Crystallography & Symmetry
patterns derived from their groups
Generator

glide: 3 mirrors

glide: 3 mirrors
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.

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.

6 essays call mirror-word. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about. Every one of this site's 393 essays names its parameters at the call site, which the standard pass of 2026-08-09 established and param-floor holds.

Where it is called

Changing this generator changes every one of these figures.

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. Operations

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.

 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. Operations

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.

 5 units of 70.53°: 7.36° left. 5 tetrahedral units of face-centred cubic metal, each the mirror image of its neighbour in a {111} plane, arranged about a common ⟨110⟩ edge. The angle between two such planes is arccos(1/3) = 70.53°, computed from the plane normals rather than quoted, and 5 of them come to 352.64°. The shaded sector is what is left over: 7.36°, or 2.04 per cent of a full turn, which must be taken up by strain, by a gap, or by a defect along the axis. What a lattice forbids

Five copies, and the gap they leave

Gold, silver and silicon grow particles with a five-fold axis down the middle, out of a lattice that forbids one. Nothing is violated: five tetrahedral pieces of ordinary face-centred metal, each the mirror image of its neighbour, come to three hundred and fifty-two and a half degrees rather than three hundred and sixty — and the seven degrees left over have to go somewhere.

 A circuit that closes on the wrong point: (1, 0). A square lattice with one extra half-column, drawn as a graph: the rows above the core have one more site than the rows below, and the core is the site at the end of the extra column. The path is 4 steps east, 4 north, 4 west and 4 south — the same number out as back — and it ends one lattice vector from where it started. Every one of the 12 circuits in the survey that goes round the core fails by that vector, and all 10 that miss it close exactly. Symmetry at work

The circuit that does not close

A defect in a crystal is usually introduced as a picture — an extra half-row of atoms, a wedge taken out. What makes a defect a crystallographic object rather than a drawing is a closure failure: walk a closed circuit through the lattice and come back to the wrong point, by an amount the lattice itself decides.

 square: 3 dislocations, 2 stable. The short lattice vectors of the square lattice, grouped into orbits under its own automorphism group of 8 operations. Two vectors of one orbit are the same defect seen from different directions, so the number of kinds of dislocation is the number of orbits — 3 out to 4 times the shortest squared length. Each orbit has its own colour. Solid arrows are stable: no pair of shorter lattice vectors adds to them with a smaller total of |b|². Dashed ones split, and 1 of the orbits do. Which splits is decided by comparing integers; that the energy goes as |b|² at all is Frank's rule and comes from elasticity, not from here. Symmetry at work

How many dislocations a lattice has

A circuit round a defect comes back to the wrong lattice point, and the amount by which it misses is a lattice vector. That much is quantised. The next question has a number for an answer: how many *different* dislocations are there? Two Burgers vectors related by an operation of the point group are one defect seen twice, so the answer is a count of orbits.

 Four angles, and the integer that picks them. Two roots at angle θ have Cartan integers whose product is 4cos²θ. Both are whole numbers and the product is below four, so it is nought, one, two or three — and each value fixes the angle between the two roots, and with it the angle between the mirrors perpendicular to them. The shaded wedge is the region the pair of mirrors folds the plane onto; the smaller it is, the larger the group they generate. What a lattice forbids

Four root systems, and the same four rotations

Two mirrors meeting at an angle generate a group. Ask that the group be finite and that a certain pairing between the mirrors come out a whole number, and the angle has only four possible values — from which the rotations that survive are of order two, three, four and six. The crystallographic restriction arrives with no lattice anywhere in the argument.

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