Crystallography & Symmetry
patterns derived from their groups
Concept

Holohedry — where it appears

The full point symmetry of a lattice, which is the ceiling on the symmetry any crystal built on that lattice can have. That single containment decides which system a class belongs to and why there are seven systems rather than thirty-two.

Named by 32 essays across 6 fields — each of them below, with the objects they name alongside it.

The five rotations a lattice will carry. One motif and every rotation a plane lattice permits: orders 1, 2, 3, 4, 6, and nothing else up to 12. Each panel turns the motif by its own operation as many times as the order allows, on the lattice that operation requires — oblique for the identity and the half turn, hexagonal for the third and the sixth of a turn, square for the quarter. The trace printed under each is the sum of the diagonal of the operation's matrix written in the lattice's own basis, and it is a whole number in every panel, which is the entire content of the crystallographic restriction. The list of orders is produced twice, once from that trace condition and once from the degree of a cyclotomic polynomial, and the figure refuses to draw if the two disagree.

The crystallographic restriction

A repeating pattern may have rotations of order two, three, four or six, and nothing else whatever. The proof is one line of arithmetic, and everything finite in the subject descends from it.

restriction · Restriction The hexagonal lattice. Every periodic pattern in the plane repeats on one of five lattices. The classification is by which point symmetries the lattice itself admits, and the five are exhaustive — a sixth would need a rotation order no lattice can carry.

The lattice underneath

Strip a pattern of everything but its repeats and a grid of points is left. That grid is not decoration — it is the object that decides which symmetries the pattern is permitted to have.

lattices · Lattice 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 five plane lattices. Every periodic pattern in the plane repeats on one of five lattices. The classification is by which point symmetries the lattice itself admits, and the five are exhaustive — a sixth would need a rotation order no lattice can carry.

Five lattices, and no others

A repeating grid can be oblique, rectangular, centred, square or hexagonal. That is the complete list for the plane, and the argument that closes it is a page long.

lattices · Lattice The round trip, on Pnma. 8 operations were generated from the standard generators of Pnma; the orbit of three points in general position was formed, the group was discarded, and 8 operations were rediscovered from the 24 points alone. The two sets are identical, which is what the figure asserts.

Forgetting a group in three dimensions

For three phases this site said its machinery was two-dimensional and decided nothing about a space group. That was true, and it was a limit rather than a principle — nothing in the decidability argument mentions the number two.

space-groups · Space groups The 48 point symmetries of a cubic lattice. Every operation that maps a cubic lattice onto itself, built as the integer matrices preserving that system's metric: 48 of them, of which 24 are proper rotations and 24 reverse handedness. The orders occurring among the rotations are 1, 2, 3, 4 — the same list the plane gives, so the crystallographic restriction does not change in three dimensions, and there is no six anywhere. The 13 rotation axes are counted from the rotations they carry rather than drawn from memory, and every rotation but the identity is checked to belong to exactly one of them.

The restriction in three dimensions

Space is roomier than the plane in every other respect, so the natural expectation is that it permits more rotation orders. It permits exactly the same five, and seeing why is more interesting than the result.

restriction · Restriction 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 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 Centring the five lattices. Each of the five plane lattices with the midpoint of every cell added, and the type of lattice that results — read off the reduced basis of the new point set rather than looked up. Every centring halves the cell area, so the original lattice is a sublattice of index two in the centred one, and every centred lattice is again one of the five. Two of the five come back as themselves and are therefore no richer for being centred. The rectangular and rhombic lattices exchange, which is what makes them one family under two descriptions. And the hexagonal lattice centred is rectangular — its holohedry falls from 12 to 4, so centring destroys the symmetry it was meant to display.

Centring, counted as a sublattice

Adding the centre of every cell to a lattice produces another lattice, containing the first with index two. Doing it to each of the five in turn shows why the list is five rather than ten, and why only one of the five has a centred description worth keeping.

lattices · Centring 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 Which classes can twin by merohedry, and how many ways. Every crystal class, with the number of twin laws its own lattice offers it. The index of the class in the point group of its lattice is the number of orientations available; 25 of the thirty-two have more than one, and the 7 holohedral classes have exactly one — their crystal already has every symmetry their lattice has, so there is nothing left over to twin by. The names along the right are the old mineralogical ones: hemihedral for half, tetartohedral for a quarter.

