Crystallography & Symmetry
patterns derived from their groups
Concept

Point group — where it appears

The group of symmetry operations that fix a point, which for a crystal is one of the thirty-two classes. Every operation of a space group has a linear part, and those parts form the point group, which is what a physical property is indexed by.

Named by 42 essays across 8 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 seventeen wallpaper groups. Every way of repeating a pattern across the plane, one cell of each. Each tile was generated from its group's operations and then examined independently to confirm it has exactly those symmetries and no others.

The seventeen

Every way of repeating a pattern across a flat surface, exhibited rather than tabulated. The number is a theorem with a finite proof, and the proof is walkable in an afternoon.

classification · Seventeen The thirty-two crystal classes. Every crystallographic point group, as a stereogram. Each was found by enumerating the subgroups of m3̅m and of 6/mmm, and each diagram is the orbit of one general direction under the group, filled where the pole is in the upper hemisphere and open where it is in the lower — which is the only thing in the picture that tells a rotation from a rotoinversion.

Thirty-two, and no others

There are exactly thirty-two ways a crystal can be symmetric about a point. Not thirty-two that anybody has catalogued — thirty-two that a finite search produces, from two starting groups, with every step of the reduction counted separately so that no two of them can quietly compensate.

point-groups · Crystal classes The merge, and what witnesses it. The two holohedries are enumerated separately and their class lists merged where the element signatures agree. That returns the right total, which is not the same as being right: 33 merges are made and every one of them is checked by constructing an explicit change of basis carrying one group onto the other.

A fingerprint that gave the right answer

The thirty-two classes were merged on a fingerprint — the census of operation types — and the fingerprint returned thirty-two, which is correct. Returning the correct answer is not the same as being entitled to it, and the difference took three wrong constructions to close.

point-groups · Crystal classes The wallpaper group p3m1. A pattern with the symmetry of p3m1, generated by applying the group's 6 operations to an asymmetric motif and repeating across the lattice. The symmetries of the result were then found independently and match the group exactly.

p3m1 and p31m

Two groups with the same lattice, the same point group and the same number of operations, differing only in where the mirrors sit. The pair is the clearest evidence that position is as much a part of a symmetry as presence.

classification · Seventeen The crystal classes 4, 4̅, 3̅, 6̅. 4, 4̅, 3̅, 6̅: the orbit of a general direction under each group, giving 4, 4, 6, 6 poles, with general positions and symmetry elements. Filled marks are poles above the plane of the page and open ones below it.

What a trace decides

Ten kinds of operation, and two integers tell them apart. The determinant and the trace name a symmetry operation completely — which is why the classification can be run on integer matrices in a lattice basis and never once ask what angle anything turns through.

point-groups · Crystal classes Doing one after another. Two symmetries of a pattern, and the one that doing both lands on. The third picture is not a new operation drawn to fit — it is the composition, and it was already in the group.

Why it is a group and not a list

The symmetries of a pattern cannot be chosen independently. Do two of them in succession and the result is forced to be a third, which is why there is no eighteenth wallpaper for anybody to invent.

operations · What symmetry is Reading 4/mmm off its own directions. Each position of 4/m2/m2/m reports one symmetry direction of the tetragonal system: the highest-order axis lying along it, and whether a mirror is perpendicular to it. Nothing is looked up — every row is computed from the group's own matrices.

Reading a class off its own axes

A Hermann–Mauguin symbol is not a name that was assigned. It is a report on three directions, read in order, and the whole of it can be derived from the group's matrices — with one genuine convention and one exception, and the exception is orthorhombic.

point-groups · Crystal classes What p3 scatters, and what the scattering shows. The structure on the left has point group 3, of order 3. The intensities it scatters, on the right, have point group 6, of order 6 — more symmetric than the thing that produced them. Reversing the sign of both indices conjugates every term in the sum and leaves the modulus alone, so a diffraction pattern always acquires a centre of symmetry, and in the plane a centre is a half turn. Both numbers are measured: the left from the operations, the right by testing each candidate against the computed intensities.

The symmetry diffraction adds

A diffraction pattern is always more symmetric than the crystal that made it. The extra symmetry is not a mistake in the experiment and no care removes it — it is a property of what a detector records, and it collapses the seventeen groups onto six.

diffraction · Accidental symmetry The crystal classes 3m, 3̅m, 6̅2m. 3m, 3̅m, 6̅2m: the orbit of a general direction under each group, giving 6, 12, 12 poles, with general positions and symmetry elements. Filled marks are poles above the plane of the page and open ones below it.

