Crystallography & Symmetry
patterns derived from their groups
Concept

Order — where it appears

How many times an operation must be applied to return everything to where it started, or how many elements a group has. In a lattice basis the first is decided by a trace, which is why only five values are available.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

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.

space-groups · Screws and glides 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.

space-groups · Screws and glides

Named alongside it

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

Screw axisChiralityCoprimeCrystallographic restrictionEnumerationFixed pointHelixIntrinsic translationLattice translationPitchRotation

All concepts