Twenty-five of the thirty-two can twin, and seven cannot

The number of twin laws available to a crystal is the index of its class in the point group of its lattice, minus one. Doing that arithmetic for all thirty-two classes takes a moment and produces a census with a sharp edge on it — the seven classes that cannot twin this way are exactly the seven that already use everything their lattice has.

applied · Twinning The fourteen Bravais lattices. All fourteen lattices: triclinic P, with 2 symmetries; monoclinic P, with 4 symmetries; monoclinic C, with 4 symmetries; orthorhombic P, with 8 symmetries; orthorhombic C, with 8 symmetries; orthorhombic I, with 8 symmetries; orthorhombic F, with 8 symmetries; tetragonal P, with 16 symmetries; tetragonal I, with 16 symmetries; rhombohedral P, with 12 symmetries; hexagonal P, with 24 symmetries; cubic P, with 48 symmetries; cubic I, with 48 symmetries; cubic F, with 48 symmetries. The corner points are the conventional cell; the points in the second colour are the centring translations, drawn at every position inside the cell rather than one per face. The cell shapes are the picture's, chosen so no two systems look alike; only the angles a system is defined by mean anything.

Twenty-five cells, and fourteen lattices

The usual picture of the fourteen Bravais lattices is a plate of fourteen boxes, which is the answer with the argument removed. The argument is one question asked of twenty-five candidates, and the question has a computable answer.

lattices · The fourteen Bravais lattices 4 lattices. 4 lattices: cubic P, with 48 symmetries; cubic C, with 16 symmetries; cubic I, with 48 symmetries; cubic F, with 48 symmetries. The corner points are the conventional cell; the points in the second colour are the centring translations, drawn at every position inside the cell rather than one per face. The cell shapes are the picture's, chosen so no two systems look alike; only the angles a system is defined by mean anything.

Forty-eight becomes sixteen

Centre one face of a cube and the four threefold axes along its body diagonals are gone. That sentence is usually offered as a fact to accept; it is a computation whose answer is a number, and the number says which lattice you got instead.

lattices · The fourteen Bravais lattices The Wigner–Seitz cell of the hexagonal lattice. Every point closer to the central lattice point than to any other. The faint lines run to the 6 neighbours whose perpendicular bisectors bound the region; every other lattice point is cut off by one of them. The cell has exactly the area of a unit cell — asserted while the figure is drawn, against √det G computed from the metric — and it carries all 12 of the lattice's symmetries, which a conventional cell need not. Nothing was chosen to build it: no basis, no axes, no convention. Two people who agree about the lattice cannot disagree about this cell.

The cell nobody chose

Every unit cell on this site is a convention, and one construction escapes the warning entirely: the region of the plane closer to one lattice point than to any other. It needs no basis, no axes and no rule — and its combinatorics are decided in integers, with the square roots confined to drawing it.

lattices · Wigner–Seitz cells Six integers that do not depend on the description. The same monoclinic lattice written in 4 different bases, each obtained from the last by an integer matrix of determinant one, and each reduced by Niggli's algorithm. Every one of them gives the same six integers — the squared lengths and twice the dot products of the reduced basis. That is what makes the reduced form a fingerprint of the lattice: two cells with no number in common are the same lattice exactly when their reduced forms agree, and the comparison has no tolerance in it.

The cell that settles the argument

Two determinations of one compound can report cells that share no number and describe the same lattice. Reduction is the procedure that decides — six integers that depend on the lattice and not on anybody's choice of axes, and that agree exactly when the lattices do.

lattices · Lattice 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 The shells of the hexagonal lattice. Every point of the hexagonal lattice within a squared distance of 24, with a circle drawn at each length that occurs. The form is x² + xy + y², and the number of points on each circle is a coefficient of the lattice's theta series: 6 at 1, 0 at 2, 6 at 3, 6 at 4, 0 at 5, 0 at 6, 12 at 7, 0 at 8. The gaps matter as much as the counts — a circle with no points on it is a length the lattice does not have, and which lengths those are is a question in number theory rather than in geometry.

How many vectors of each length

Counting the lattice points at each distance from the origin turns out to be a question about divisors, and the answer explains something a crystallographer meets every day: why a cubic powder pattern has no line at seven.

lattices · Lengths 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 The region every plane lattice lands in. The shape of a plane lattice is one complex number, τ, and every lattice can be brought by a change of basis into the region shaded here: the strip between 0 and a half, outside the unit circle. Its interior is the oblique lattices. Its left edge is the rectangular ones, its arc and its right edge the centred rectangular ones, and its two corners are the square lattice at i and the hexagonal lattice at ρ. Five kinds, and they are a region, three arcs and two points rather than five things of one sort. The region is unbounded upwards, where the cell gets longer and thinner without limit.

