Crystallography & Symmetry
patterns derived from their groups
Concept

Arithmetic crystal class — where it appears

A point group together with the lattice it acts on, taken up to a change of integer basis — thirteen in the plane, seventy-three in space. Each holds exactly one symmorphic space group, which is why those two counts agree.

Named by 14 essays across 3 fields — each of them below, with the objects they name alongside it.

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.

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.

space-groups · Space groups The space groups in class mm2 (P), counted. Every way of attaching translations to the generators of mm2 (P): 64 assignments close into a group of the right size, 16 survive moving the origin, and 10 survive relabelling the axes — which is the number the International Tables record for this class. the ten primitive orthorhombic groups with a polar axis.

Sixteen candidates, ten groups

One point group, one lattice, and every consistent way of attaching translations to it — enumerated in full. The count comes out at sixteen, and then at ten, and the step between the two numbers is a decision about what "the same group" means rather than an arithmetic fact.

space-groups · Space groups 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 Thirteen ways to hold a lattice. Every finite group of integer matrices in two dimensions, up to a change of integer basis: 13 of them. Ten different abstract groups appear, and three of the ten hold a lattice in two inequivalent ways — a mirror along an axis or along a diagonal, and the same for 2mm and for 3m. The enumeration is a search: every subgroup of the two maximal holohedries, merged by conjugacy under integer matrices of determinant ±1, with the answer checked for not depending on how wide the search was.

Thirteen ways to hold a lattice

The crystallographic restriction is about one matrix. A crystal has a whole group of them acting on one lattice at once, and asking how many such groups there are gives thirteen — not the ten of the plane point groups, and not the seventeen of the plane groups.

restriction · Restriction Why the seventeen is a number at all. The classification is finite because three counts in a row are finite, and the first two are where the work is. Finitely many lattice types, because a lattice's symmetry group is a finite group of integer matrices; finitely many such groups, by Minkowski's lemma and his bound; and finitely many ways to attach translations to each, which is the extension problem. Every step is a count this site makes elsewhere — five, thirteen, seventeen — and this is the reason each of those searches was allowed to stop.

Why there is a list at all

Five lattices, seventeen groups, thirty-two classes, two hundred and thirty. Every one of those counts came out of a search that had to know when to stop, and the reason it could stop is a divisibility Minkowski proved in 1887.

restriction · Finiteness Modulo 3 injective on all thirteen, modulo 2 on 5. Minkowski's lemma says the kernel of reduction modulo an integer of at least three is torsion-free, so a finite group of integer matrices is carried faithfully into a finite group of matrices over ℤ/3 — which is why the classification is finite, before any bound is computed. The middle column checks it on every finite subgroup of GL(2,ℤ) there is: thirteen classes, no collapses. The right column is the case the lemma has to exclude. Modulo 2, minus the identity is the identity, and 8 classes lose operations.

Reduction modulo three

A finite group of integer matrices survives being reduced modulo three: no two of its operations collide. That single fact proves the classification finite without computing any bound — and modulo two it is false, refuted by the inversion centre.

restriction · Finiteness p4: the map comes back. p4 written on two bases related by an integer matrix of determinant one, and about two origins. The two descriptions share no coordinate; they are the same group. The matrix and the origin shift were then recovered from the two operation sets alone — which is what Bieberbach's theorem promises, carried out as a search over the integer matrices and the origins the lattice permits, and checked by applying what was found.

The same group means the same pattern

Seventeen patterns is not the same statement as seventeen groups. Two patterns that look nothing alike could in principle have symmetry groups that are abstractly the same, and then the classification would be a classification of drawings. Bieberbach's theorem says they cannot — and the affine map that proves it can be recovered from the two operation sets alone.

restriction · Finiteness 18 extension classes, 17 groups. Each of the thirteen arithmetic classes with the number of ways translations may be attached to it — its cohomology — the shape of that group, and how many distinct plane groups the classes come to once the changes of basis that are mere relabellings are quotiented out. The two columns differ in exactly one row, 2mmp, where four extension classes are three groups because two of them are the same group with the axes swapped. No lattice is drawn anywhere in this computation.

Seventeen, without a picture

Every other count of the plane groups has a plane in it — a pattern generated, a domain folded, an orbifold's curvature spent. The same seventeen come out of pure algebra: attach translations to a point group, keep the assignments that close, throw away the ones that differ only by where the origin was put, and add up over the thirteen arithmetic classes.

classification · Cohomology Two, seventeen, two hundred and thirty, and then. The number of arithmetic crystal classes and the number of crystallographic groups in each of the first six dimensions, with the second divided by the first. The classes multiply by between five and fourteen a dimension; the groups multiply by much more, and the quotient — how many groups an average class carries — goes 1.00, 1.31, 3.15, 6.74, 36.5 and 339. The last column says what is derived on this page and what is quoted: the plane in full, six of the seventy-three classes in space, and nothing at all above three dimensions, where the counts come from machine enumerations of the 1970s onwards.

Finitely many is not few

Bieberbach's third theorem says each dimension holds finitely many crystallographic groups and gives no idea how many. The counts are 2, 17, 230, 4783, 222018 and 28927922, and dividing them by the number of arithmetic classes says which of the classification's three steps supplies the explosion — the step that attaches translations, not the one that finds the matrix groups.

restriction · Finiteness How the space groups are spread over their classes. For each number of groups, how many arithmetic classes carry exactly that many space-group types, in space and in the plane, every count derived from the classes' cohomology and the normaliser's action on it. In the plane ten of the thirteen classes carry one group. In space twelve of the seventy-three carry one, thirty-five carry two and fifteen carry four, so the typical class carries two; the long tail is the orthorhombic and tetragonal classes with eight, ten and sixteen.

Two hundred and nineteen over seventy-three

The space groups average three to an arithmetic class, and an average of three could be a few classes carrying dozens or every class carrying a handful. Derived class by class from the cohomology and its normaliser, the answer is the second: twelve classes carry one group, the median carries two, and the largest carries sixteen. Sixty-one of the seventy-three admit a non-symmorphic group, and the step that merges relabellings, which the plane barely uses, removes more than a quarter of the list.

restriction · Finiteness 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.

Group extensionPoint groupScrew axisClassificationCoboundaryCocycleEnumerationIntrinsic translationSymmorphicGlide planeInteger matrixMinkowski bound

All concepts