operation carrying points out of the window cannot be tested at all; cm over-reports, admitting operations the pattern does not have. The groups that take longest to settle are the ones distinguished by a glide, which moves a point half a cell before anything can be compared.

How much pattern is enough

Every claim here about a pattern's group is a claim about an infinite pattern. A reader sees a patch. Measuring what a finite window can decide gives a number — about one cell's radius — and two opposite ways of being wrong on the way there.

classification · Seventeen glide: 3 mirrors. A glide of the plane, drawn together with the mirrors it is a product of. The first shape is the motif; the pale ones are what each mirror in turn produces; the last is the image the motion itself gives. There are 3 mirrors, which is the smallest number that can produce this kind of motion, and their product was formed and compared with the motion before the figure was drawn.

Three reflections, and never four

Every motion of the plane is a product of mirrors, and the number needed is never more than three. That count is not a curiosity about mirrors — it is the classification of the four motions written as an integer, with the parity of the number deciding handedness and the geometry of the last two mirrors deciding everything else.

operations · Composition p1 folds into a torus. The cell of p1 with its edges marked as the group joins them: both pairs by a plain translation, both arrows the same way round. Gluing top to bottom gives a tube and gluing its ends gives a torus. Nothing in p1 holds a point still, so the surface has no marked points and its first homology is two copies of the integers.

The two that fold into a surface

Fold a wallpaper pattern along its own symmetries and what is left is usually a shape with corners and edges nobody drew. For two of the seventeen it is a plain surface with no marks on it at all — a torus and a Klein bottle — and which two is decided by a single question asked of every operation.

classification · Flat space pg scatters as pmm. The orbit of a motif under pg, and the set of all ordered differences between its points brought to a common origin. The vector set was handed to the detector with no indication of where it came from, and came back as pmm: 31 peaks from 6 atoms, on the same lattice.

Seventeen groups, seven vector sets

A map of interatomic vectors is more symmetric than the structure it came from, twice over: it always acquires a centre, and it loses every translation part. So a glide becomes a mirror, seventeen plane groups collapse onto seven — and the collapse is verified by handing the vectors to a detector that has never heard of Patterson.

diffraction · The Patterson function Four ways to lay a second row on the first. The same shape four times, with the upper row related to the lower one by a translation, a half turn, a glide and a mirror. In each case the upper row is pushed down until it touches, and the number is the density that results. The three that keep the shape the same way round come out within two per cent of one another; the mirror packs at 78 per cent of the best of them — 22 per cent less dense — because it presents a protrusion to a protrusion. Stated the other way round, the best of the four is 28 per cent denser than the mirror; the two percentages are the same measurement against two different bases, and neither is the other.

The four plane groups a molecule packs in

A molecule is not a disc: it has bumps and hollows, and packing it tightly means getting one molecule's bump into another's hollow. A mirror puts a bump against a bump. Filter the seventeen by that one observation and four survive — and the space groups the structural literature is mostly made of are the three-dimensional version of the same four.

applied · Packing The seven friezes rolled into cylinders are the seven axial families. Each of the seven frieze groups drawn on a strip 3 cells long, beside the same strip rolled into a cylinder so that its ends meet. A translation by one cell becomes a rotation by a 3th of a turn about the axis, a mirror across the strip a mirror containing the axis, the centre line a mirror perpendicular to it, a half-turn in the strip a half-turn about a horizontal axis, and a glide a rotation by half a cell's angle combined with that perpendicular mirror. Each cylinder's symmetry group was built from the rolled strip and again from the family's own generators, and the two agree. At n = 3 the orders are 3, 6, 6, 6, 6, 12, 12, and the last column names the crystal class each member is, coloured by whether it is proper, contains the centre, or is neither.

Seven friezes round a cylinder

A point group with one principal axis belongs to one of seven infinite families, and there are seven frieze groups. They are the same seven. Draw a frieze on a strip, roll the strip into a cylinder, and every translation becomes a turn about the axis and every glide a rotoreflection.

restriction · Finite groups p2's symmetries, sorted into classes by the group itself. A pattern with the symmetry of p2 over 2 by 2 cells, with its rotation centres and mirror lines marked in the International Tables' shapes and coloured by conjugacy class in the infinite group: two marks share a colour exactly when some operation of the group carries one element onto the other. Where rotations of several orders share a centre, the mark is the highest order's and so is its colour. Glides are not drawn. Classes counted: half-turns: 1 in the quotient, 4 in the group.

Two mirrors a coset cannot tell apart

Taken modulo its lattice a wallpaper group is finite, and its conjugacy classes are easy to list. But a coset holds every mirror of one direction at once, and the group itself keeps apart mirrors the list merges: pm has two classes of mirror, p2 four classes of half-turn, p3 six classes of rotation. Deciding which is which is Dehn's conjugacy problem, and for these groups it comes down to whether one vector lies in one lattice.

operations · What symmetry is

Named alongside it

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

Plane groupDecidabilityDetectionFixed pointFrieze groupHalf-turnMeasurementOrbifoldRound tripAbelianisationAccidental symmetryAutocorrelation

All concepts