Generator

Every screw axis a lattice permits

All eleven screw axes a lattice permits, each drawn as the helix it is: 2₁, 3₁, 3₂, 4₁, 4₂, 4₃, 6₁, 6₂, 6₃, 6₄, 6₅. A turn of 2π/n followed by an advance of m/n of the repeat, so that n turns land exactly m cells along. 3 of those drawn are its own mirror image; the rest come in left- and right-handed pairs.

7 essays call screw-axis. 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.

P2₁/c, in the two diagrams the Tables print. Space group P2₁/c, number 14, projected down c on a primitive monoclinic cell. The symmetry elements drawn: 2 2₁ screw axes, 2 glide planes, 4 inversion centres. 4 general positions, the orbit of one point, each labelled with its height along c and marked with a comma where the operation that produced it reversed handedness.

Into space

The step a flat surface has no room for

Seventeen becomes two hundred and thirty, and the factor of thirteen is not a matter of having more directions to work in. It comes from an operation the plane cannot hold — a translation that no choice of origin will remove.

The translation of every operation, split in two. Every operation of P2, P2₁, Pm and Pc other than the identity, with its translation split into the intrinsic part — one n-th of the sum of the operation applied to itself n times, which no choice of origin can remove — and the location part, which is only a statement about where the origin was put. 2 of the 4 operations shown have a non-zero intrinsic part, and those are exactly the screws and the glides.

Into space

The half of a translation that is not a choice

Every operation's translation splits in two — a part that belongs to the operation and a part that only records where somebody put the origin. Almost everything peculiar about space groups is a consequence of that split, including why there are two hundred and thirty rather than seventy-three.

2 screw axes. 2 of the eleven screw axes a lattice permits, each drawn as the helix it is: 3₁, 3₂. 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.

Into space

Eleven groups that are their own reflection's rival

There are two hundred and thirty space groups, and there are two hundred and nineteen. Both numbers are correct and they answer different questions, and the eleven that separate them are the reason a crystal can be built one way round and not the other.

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.

Into space

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.

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

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.

P2₁2₁2₁: the sections its symmetry forces. The Patterson cell of P2₁2₁2₁ with the sections marked. Each operation (M, t) sends an atom at x to Mx + t, so the vector between them is (I − M)x − t; where I − M is singular that vector cannot leave a plane, and the plane's equation comes from the left null space in integers. This group has 3 such operations, giving the sections w = 0.5, u = 0.5, v = 0.5. A heavy atom's vector to its own image is somewhere on one of them, which is what made structure solution possible before computers: a plane can be searched by eye and a volume cannot.

How it is known

Where symmetry stacks the vectors

A Patterson map of a real structure is a blur with thousands of overlapping peaks. A screw axis rescues it: the vectors between symmetry-related atoms cannot leave a plane, so the search for a heavy atom is a search of a section rather than of a volume — and which plane it is falls out of the operation's matrix in integers.

4_1: which index gives which group. The isomorphic subgroups of a 4₍1₎ screw group, index by index. An index sharing a factor with 4 gives nothing — the translation cannot be written on the new cell at all — and the rest give a screw whose index is the old one times the inverse of p modulo the axis order. So the answer alternates: some indices give the group back and others give its mirror image, and which is which is decided by p modulo the order of the axis.

Into space

A bigger cell, and sometimes the mirror

An isomorphic subgroup gives up nothing but scale — the same group again on a coarser lattice. In space the screw axes sharpen the question, and the answer contains a surprise: a cell three times taller holds the group's enantiomorphic partner, so a left-handed screw contains a right-handed one with nothing done to the crystal but a change of description.

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