Crystallography & Symmetry
patterns derived from their groups
Concept

Orientation — where it appears

Whether a motion preserves handedness or reverses it, which is the sign of its determinant. It is a homomorphism onto a group of order two, so a group either preserves handedness throughout or does so in exactly half of its operations.

Named by 6 essays across 4 fields — each of them below, with the objects they name alongside it.

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 625 tiles, 32 directions. The subdivision applied 4 times to one right triangle with legs 1 and 2, giving 625 tiles of one shape and size. They point in 32 distinct directions — the tint follows the direction — and the count grows every time the rule is applied, without bound.

The tiling that points every way

A Penrose tiling never repeats and its tiles still point in only ten directions, which is why its diffraction pattern has ten-fold symmetry. One triangle, cut into five copies of itself, breaks that — and the difference between it and a tiling with eight directions is which diagonal of one small rectangle gets drawn.

aperiodic · Aperiodic p4m: 4 and 4. The standard motif — three points in no particular arrangement — repeated by p4m, with each copy coloured by the sign of the area of the triangle it makes. 4 copies have one sign and 4 the other, because the group contains an operation that reverses orientation. A structure built from one enantiomer cannot sit here: the group would put its mirror image in the same crystal. The colours were computed from the coordinates rather than assigned.

The groups a single hand may sit in

A protein is built from one enantiomer of every amino acid, and a crystal of it contains nothing else. That single fact deletes most of the classification at a stroke: any operation reversing orientation would put the other hand in the same crystal. The criterion is one line of arithmetic, and in the plane the enumeration is complete — five of the seventeen.

space-groups · Chirality No two cubic grains are more than sixty-three degrees apart. For each proper class: how many rotations describe one misorientation, the largest disorientation there is, and the mean over uniformly random orientations. The maximum is found by sampling and then climbing locally, so it is a lower bound that has stopped moving rather than a solved value — and it lands on the numbers the literature records.

The angle two grains differ by

A crystal's axes are not labelled, so a relative orientation between two grains has as many descriptions as the symmetry allows — five hundred and seventy-six of them for a cubic crystal — and their rotation angles run from a few degrees to more than a hundred and seventy. The honest answer is the smallest, and its largest possible value is a number: no two cubic grains are more than sixty-three degrees apart, whatever anybody does to them.

applied · Interfaces Seventeen plane groups, and one chiral sheet over each. The seventeen plane groups, whether each is chiral as a pattern in the plane, how many sheets can be built over it by giving each operation a sign on the sheet's normal — 63 in all — and which of those sheets is chiral in space. There is always exactly one. For the five groups chiral in the plane it is the sheet whose two faces differ and nothing turns it over. For the twelve achiral in the plane it is the sheet turned over by exactly the operations that reverse orientation in the plane, so that every mirror line becomes a half-turn axis lying in the sheet.

Chiral in the plane is not chiral in the room

A pattern with mirrors all over it can be a sheet with a hand, and a pattern with no mirror can be a sheet without one. Whether a layer is chiral depends on what each of its operations does to the side of the sheet, and over every one of the seventeen plane groups exactly one sheet is chiral in space.

space-groups · Chirality A screw out of two rotations that have none. Four pairs of located rotations of space, composed, with the result read back as a screw: its angle, and its pitch, which is the part of its translation lying along its own axis. Axes that meet give a rotation and no translation at all, because the point where they meet is fixed by both. Parallel half-turns give a translation. Skew axes give a screw — a motion with a translation in it, out of two motions with none — and the translation is twice the distance between the two axes. Nothing in either factor moves anything along the product's axis, and the product does.

The axis a product lies on

Two rotations of space about axes that do not meet compose to a screw — a motion with a translation in it, out of two that have none. The translation is twice the distance between the axes and the angle twice the angle between them, and the screw's own axis is not somewhere arbitrary: it lies on the two axes' common perpendicular, at a place the arithmetic gives.

operations · Composition

Named alongside it

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

ChiralityCompositionFixed pointPoint groupSohncke groupSpace groupSymmetry operationAperiodicityCartan dieudonneCoincidence site latticeCosetCrystal class

All concepts