The space every lattice lives in

Five lattices in the plane is the number of *kinds*. The number of lattices is a continuum — and it has a shape: one two-dimensional region with two corners, three edges and an interior, where the five kinds turn out to be a region, three arcs and two points rather than five things of one sort.

lattices · Moduli P222: 16 descriptions of one structure. P222 has 4 operations in a cell. 8 origins leave every one of them exactly where it was, and 8 linear parts of the lattice's holohedry normalise the group, so its Euclidean normaliser has 64 elements per cell and the index is 16. That index is the number of coordinate lists that describe one and the same arrangement of atoms. Each was applied to a motif and the resulting point sets compared: the numbers differ and the sets are identical, which is the check that makes the count mean anything.

One crystal, and sixteen coordinate lists

Two structure reports can disagree in every number and describe the same arrangement of atoms, because a space group does not fix its own origin or its own axes. How many genuinely different lists there are is the index of the group in its Euclidean normaliser — a number, computable, and the thing a structural database has to divide out before it can say two entries are the same compound.

operations · Normalisers Sublattices of index n, in space. How many sublattices a three-dimensional lattice has at each index, beside the plane's answer, with the Hermite enumeration and the coefficient of ζ(s)ζ(s−1)ζ(s−2) in separate columns. The two are computed by routines sharing no code, and a row where they disagreed would be a failure rather than a result. The last column counts the ones that survive every operation of the cubic group, and it is almost always empty.

The three that stay cubic

A lattice in space has far more sublattices than one in the plane — 651 of index sixteen against 31 — and almost none of them keeps the symmetry it came from. The ones that do exist at indices m³, twice m³ and four times m³, there is exactly one at each, and they are the primitive, face-centred and body-centred cubic lattices, arrived at by asking which sublattices keep a symmetry rather than by enumerating centrings.

lattices · Sublattices square: 3 dislocations, 2 stable. The short lattice vectors of the square lattice, grouped into orbits under its own automorphism group of 8 operations. Two vectors of one orbit are the same defect seen from different directions, so the number of kinds of dislocation is the number of orbits — 3 out to 4 times the shortest squared length. Each orbit has its own colour. Solid arrows are stable: no pair of shorter lattice vectors adds to them with a smaller total of |b|². Dashed ones split, and 1 of the orbits do. Which splits is decided by comparing integers; that the energy goes as |b|² at all is Frank's rule and comes from elasticity, not from here.

How many dislocations a lattice has

A circuit round a defect comes back to the wrong lattice point, and the amount by which it misses is a lattice vector. That much is quantised. The next question has a number for an answer: how many *different* dislocations are there? Two Burgers vectors related by an operation of the point group are one defect seen twice, so the answer is a count of orbits.

applied · Defects monoclinic: 9 twin laws, 1 of them exact. The twin laws of a monoclinic lattice: a two-fold about the row [uvw] paired with the plane (hkl) it is meant to be a mirror in, kept when the twin index is at most six and the obliquity at most six degrees, which are Friedel's own limits and are a convention rather than a theorem. 9 of the 9 are listed. The index n is how many lattice nodes there are per node the operation restores, computed from the integers and checked against the sublattice built from the plane and the row. The obliquity is the angle between the row and the plane's normal: 1 of these laws have none, and for those the operation restores a sublattice exactly.

The index and the angle a twin misses by

Whether a crystal will twin on a given operation is decided by its lattice, not by its structure. Two numbers decide it: how many lattice nodes there are per node the operation restores, and how far the operation is from being a symmetry at all. Both are computed from integers, and one of them is a fiction that has to be labelled as one.

applied · Twinning One net, six descriptions, four different answers about its symmetry. The honeycomb written against six bases of ℤ², all of them the same net. The detector tests each lattice type's holohedry in standard position, so a symmetry written against another basis is a matrix that is not in the list and is never tried — and the answer comes back as p6m, or an unnamed group of order four, or p2, or cmm, depending on how the voltages were typed. The metric column is the form the net's own edges make, inverted; the reduced column is that form after Lagrange–Gauss reduction, and it is the same in every row, which is what makes the last column a property of the net.

The symmetry a net was written with

A net has no coordinates, so its symmetry is whatever its best drawing has. This collection measured that by handing the drawing to a detector — and the detector tests a fixed list of matrices, so the answer depended on which pair of translations the voltages had been written against. The honeycomb came back as p6m, or p2, or cmm, or nothing, one net and four answers.

applied · Nets 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 Twenty-two halvings the fourteen lattices permit. Every lattice has exactly seven subgroups of index two, whatever its shape. The third column is how many of the seven the lattice's own group carries onto themselves, and the fourth is how many of those survive as distinct types once a change of basis within the type is allowed to identify them. The running total ends at twenty-two, which with the fourteen grey lattices is the thirty-six magnetic Bravais lattices — and the row that ends at zero is the face-centred cubic lattice.