3m1 and 31m are one class

This site has an essay arguing that p3m1 and p31m are genuinely different groups. As point groups the same two objects are one class — and the two subgroups are each normal in the hexagonal holohedry, so nothing in the lattice relates them. What does is a rotation of thirty degrees.

point-groups · Crystal classes p3, twinned. p3 twinned by a rotation. To the left of the composition line the motif sits where p3 puts it; to the right every copy has been carried over by the twin law, which is one of the 3 operations the hexagonal lattice has and p3 does not. 24 images on the left, 24 on the right, and the lattice runs through the line unbroken — which is exactly why a twinned crystal looks like a single one.

A twin is a symmetry the lattice has and the crystal does not

Two orientations of one structure, grown together across a boundary the lattice runs straight through. The operation relating them cannot be a symmetry of the crystal, or there would be nothing to see, and it must be a symmetry of the lattice, or the boundary would be a crack — which leaves exactly a coset, and a short computable list.

applied · Twinning The seventeen sorted by lattice: 2, 5, 2, 3, 5. The five plane lattices, each drawn from the basis every other figure here uses, with the wallpaper groups that sit on it and the order of each against its lattice's holohedry. The counts are 2, 5, 2, 3, 5, which is seventeen again, arrived at by a different route from the case analysis on rotation order. Two relations hold and both are checked. Every group's order divides its lattice's holohedry, because an operation has to map the lattice onto itself before it can map the pattern onto itself — which is why a quarter turn has nowhere to live but a square lattice. And the converse fails on every one of the five: each lattice carries at least one group whose order falls short of what the lattice offers, so knowing the lattice narrows the group to a handful of candidates and never to one. The pairs printed in the accent colour are the groups that take everything their lattice permits.

The classification proof, one branch at a time

Seventeen is a theorem, and the argument that establishes it is a finite case analysis that fits on a few pages. Working through it is the difference between knowing the number and knowing why there is no eighteenth.

classification · Seventeen The thirty-two, by crystal system. 32 classes in 7 crystal systems. Each column is one crystal system and each cell one class, ordered by the number of operations it holds. Nothing here is tabulated: the classes come from the enumeration and the marking from a character sum over each group.

The holohedry is the ceiling

A crystal never has more point symmetry than its lattice. That single containment decides which system a class belongs to, why there are seven systems and not thirty-two, and why a lattice can be more symmetric than the crystal sitting on it — which is the usual case rather than the exception.

point-groups · Crystal classes The icosahedral group, counted. The sixty rotations of an icosahedron, found by trying every map that sends one adjacent pair of vertices to another and keeping those that carry the whole vertex set onto itself. They fall into 6 axes of order 5, 10 axes of order 3, 15 axes of order 2 — and the fivefold axes are the reason this group cannot be the point group of any crystal, since no three-dimensional lattice admits a rotation of order five. Quasicrystals have it anyway, which is what made 1982 an argument rather than a measurement.

Icosahedral symmetry

Sixty rotations, six fivefold axes, and no lattice in three dimensions that can hold any of them. It is the point group a crystal is forbidden, and the one the first quasicrystal turned out to have.

aperiodic · Quasicrystals The eleven Laue classes. Adjoining the inversion to each of the thirty-two crystal classes collapses them onto 11 groups. Friedel's law says a diffraction experiment sees the crystal and its inverse alike, so this — and not the crystal class — is what a diffraction pattern's symmetry reports. The highlighted symbol in each row is the class that is already its own Laue class, which is to say the centrosymmetric one.

The eleven a diffraction pattern reports

A diffraction experiment cannot tell a crystal from its inverse. So the thirty-two classes collapse to eleven before a single reflection is indexed, and a structure determination begins by answering a different question from the one it was asked.

point-groups · Diffraction Neumann's principle for elastic constants in mmm. Each bar is one operation's contribution to the character of the representation the elastic constants live in — the stiffness of the crystal in every direction and shear. The identity contributes the unconstrained count of 21; every other operation of mmm subtracts from it, and the average over all 8 is 9, which is how many independent components the property may have. The sum is exact in integers and is taken in the lattice basis, so no Cartesian frame is chosen anywhere in it.

Neumann's principle, as one sum

A physical property of a crystal must be unchanged by every symmetry the crystal has. That is a whole subject in one sentence, and it reduces to arithmetic: how many independent components a property may have is a character averaged over the point group, exact in integers.

point-groups · Neumann's principle A spontaneous vector in 222. Averaging each of the three axes over the 4 operations of 222 leaves 0 independent components. No direction survives, so the class permits no spontaneous polarisation at all — which is a statement about what is forbidden, not about any measurement.

