Crystallography & Symmetry
patterns derived from their groups
Concept

Intrinsic translation — where it appears

The part of an operation's translation lying along its own axis or plane, which no choice of origin removes. It is the whole difference between P2 and P2₁, and computing it is how a screw is told from a rotation without looking at where the axis sits.

Named by 13 essays across 3 fields — each of them below, with the objects they name alongside it.

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.

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.

space-groups · Space groups 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 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.

space-groups · Screws and glides Each arithmetic class holds exactly one symmorphic group. The arithmetic crystal classes counted here in full, with the number of space groups each produces and the number of those that are symmorphic — that is, that have an origin at which every operation's translation part vanishes. The right-hand column is one in every row, over 25 groups in all, and each one is computed from the class, not copied from a table. That is the bijection behind the number 73: there are seventy-three arithmetic crystal classes in three dimensions and seventy-three symmorphic space groups, and the correspondence is this one, class by class.

One symmorphic group per class

Every arithmetic crystal class holds exactly one space group in which some origin clears every translation part at once. That bijection is why there are seventy-three symmorphic space groups and seventy-three arithmetic classes, and it is checked here class by class rather than counted.

space-groups · Space groups 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.

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.

operations · What symmetry is 75 rod groups over 27 axial classes. Each axial crystal class, with the number of rod groups it carries: every consistent choice of translation along the axis, in every way the class can sit on the rod, with two groups counted as one when a shift of the origin along the rod or a turn about it carries one onto the other. The total is 75, and every row as well as the total agrees with the International Tables, which are compared with this enumeration rather than used to produce it.

Seventy-five ways to be a thread

Eighty layer groups and seventy-five rod groups are usually quoted, both as numbers from the literature. The second is derived here, class by class — and the total alone turned out to be no check at all, because two errors of six groups each give seventy-five as well.

classification · Subperiodic Ten ways for space to be flat. The thirteen groups, with each mirror-image pair counted once, because a shape and its mirror image are the same shape. 3 of the ten arrive that way — the three-fold, four-fold and six-fold screws, which are the enantiomorphic pairs this collection already counts among the two hundred and thirty. Six of the ten are orientable and four are one-sided.

Ten ways for space to be flat

Thirteen of the two hundred and thirty space groups hold no point still, and folding space along one of them gives a shape with no curvature anywhere. There are ten such shapes, not thirteen, and the difference is the same eleven pairs that separate 230 from 219.

space-groups · Flat space 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.

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.

space-groups · Isomorphic subgroups 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.

space-groups · Screws and glides One group without an axis, and as many as the order with one. For a rotation of each order that an integer matrix can have in a small dimension, the number of space groups its arithmetic class admits — computed from the cohomology rather than enumerated. A rotation acting on the smallest lattice that will hold it fixes no direction and admits exactly one group: the symmorphic one, with no screw. Add a direction it leaves alone and the count becomes the order of the rotation, and the extra groups are its screws. The four-fold with an axis gives four, which are P4, P4₁, P4₂ and P4₃; the five-fold with an axis gives five, in five dimensions, where no published table exists to check it against.

The screw a dimension does not have

The extension count is a machine that runs in any dimension, and the seventeen were the case where every step could be checked against a list arrived at four other ways. Run on a cyclic point group it has a closed form two lines long — and it says a five-fold screw axis does not exist in four dimensions, which is a prediction rather than a check.

classification · Cohomology Tight where there is an axis and vacuous where there is not. The order of a point group annihilates its cohomology, so every intrinsic translation is a multiple of one over that order — the bound every account of the subject quotes. What occurs is one over the exponent of the cohomology, which divides the bound. For a rotation with a direction it fixes the two agree exactly: a four-fold screw does need quarters and a six-fold sixths. For a rotation acting with no fixed direction the exponent is one — the cohomology is trivial and no fraction occurs at all — so the bound is slack by the whole order. The same bound is sharp and useless in the same table.

The denominator a group actually needs

The order of a point group annihilates its cohomology, so every intrinsic translation is a multiple of one over that order — a bound every account of the subject quotes. What occurs is one over the exponent, which divides it. The same bound turns out to be attained exactly and to be slack by its whole size, in two rows of one table, and what decides which is whether the rotation fixes a direction.

classification · Cohomology Seventy-three classes, seventy-three Smith forms. Every arithmetic crystal class of space, grouped by lattice system, with the cohomology group read off the Smith normal form of the integer conditions its Cayley graph imposes, shaded by the group's exponent. Twelve classes have nothing; fifty-three have exponent two, among them the orthorhombic P holohedry with ℤ2 to the sixth; three have exponent three, three have exponent four, and two — the hexagonal 6 and 622 on the primitive lattice — have exponent six. Nothing larger occurs.

Seventy-three Smith forms

The cohomology group that counts a crystal class's intrinsic translations has a closed form only when the point group is cyclic. For every other class it is still one integer matrix away: the conditions the Cayley graph's cycles impose, reduced to Smith normal form, leave the group on the diagonal. Run on all seventy-three classes of space, the diagonal says the largest exponent is six, that a group of order forty-eight needs nothing finer than halves, and that no class needs a denominator its operations do not already have.

classification · Cohomology I222 and I2₁2₁2₁: the same axes, apart. The two-fold rotation axes of the two space groups on the body-centred class of 222, drawn in one conventional cell seen down its third edge: axes along that edge as diamonds, axes along the first edge as solid lines and along the second as dashed lines, with the height of each written beside it. Both groups have pure two-fold rotation axes in all three directions. In the symmorphic group some axes of different directions cross, at the origin among other places; in the other none does, because the heights of the axes in each direction are a quarter out of step with the positions of the axes in the others.

Two groups no single operation reveals

A space group is non-symmorphic when no choice of origin removes all of its translations at once, and every textbook example has a screw or a glide to show for it. Two of the two hundred and thirty do not need one. Every operation of I2₁2₁2₁, taken on its own, is also a plain rotation about some axis; so is every operation of I2₁3. What makes them non-symmorphic is that their rotation axes never meet — and no diffraction extinction can tell them from I222 and I23.

classification · Cohomology

Named alongside it

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

Screw axisArithmetic crystal classCocycleGlide planeGroup extensionCoboundaryEnumerationFixed pointLattice translationOrigin shiftSymmorphicCayley graph

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