Crystallography & Symmetry
patterns derived from their groups
Concept

Index — where it appears

How many cosets a subgroup has in its parent, which is also how many times larger the smaller group's fundamental domain is. It counts the domain states a symmetry-lowering step produces and the twin laws a crystal class has available.

Named by 37 essays across 7 fields — each of them below, with the objects they name alongside it.

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 p4 inside p4m, by area. A fundamental domain for p4m beside one for p4, drawn by the same construction on the same grid. p4 sits inside p4m with index 2: it has 8 ÷ 4 = 2 times as few operations to rebuild the pattern with, so it needs 2 times as much of the cell to rebuild it from — measured here at 1.51 times as many samples. The two domains are one region and 2 copies of it, and the ratio between them is the index rather than a coincidence of shape.

Domains of a subgroup

A group with half the operations needs twice as much of the cell to rebuild the pattern from. That single sentence is the index arithmetic of the whole classification, and it turns the containments among the seventeen into a statement about area.

operations · Fundamental domain m3̅m → 4mm: 6 domain states. The transition from class m3̅m to class 4mm loses 40 of the parent's 48 operations, so the child has index 6 and the crystal comes apart into 6 domain states. Each colour is one state — one coset of 4mm in m3̅m — and each holds the same 8 poles. The lost operations are what carries one state onto another, and they survive in the crystal as the relation between its domains rather than as symmetries of any part of it. The descent changes the crystal system, so the states differ in shape as well as in orientation and the transition is ferroelastic.

How many domains a transition makes is an index

Cool a crystal through a symmetry-lowering transition and it has to choose one of several equally good low-symmetry arrangements. Different parts of it choose differently, and the number of available choices is the index of the new group in the old one — a number available before any crystal is grown, and one of the few predictions in this field that is a count rather than a bound.

applied · Domains Everything class m3̅m can descend to. The 25 crystal classes that are subgroups of m3̅m, arranged by order, with the 56 maximal steps between them drawn as edges. The order of each row is printed down the left, so the index of any step is the ratio of the two rows it joins. A symmetry-lowering transition can only be continuous when it goes down one of these edges, and the index on the edge is the number of domain states the transition produces. A descent of several steps is possible but has to happen discontinuously or through the intermediate classes.

The descent of symmetry is a lattice, not a tree

Which classes a crystal can fall to when it loses symmetry, drawn as a graph with the index on every edge. It is routinely called a tree and it is not one — a class can be reached from its parent by several different routes of the same total index, and which route a material takes is a physical question the diagram deliberately leaves open.

applied · Domains The subgroups of p4m of index 2. p4m has 7 subgroup(s) of index 2 with cyclic quotient. 3 of them keep every translation and lose operations — the lattice is untouched and the pattern loses a symmetry at every point. 4 keep every operation and lose translations, and each is named beside the basis of the sublattice it keeps, written in the parent's own axes. Each subgroup is the kernel of a homomorphism onto a cyclic group, found by enumeration; each name is found by searching changes of basis and origin until the operation sets match exactly.

Two ways down from a group

A pattern can lose a symmetry by giving up an operation or by giving up a translation, and the two are different in kind. Sorting the seventy-four subgroups of index two among the seventeen splits them twenty-nine to forty-five — and a containment test that compares operations modulo one shared lattice can only see the twenty-nine.

operations · Subgroups An antiphase boundary in p4. Where two antiphase states meet. Above the line the species alternate one way and below it the other, so at the boundary two cells of the same species sit next to one another and the ordering is out of step. This is a domain wall with no change of orientation across it: the crystal is not twinned, its lattice is undisturbed, and diffraction sees it only in the width of the superlattice reflections.

The domains a lost translation makes, which nothing optical can see

An ordering transition can leave the crystal class untouched and take away translations instead. The domains that result have the same orientation, the same shape and the same optical properties as each other, and where two of them meet the ordering is simply out of step — a boundary with no change of direction across it and no way to find it except by looking at the ordering itself.

applied · Domains The seventeen, arranged by what they can lose. Each group at the height of its own order, joined to every maximal subgroup that keeps all of its translations. Reading downwards is a crystal losing operations at a phase transition. The edges are the maximal ones only — every other containment is a path through these — and the whole graph is enumerated by closing every subset of each group's operations, so nothing is here because a table said so.