The ten with a direction of their own

A crystal has a spontaneous electric polarisation only if some direction is left completely alone by every one of its symmetry operations. Ten of the thirty-two classes have such a direction, and the same ten are computed here twice by routes that share nothing but the group.

point-groups · Properties Every property count, in every class. For each of these 32 classes, how many independent components a property may have: elastic constants from 21 down to 3, piezoelectric moduli from 18 down to 0, dielectric tensor from 6 down to 1, pyroelectric vector from 3 down to 0, gyration tensor from 6 down to 0. Every number is a character averaged over the class's own operations, computed in the lattice basis with integer arithmetic. A zero means the symmetry forbids the property outright; a positive number means it does not forbid it, which is a weaker statement than it is usually read as.

A filter of great precision and no predictive power

The whole table in one place — thirty-two classes, six properties, 192 exact integers. What it settles, what it merely permits, and why knowing which of the two is happening at any moment is the entire skill of using it.

point-groups · Properties 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 625 tiles, 32 directions. The subdivision applied 4 times to one right triangle with legs 1 and 2, giving 625 tiles of one shape and size. They point in 32 distinct directions — the tint follows the direction — and the count grows every time the rule is applied, without bound.

The tiling that points every way

A Penrose tiling never repeats and its tiles still point in only ten directions, which is why its diffraction pattern has ten-fold symmetry. One triangle, cut into five copies of itself, breaks that — and the difference between it and a tiling with eight directions is which diagonal of one small rectangle gets drawn.

aperiodic · Aperiodic Thirty-two classes, eighteen groups. Every abstract group the thirty-two crystal classes realise, with the classes that realise it. 8 of the eighteen carry more than one class, and the largest collision is the four hexagonal classes that are all the dihedral group of order twelve. Nothing here is looked up: two classes are put in the same row when a search over images of a generating set finds a bijection preserving multiplication, and the search is finite because a generating set is small and the elements it may map to are the ones of the same order.

Thirty-two classes, eighteen groups

An inversion centre, a mirror and a two-fold rotation are three of the most different things a crystal can have, and they are the same group of order two. Forget the matrices and keep the multiplication table, and the thirty-two classes collapse to eighteen.

point-groups · Crystal classes The seven groups a uniform field can have. Five of them have an axis and two do not. A cone has every rotation about its axis and mirrors containing it; a cylinder adds the mirror across the axis and the two-folds that go with it; turning either one destroys the mirrors that would reverse the turn. The sphere and the sphere made of something with a handedness are the two with no axis to speak of. Each drawing is the definition: the group is the set of motions leaving the picture unchanged.

The seven groups a field can have

Every group in this collection so far has been finite, because a lattice forbids the alternatives. A uniform field has no lattice: rotate it about its own axis through any angle at all and nothing has changed. There are exactly seven such groups, and they come out of the same closure argument that turns sixteen frieze candidates into seven.

point-groups · Curie 4mm: how many independent invariants there are at each degree. The number of independent polynomial invariants of the plane point group 4mm, one bar per degree from 0 to 8. The heights are the coefficients of the Molien series, computed from an integer recursion on the trace and determinant of each operation. A dot above a bar marks a degree where the same number has been computed a second way, by averaging every monomial of that degree over the group and taking the rank of the result — a computation sharing no code with the first. The two agree at every degree checked. Degrees where the bar is absent hold no invariant at all: for 4mm that is every odd degree below the first invariant, and it is a statement about which functions the group refuses to leave alone.

How many invariants of each degree

A group moves the plane about, and some polynomials do not notice. How many independent ones there are at each degree is a sequence of integers, computed here by a recursion on traces and again by averaging every monomial — two routes that share no code and agree everywhere.

point-groups · Invariants Six of the ten plane classes have a free invariant ring, and four do not. Every plane point group, with the degrees of the generators of its invariant ring, whether the ring is free, and the relation where it is not. The six generated by their own reflections — 1, m, 2mm, 4mm, 3m and 6mm — have two generators whose degrees multiply to the order of the group, which is Chevalley's theorem checked rather than quoted. The four without reflections — 2, 4, 3 and 6 — need three generators in two variables, so one polynomial relation ties them together, and the degree that relation appears at is printed at the right of its row. Nothing here is a lookup: the generators are found degree by degree as the invariants the earlier ones do not reach, and the relation is the kernel of the map back to polynomials.