The halving a lattice will not permit

Admit time reversal and a lattice splits into points that leave the moments alone and points that reverse them. The second set is a coset of a subgroup of index two, and every lattice has exactly seven of those, whatever its shape. What differs is how many of the seven the lattice's own symmetry survives — and the face-centred cubic lattice survives none of them.

lattices · Magnetic Seventy-three arithmetic classes, from fourteen groups. Every subgroup of every lattice's own group, split by whether the subgroup's own Bravais group is that lattice's. The ones that are not belong to a lower lattice and are counted there, which is what stops the same class being counted twice. The running total ends at seventy-three, and no conjugacy in GL(3, ℤ) was ever decided.

Seventy-three, without a search

The unit a space group is built from is a point group together with the lattice it acts on, and there are seventy-three of them. Getting there looks like it needs conjugacy in GL(3,ℤ), which is a search this collection tried and abandoned. It does not: every finite group of integer matrices carries a canonical larger group that says which lattice it belongs to, and once that is computed the search has nothing left to do.

point-groups · The fourteen Bravais lattices Four groups whose description count a metric can raise. Every plane group, the number of ways of writing one arrangement down on the lattice the group requires, and the number on the most symmetric lattice it may sit on. Eight groups already occupy the most symmetric lattice available to them and have nowhere to go. Four have a metric that raises the count, by two and in one case by six. The starred rows belong to groups whose normaliser has a free direction, where the quantity is a count of grid points rather than an index and cannot be compared.

The normaliser is not a function of the group

How many ways there are of writing one structure down is computed from the group and printed in a table beside its name. It is not a property of the group. Draw a p2 pattern on a hexagonal cell rather than an oblique one and the number goes from four to twenty-four, with nothing done to the group at all.

operations · Normalisers One achiral motif in p4, chiral along one line and achiral along two. The same motif — three points with a mirror and no other symmetry — repeated by p4, the plane group of quarter-turns with no mirror, and placed with its mirror along three different lines of the square lattice. Along the first line the pattern has no operation that reverses orientation: it is chiral, although every piece of it is achiral. Along the second and third the motif's mirror is also a mirror of the whole pattern, and the detected groups are p4m and p4g; the mirror lines of each pattern are drawn. All three verdicts come from detecting the symmetry of the points and agree with whether the motif's mirror normalises p4.

A hand made of pieces that have none

Quartz is built from tetrahedra that have no handedness, and every quartz crystal is left-handed or right-handed anyway. Put a piece with a mirror into a pattern whose group has none, and the pattern keeps the piece's mirror only if that mirror lies on one of a few lines the group's normaliser draws. Anywhere else, the arrangement has a hand its parts do not.

space-groups · Chirality Seventy-two positions and the sets they fall into. Every plane group with the number of its Wyckoff positions, and the number of sets those positions fall into when the positions a normaliser exchanges are counted once: on the cell the group requires, on the most symmetric cell it may sit on, and under every change of basis carrying the group onto itself. 72 positions become 53 sets on the required cells and 51 on the best ones, and the last column never goes lower. The two groups a special cell changes are pmm and cmm.

The same site under two names

A structure report puts each atom on a Wyckoff position, and two correct reports of one crystal can name different positions. The positions that can trade places are exactly the ones the normaliser exchanges — and which those are depends on the cell as measured, not only on the group.

operations · Normalisers Free going down, two conditions going up. The edge between p2 and p4 in the diagram of maximal translationengleiche relations, read in both directions. Downwards it costs nothing: the quarter-turns are discarded and the lattice is exactly the lattice that was there, so every p4 pattern contains a p2 pattern. Upwards the added quarter-turns must carry the lattice onto itself, which forces the cell to have equal edges at a right angle — two conditions on a general oblique cell, which has only two parameters to give. So a p2 structure has a p4 supergroup exactly when its measured cell happens to be square, and the question is about the metric rather than about the group.

Going up costs the cell a parameter

The usual asymmetry — finitely many maximal subgroups below, infinitely many minimal supergroups above — is false in both halves for a plane group. Both directions are infinite and equinumerous index by index. The real asymmetry is that 17 of the 31 edges cost the lattice a parameter going up and nothing going down.

operations · Subgroups 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

Named alongside it

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

Bravais latticeLattice automorphismMetric tensorPoint groupCentringCrystal classIndexNormaliserSubgroupSublatticeGram matrixUnimodular matrix

All concepts