Crystallography & Symmetry
patterns derived from their groups
Series

Screws and glides — the series

6 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The 4₁ screw axis. 1 of the eleven screw axes a lattice permits, each drawn as the helix it is: 4₁. A turn of 2π/n followed by an advance of m/n of the repeat, so that n turns land exactly m cells along. 0 of those drawn are its own mirror image; the rest come in left- and right-handed pairs.

    Turning and climbing at once

    A rotation has a fixed point and a screw has none. That sounds like a small difference and it is the reason a space group is not a point group with extra letters, the reason two hundred and thirty is not seventy-three, and the reason a helix can be a crystal.

    part 1 · space-groups
  2. The eleven screw axes, enumerated. An n-fold axis admits a screw for every m from 1 to n − 1, so the orders a lattice permits — 2, 3, 4, 6 — give 11 screws in all. Each row shows the fraction of a cell one turn advances, plotted between zero and one. 3 of them are their own mirror image, which happens exactly when m is half of n; the others pair off into left- and right-handed twins.

    Eleven ways to turn while climbing

    Five rotation orders survive the crystallographic restriction, and each of them admits a screw for every whole number of cells its turns can amount to. That is a sum with four terms, and it is why there are eleven screw axes rather than some other number.

    part 2 · space-groups
  3. The five kinds of glide plane. All five glide letters: a, sliding by a/2; b, sliding by b/2; c, sliding by c/2; n, sliding by (a+b)/2, (b+c)/2 or (a+c)/2; d, sliding by (a+b)/4, (b+c)/4 or (a+c)/4. 3 of those drawn are axial — the slide is half of a single cell vector lying in the plane. A glide's square is a pure translation, so its slide can only be half of a lattice vector, and the quarter-cell d glide exists only where centring has already made the half-diagonal a lattice vector.

    Reflect, then slide by half of something

    A glide's slide must double to a lattice vector, which leaves three candidates in any plane and a fourth that exists only where centring has already made a half-diagonal into a lattice vector. Five letters, and the fifth is the one the enumeration explains.

    part 3 · space-groups
  4. The symmetry elements of Ccmm. Space group Ccmm, number 63, projected down c on a C-centred orthorhombic cell. The symmetry elements drawn: 4 2-fold rotation axes, 6 glide planes, 12 2₁ screw axes, 2 mirror planes, 8 inversion centres. 2 kind(s) of element in this group have no line or mark in a projection down c — an axis at an angle to the page, or a plane lying parallel to it — and are not in the picture.

    The operations nobody put in

    A group is not a list of generators. Compose two of them and something arrives that neither contained — a screw where there were only mirrors, a glide where there was only a mirror and a centring vector — and in three dimensions most of a group's operations get there this way.

    part 4 · space-groups
  5. Aem2: one plane, two glides. The plane of Aem2 that carries two operations, drawn edge-on with each slide beside it. The two differ by the A-centring translation, which lies inside this plane — that is the whole condition for a plane to carry two, and it is why no primitive group has one. Both slides are axial, b and c, so neither letter has a claim on the symbol, and before 1992 the Tables simply chose. The letter e is the choice being refused.

    The plane that carries two glides

    A plane can hold two glide operations at once, with slides that have no claim on each other. Every symbol printed before 1992 chose one of them, so the name recorded a convention rather than a group — and the International Tables invented a letter to stop it.

    part 5 · space-groups
  6. Two dimensions to be ambiguous in, and one. Why a plane can carry two operations and a line cannot, side by side. Two reflections sharing a plane differ by a translation lying in that plane, and their slides are vectors in the plane — a two-dimensional space, in which a centring vector need not be a multiple of the slide. So the two slides can be genuinely different glides, b against c, and in 1992 the International Tables invented the letter e for the case where neither has a claim. Two rotations sharing an axis differ by a translation along that axis, because anything across it would move the line; their intrinsic parts are vectors along the line, a one-dimensional space in which every lattice vector is a whole multiple of the shortest. So the two differ by a whole number of repeats and are the same screw. The plane has one dimension of freedom left over and the line has none.

    A line carries one screw

    A plane can hold two glide operations at once, and in 1992 the International Tables invented a letter for the case where neither has a claim. The same question put to an axis has the opposite answer: across 3,388 axes, not one line carries two — and the reason is that a slide has two dimensions to be ambiguous in and an intrinsic translation has one.

    part 6 · space-groups

All series