The groups whose invariants are free

Six of the ten plane classes have an invariant ring generated by two polynomials with no relation between them, and the six are exactly those generated by their own reflections. The degrees of those generators multiply to the order of the group, and their excess counts the reflections.

point-groups · Invariants Every subgroup of index two is normal; at index three most are not. For each of the seventeen plane groups, its abelianisation and the number of normal subgroups of each small index against the number of subgroups of that index. The index-two column is complete every time, because the left and right cosets of a subgroup of index two are the same pair of sets. At index three and four the two numbers part, and the gap is what normality costs: a subgroup that is carried to a different subgroup by some operation of the group it sits in.

The quotient each normal subgroup leaves

Two hundred and eighty-one subgroups of index four across the seventeen plane groups, and ninety-seven of them normal. Which ones, and what is left when they are divided out, needs no enumeration at all: below order six every group is abelian, so a normal subgroup of small index is a subgroup of the abelianisation and its quotient is decided by a product of greatest common divisors.

operations · Subgroups One framework has a count of zero, one mechanism and one self-stress. Every net this collection has a placement for, as a periodic bar-and-joint framework in a fixed cell: its point group, the joints and bars of one cell, the scalar Maxwell count 2n − e − 2, and the mechanisms and self-stresses found exactly from the rank of the rigidity matrix. The scalar count is always the difference of the last two, which is Maxwell's identity — and the bathroom net is the row that shows what the identity costs: nought equals one minus one, and a framework that reads isostatic moves.

The mechanisms a count cannot see

Maxwell's count subtracts constraints from freedoms, and a mechanism and a state of self-stress cancel in the subtraction — so a framework with one of each reports the same number as a rigid one. The bathroom net reports nought and moves. Doing the same subtraction with representations instead of numbers separates them, because a mechanism and a self-stress cancel only when they belong to the same representation.

applied · Rigidity 21 superspace groups in (2+1) dimensions, from 31 names. The whole count, in the order it is built. Thirteen arithmetic classes of the plane; six of them admit an incommensurate wavevector; those six give ten sign assignments; each assignment contributes the plane cohomology times the internal cohomology, which is thirty-one names; and the names are merged by the changes of basis that are relabellings — a change of the plane basis, which moves the sign assignment and the internal cocycle with it, and the choice of q against −q. The last row is what the count would be if the two factors were quotiented separately, which over-counts because the merge is not independent of the internal part.

Superspace groups in the plane

A modulated crystal has no space group, and in a space of one more dimension it has one. Counting them in the plane is the seventeen's own extension arithmetic with a third coordinate on which the point group acts by a sign — and the sign has to be plus or minus exactly, which kills the three-fold, four-fold and six-fold classes before a single extension is counted.

aperiodic · Modulation Four of the seventeen have a centre, and they are the four with no rotation. For each plane group: the order of its point group, how many of its operations are rotations, the lattice vectors every operation of the point group fixes, and the centre those vectors make. A central element must commute with every translation, which forces its linear part to be the identity — so the centre is a group of translations, and a translation is central exactly when the point group leaves it alone. A rotation leaves nothing alone but zero.

The four groups with a centre

An element that commutes with everything has to commute with every translation, and that forces its linear part to be the identity. So the centre of a plane group is a group of translations — the ones its point group leaves alone — and a rotation leaves nothing alone but zero. Four of the seventeen have a centre and thirteen have nothing at all.

operations · Composition No two cubic grains are more than sixty-three degrees apart. For each proper class: how many rotations describe one misorientation, the largest disorientation there is, and the mean over uniformly random orientations. The maximum is found by sampling and then climbing locally, so it is a lower bound that has stopped moving rather than a solved value — and it lands on the numbers the literature records.

The angle two grains differ by

A crystal's axes are not labelled, so a relative orientation between two grains has as many descriptions as the symmetry allows — five hundred and seventy-six of them for a cubic crystal — and their rotation angles run from a few degrees to more than a hundred and seventy. The honest answer is the smallest, and its largest possible value is a number: no two cubic grains are more than sixty-three degrees apart, whatever anybody does to them.

applied · Interfaces Four angles, and the integer that picks them. Two roots at angle θ have Cartan integers whose product is 4cos²θ. Both are whole numbers and the product is below four, so it is nought, one, two or three — and each value fixes the angle between the two roots, and with it the angle between the mirrors perpendicular to them. The shaded wedge is the region the pair of mirrors folds the plane onto; the smaller it is, the larger the group they generate.