The descent with no shortcut

A subgroup can give up operations, or it can give up translations. Hermann's theorem says that a *maximal* subgroup does one or the other and never both at once — which is why a crystal losing symmetry can be followed one clean step at a time, and why every route from p6m down to p1 has exactly three steps.

operations · Subgroups Two-colourings of the seventeen. How many ways each of these 17 plane groups can be two-coloured so that every symmetry either preserves the colours or exchanges them. 74 in all, each one a subgroup of index two enumerated by trying every assignment of colours to a generating set and keeping the assignments that turn out to be consistent. p3 admits none: a homomorphism onto a group of order two has nothing to send a three-fold rotation to but the identity, and once the rotation and its conjugates are killed nothing is left to reverse the colours. pmm admits the most, with 15. Every count is one less than a power of two because the homomorphisms of a group onto the two-element group are the non-zero elements of a vector space over that field.

Two colours, and a symmetry that swaps them

A chessboard and a grid of identical squares have the same group, which is plainly not what anybody sees. Admitting the colour swap as an operation gives a finer classification — and one of the seventeen turns out to admit no two-colouring at all.

classification · Ornament Sublattices of index n in the plane. For each index up to 12: the number of sublattices found by building every Hermite normal form of that determinant, and the number the Dirichlet series ζ(s)ζ(s−1) predicts — the sum of the divisors in the plane, and a longer sum in space. The two columns are computed by routines that share no code, and the figure does not appear at all if any row disagrees.

How many ways there are to thin a lattice

A sublattice of index n keeps one lattice point in n, and there is never only one way to do it. In the plane the number of them is the sum of the divisors of n; in space it is a longer sum; and both are counted here by writing every one of them down.

lattices · Sublattices How many ways each class can lose symmetry. Every crystal class, with the number of distinct classes it can descend to — 247 parent-and-child pairs in all across the thirty-two, counted up to conjugacy in the parent, which is the equivalence that says two descents differing only by which axis was chosen are one transition. The count rises steeply with the order of the parent, which is why the cubic and hexagonal holohedries dominate the list of materials with rich domain structures.

Two hundred and forty-seven descents, or two hundred and twelve

How many distinct ways can a crystal lose symmetry? Counting parent-and-child pairs up to conjugacy in the parent gives 247. The standard enumeration in the ferroics literature gives 212, and the operation that merges the extra thirty-five turns out to be a rotation through forty-five degrees — which no lattice may have, and which no integer matrix in a lattice basis can therefore express.

applied · Domains Σ5: two square lattices at 36.87°. Two square lattices, one turned through 36.87° about a shared point. At this angle one point in 5 lands exactly on a point of the other lattice — 29 of the 149 drawn — and those shared points are themselves a lattice, the coincidence site lattice, of index 5. The angle comes from tan(θ/2) = 1/3, and Σ is the odd part of 3² + 1² = 10. Nothing here is measured: whether a point is shared is decided by an integer congruence.

Turn a lattice against itself and almost nothing lines up

Two copies of one lattice rotated about a shared point share that point and, at almost every angle, no other. At a discrete set of angles they share a whole sublattice — one point in three, or five, or seven — and a grain boundary built on such an orientation costs a fraction of what a general one costs, because a fraction of the atoms are already where both sides want them.

applied · Interfaces Which indices have a square sublattice. For each index up to 26: how many sublattices of the square lattice are themselves square, found by testing whether the quarter-turn maps each one onto itself; the same count as a sum over divisors, +1 for each divisor one more than a multiple of four and −1 for each one less; and the ways of writing the index as a sum of two squares. The three agree at every row, which is Fermat's theorem — and it says that 3, 7 and 11 have no square sublattice at all while 5, 13 and 17 have two.

The sublattices that stay square

A sublattice of the square lattice is itself square exactly when its index is a sum of two squares — so index five has two and index seven has none, and which superstructures a surface can form is decided by a theorem of Fermat's about primes.

lattices · Sublattices Every coincidence index is odd. The rotations that bring a cubic lattice into coincidence with itself, up to index 25: 17 distinct relations across 12 indices, each found by enumerating integer quaternions and each index computed as the size of a sublattice rather than from the usual formula — the two are then required to agree. 5 of the 12 indices carry more than one relation, which is why the tables write 13a and 13b. Every index is odd. That is not a feature of this range: a rational orthogonal matrix written in lowest terms has an odd denominator, so the factors of two always cancel.

Every coincidence index is odd, and in the plane most of them do not exist

The indices at which two copies of a cubic lattice share points are 3, 5, 7, 9, 11 and every odd number after them. There is a two-line proof that no even index can occur. Ask the same question about a square lattice and the answer is a different list entirely, governed by which numbers are sums of two squares — so Σ3, which is the commonest boundary in every metal, has no plane analogue at all.

applied · Interfaces Subgroups of index 3, across the seventeen. Every subgroup of index 3 with cyclic quotient in each of the seventeen plane groups, sorted into the two kinds: 4 keep all the translations and lose operations, 22 keep all the operations and lose translations, and the total is 26. The split is decided by whether the homomorphism onto ℤ3 kills the two lattice translations, which is a property of the kernel and not a judgement. Every one of them is found by enumeration inside the finite quotient by 3Λ, and the count for the whole classification is a measurement.

Three colours, and why most patterns cannot have them

Seventy-four of the seventeen plane groups' subgroups have index two, and every group but one has at least one. At index three there are twenty-six, and ten of the seventeen have none at all — because a symmetry of order two cannot survive being asked to permute three colours.

classification · Colour One hundred and twenty-two magnetic point groups. The three kinds, counted. Thirty-two ordinary groups, which contain no primed operation; thirty-two grey groups, which contain time reversal on its own and are the symmetry of anything magnetically disordered; and fifty-eight black-and-white groups, one for each way of splitting a class into a subgroup of index two and its complement. The last number is the one that has to be computed: the index-two subgroups are found by closure inside each class, reduced up to conjugacy, and reduced once more by an equivalence that needs a rotation no lattice may have. 32 + 32 + 58 = 122, and every term is a measurement.

The operation that reverses time

A magnetic moment is a current loop, so running time backwards reverses it and moves nothing. Admitting that as a symmetry operation turns the thirty-two crystal classes into a hundred and twenty-two — and eight of the merges needed to reach that number require a rotation no lattice may have.

point-groups · Magnetic Subgroups of index two, three and four. Every plane group with the number of subgroups it has at each small index, counted by enumerating the transitive actions on that many points. The zeros are the interesting entries: p3 has no subgroup of index two and the four-fold groups have none of index three, because a subgroup of index n gives an action on n points and the group has to have a quotient that can act. A rotation of order three has nowhere to go in a set of two, and one of order four has nowhere to go in a set of three that is not the identity — so the index is constrained by the point group before any geometry is done.

How many subgroups of index three

Taking operations away and closing what is left finds the maximal subgroups and stops there. Counting instead the ways a group can act on three points finds all of them — and finds that a four-fold group has none of index three at all.

operations · Subgroups 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 p = 2: 1, 3, 6, 12, 24 vertices at each distance. Every sublattice of index a power of 2, up to scale, joined when one contains the other with index 2. From the whole lattice there are 3 ways down, because a sublattice of index 2 is a line over the field of 2 elements and there are 3 of those; from each of those there are 3 again, one of which is the way back. So the counts are 1, 3, 6, 12, 24 — that is (2 + 1)·2^(k−1) — and the graph has no cycles, both of which are checked on every vertex whose whole neighbourhood was grown rather than read off the picture. The object is the Bruhat–Tits tree of the p-adic plane, and it is what the set of sublattices is rather than how many there are.

Every way down, and no way round

There are as many sublattices of a given index as the index has divisors, and counting them is where that essay stopped. This one asks what they are to each other, and the answer is a shape: an infinite tree in which every vertex has exactly p + 1 neighbours and no path ever comes back.

lattices · Sublattices At which indices a group contains a copy of itself. A filled circle where the group has a subgroup of that index which is the same plane group again. The groups with no rotation past a half-turn take every index — the lattice can be stretched along one direction by any factor. The four-fold groups take the sums of two squares and the three- and six-fold groups take the Loeschian numbers, because a sublattice invariant under a quarter or a third of a turn is an ideal in the Gaussian or Eisenstein integers and its index is a norm. The groups with mirrors take fewer still, and p4g takes only the squares.

The same group in a bigger cell

A subgroup usually gives something up. An isomorphic subgroup gives up nothing but scale — the same plane group again, on a coarser lattice — and the indices at which that is possible turn out to be the values of a quadratic form.

space-groups · Isomorphic subgroups Sums of two squares, arriving as superstructures. Which indices admit a sublattice of the same shape as the square lattice, drawn as a bar per index whose height is how many there are. The pattern is not a pattern about lattices at all: an index works exactly when it is a sum of two squares, because a similar sublattice of the square lattice is multiplication by a Gaussian integer and its index is that integer's norm. The indices that work up to 30 are 1, 2, 4, 5, 8, 9, 10, 13, 16, 17, 18, 20, 25, 26, 29, and the same list is produced here a second time by factorising rather than by searching, with the two required to agree.