Four root systems, and the same four rotations

Two mirrors meeting at an angle generate a group. Ask that the group be finite and that a certain pairing between the mirrors come out a whole number, and the angle has only four possible values — from which the rotations that survive are of order two, three, four and six. The crystallographic restriction arrives with no lattice anywhere in the argument.

restriction · Restriction 12 of the thirty-two classes have a free invariant ring. Every crystal class with its order, the number of its operations that are reflections, whether its ring of invariant polynomials is free, and the degrees of the generators when it is. A reflection here is an operation of determinant minus one whose fixed set is a plane; an inversion centre has determinant minus one and fixes only the origin and is not one. The classes with a free ring are exactly the classes generated by their reflections, which is Chevalley's theorem checked rather than quoted.

Twelve of the thirty-two are free

A crystal class leaves some polynomials alone, and the ones it leaves alone form a ring. For twelve of the thirty-two classes that ring is generated by three polynomials with no relation between them, and for the other twenty it is not — and the twelve are exactly the classes generated by their mirror planes. The two verdicts are computed by routes sharing no code, and an inversion centre is not a mirror.

point-groups · Invariants A spin needs two full turns to come back. The number a rotation about a fixed axis multiplies a state by, against the angle turned through. A vector — anything of integer spin — is back where it started after one full turn; a spin-one-half state is multiplied by minus one and needs a second turn. So the operators acting on such a state do not form the rotation group: a full turn is an operation distinct from doing nothing, and the group is twice as large.

Two turns to come back

A rotation through a full turn does nothing to a crystal and multiplies a spin-one-half state by minus one, so the group acting on such a state is not the point group but a group twice its size. Building those eleven double groups from quaternions and averaging a random operator over each gives the degeneracies a spin may have — and shows that the doubling everybody calls Kramers' is time reversal's doing and not the double group's.

point-groups · Representations Thirty-two classes, from fourteen Gram matrices. The five hundred and ten subgroups sorted by how many operations of each kind they contain — a determinant and a trace decide which of the ten kinds a matrix is. Thirty-two answers come out, and they are the thirty-two crystal classes: matched against the construction elsewhere in this collection by signature rather than by name, since nothing here names a point group.

Thirty-two from fourteen matrices

Write the fourteen Bravais lattices as Gram matrices, ask each one which integer matrices preserve it, and take every subgroup of every answer: five hundred and ten of them. Sort those by how many operations of each kind they contain — which a determinant and a trace decide — and thirty-two answers come out. They are the crystal classes, from a construction in which no point group is ever named.

point-groups · The fourteen Bravais lattices Cubic means the Sylow 3-subgroup is not normal. The thirty-two sorted two ways at once: by crystal system and by whether the Sylow 3-subgroup is normal. Two of the four boxes are empty, so the two properties coincide exactly. That gives 'cubic' a definition with no geometry in it — a class is cubic when its threefold subgroups are conjugate to each other rather than unique — and it explains why a cubic class has no principal axis: a group cannot single out one member of a conjugate family.

How many axes there are is a Sylow count

Sylow's theorems say that the subgroups of prime-power order in a finite group are all conjugate and that how many there are is congruent to one modulo the prime. Applied to the thirty-two crystal classes that arithmetic counts axes: the number of Sylow 3-subgroups is the number of equivalent threefold directions, it is four in exactly five classes, and those five are the cubic ones. So cubic has a definition with no geometry in it.

point-groups · Crystal classes The table of marks of 4mm. Every conjugacy class of subgroup of 4mm, against every other. The entry is the number of cosets of the column's subgroup that the row's subgroup holds still. The first row is the identity, which fixes everything, so it is the size of each coset space; the last column is the whole group, whose only coset is fixed by everybody.

The table that decides every action

Burnside's lemma counts orbits and stops there — two completely different actions with the same orbit count are indistinguishable to it. The object that settles the whole question is a square table whose entries count fixed cosets: lower triangular because a subgroup fixes no coset of anything smaller, positive on the diagonal because it fixes its own, and therefore invertible. Inverting it turns a list of fixed-point counts back into the orbits themselves.

point-groups · Counting The seven friezes rolled into cylinders are the seven axial families. Each of the seven frieze groups drawn on a strip 3 cells long, beside the same strip rolled into a cylinder so that its ends meet. A translation by one cell becomes a rotation by a 3th of a turn about the axis, a mirror across the strip a mirror containing the axis, the centre line a mirror perpendicular to it, a half-turn in the strip a half-turn about a horizontal axis, and a glide a rotation by half a cell's angle combined with that perpendicular mirror. Each cylinder's symmetry group was built from the rolled strip and again from the family's own generators, and the two agree. At n = 3 the orders are 3, 6, 6, 6, 6, 12, 12, and the last column names the crystal class each member is, coloured by whether it is proper, contains the centre, or is neither.

Seven friezes round a cylinder

A point group with one principal axis belongs to one of seven infinite families, and there are seven frieze groups. They are the same seven. Draw a frieze on a strip, roll the strip into a cylinder, and every translation becomes a turn about the axis and every glide a rotoreflection.

restriction · Finite groups The rectangle a (4, 2) tube is rolled from. A patch of honeycomb turned so that the rolling vector C = 4a₁ + 2a₂ lies along the page. C has length √28 ≈ 5.292; the shortest lattice vector perpendicular to it, T, has length 4.583; and the rectangle on the two holds 28 hexagons and 56 atoms. Rolling the rectangle so that its left and right edges meet makes one repeat of the tube. The two lines through the corner are the sheet's mirror directions nearest C: the zigzag direction along a₁ and the armchair direction thirty degrees from it. C makes an angle of 19.11° with the first and lies on neither.

The tube has a screw no lattice allows

Roll a honeycomb along one of its lattice vectors and the tube turns and climbs with a screw of order 14, 98 or 794 — orders the flat sheet could never have. The rolling keeps the sheet's translations and spends them on turns, and it keeps the sheet's mirrors only along two directions, which is why almost every carbon nanotube comes in two hands.

restriction · Finite groups Whether a rolled sheet ever comes back round. Three plane lattices, each with the same rolling vector C = 3a₁ + a₂ drawn from the origin and the line through the origin perpendicular to it. A translation of the rolled pattern straight up the tube, with no turn, is a lattice vector on that line. The square lattice has one, marked T, and the tube repeats every 10 turns. The general rectangular lattice has none in this direction — only along its cell edges — and the general oblique lattice has none in any direction at all, so its rolled pattern climbs forever without returning to the same angle.

Most sheets roll into a tube that never repeats

Rolling the honeycomb along a lattice vector always gives a tube with a repeat, and that is a property of the honeycomb rather than of rolling. Over the seventeen plane groups, 567 of 1,008 rolling directions give a tube with no translation along its axis at all — and every direction of an oblique pattern is one of them.

restriction · Finite groups A thread's two signs, and the four kinds of operation. Every operation of a rod group carries the axis to itself, so it does two independent things: it keeps or reverses the direction along the thread, by a sign σ, and it keeps or reverses the handedness of the plane across the thread, by the determinant of a 2 × 2 matrix. The determinant in space is the product, so the shaded cells are the proper operations — a turn or screw about the axis, and a half-turn crossing it — and the unshaded ones are the improper. A rod group is chiral when all of its operations sit on the shaded diagonal, and polar along its axis when all of them sit on the top row. The two conditions pick out different diagonals of the same square, which is why neither implies the other.

A thread's hand is not a choice

A sheet's handedness in space depends on a sign that the plane pattern does not fix, so one plane group carries several sheets and exactly one of them is chiral. A thread has no such freedom: 32 of the 75 rod groups are chiral, they sit over 9 of the 27 axial classes, and which they are is settled before any structure is drawn. Only its direction depends on how the class lies along it.

space-groups · Chirality 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 Seven indices in 40, and one lattice at each. For every index to 40, how many sublattices of a cubic lattice there are and how many of them the full cubic point group carries to itself. The first number runs into the hundreds; the second is nought at almost every index and one at 1, 2, 4, 8, 16, 27, 32. A cubic point group is forty-eight conditions on a sublattice, and forty-eight conditions leave very little. The richest point group in three dimensions has the poorest arithmetic of copies, and the two are the same fact.

The richest group has the poorest arithmetic

A plane group with no symmetry has seven hundred and sixty-two sublattices to grow by; one with a six-fold axis and mirrors has eight. Take the trend to its end in space and a cubic group has six to index forty — one at every cube, one at twice a cube, one at four times a cube, and nothing anywhere else. Every operation of a point group is a condition, and forty-eight conditions leave almost nothing.

space-groups · Isomorphic subgroups 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

Named alongside it

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

Crystal classCharacterEnumerationHolohedrySubgroupCrystallographic restrictionChiralityConjugacy classRepresentationScrew axisSymmorphicArithmetic crystal class

All concepts