The sublattices that are the same shape

Thinning a lattice usually changes its shape. Sometimes it does not: the sublattice is the parent rotated and scaled, and a drawing of it alone would be a drawing of the parent. Which indices allow it turns out to be a question Fermat answered in 1640.

lattices · Sublattices 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 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 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 The alias accounts for every line and predicts more. The observed lines above, and below them the grid of a supercell that explains all of them. The full ticks are the observed lines, which the alias reproduces exactly; the faint ones are lines the alias predicts and nobody saw. That second set is the only thing that separates the two cells, and it is why an indexing criterion has to charge for unobserved lines rather than measure agreement.

Every alias is a supercell

A cell that explains every line of a powder pattern is not a near miss and not a coincidence: its reciprocal grid contains the true one, which means its own cell is a superlattice of the true cell. So the ambiguity of indexing is the arithmetic of superlattices, and it can be counted — two cells with one unknown, sixteen with two, sixty-two with three, all of them accounting for the same twenty lines exactly.

applied · Indexing One group refuses two colours and three refuse three. The two counts side by side, with the rows that refuse a number of colours marked. p3 is the only group with no two-colouring; p4, p4m and p4g are the only ones with no three-colouring. Neither list is a subset of the other and both come from the same arithmetic — a rotation order that divides nothing the symmetric group has.

What a half-turn does to three colours

Ten of the seventeen plane groups have no three-colouring, because a half-turn cannot permute three colours cyclically — that is the first rung of this ladder and it is true. Drop the word cyclically and the answer changes completely: a half-turn permutes three colours perfectly well by swapping two and fixing one, and only the three four-fold groups refuse three colours at all.

classification · Colour How many similar sublattices the cubic lattice has at each scale. Every integer matrix satisfying MᵀM = α²I, counted up to the lattice's own point group by marking orbits rather than dividing. The even scales are drawn apart because they are the ones that give nothing new: a factor of two in the scale never produces a shape the smaller scale did not already have.

The shapes a lattice in space can thin to

In the plane, which indices admit a sublattice of the same shape is a question about which integers a quadratic form represents, and Fermat answered it. In space the question collapses: taking determinants shows the index is always a perfect cube, so there is nothing to represent. What is left is how many there are at each cube — and for a hexagonal lattice, whether there are any at all depends on one number.

lattices · Sublattices Which indices a screw axis contains itself at. The fifteen kinds of axis a space group may have, against the index of the sublattice taken along the axis. A filled cell is an index at which the axis contains a copy of its own kind; the darker cells are the indices at which what comes back is the mirror image instead. A pure rotation axis is filled everywhere and a screw is not, and which indices a screw loses is decided by one congruence rather than by any geometry.

A screw that contains its own mirror image

No operation of a crystal turns a right-handed screw axis into a left-handed one — that is what makes the eleven enantiomorphic pairs pairs. And yet a 4₁ axis contains copies of 4₃ as subgroups, at every index congruent to three modulo four. One congruence decides both which indices are possible and which hand comes back, and it is the same congruence for all fifteen kinds of axis.

space-groups · Isomorphic subgroups 19 site symmetries, and the counts each can impose. Every distinct site symmetry across the space groups this site builds, named by the multiset of its operation types, with the orientation counts a disordered molecule there may take. The counts are the indices of the site group's subgroups, computed by closing subsets under multiplication rather than looked up. Nearly every row offers every divisor of its order. One does not: a site of order twelve whose group is the tetrahedral rotation group refuses an orientation count of two, because that group has no subgroup of order six.

How many orientations a disorder needs

A molecule at a site with more symmetry than it has resolves the contradiction by occupying several orientations at once. How many is not fitted: it is the index of the molecule's symmetry in the site's, so the occupancy is the reciprocal of a whole number — and one site in the census refuses a divisor of its own order.

restriction · Local symmetry 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 Disorder models against occupancies, site by site. Every distinct site symmetry in the space groups built here — 19 of them — with its order, its number of subgroups, the number of distinct disorder models, which are the subgroups up to conjugacy by the operations of the site symmetry, and the number of different occupancies those models can have. The last column is the largest number of models that share one occupancy. In all, 162 models share far fewer occupancies; the most crowded is 4/mmm, where 11 different models all give an occupancy of 1/4. At every site, the classes of operation a model keeps separate it from every other model with the same occupancy.

The occupancy does not name the disorder

A molecule disordered on a special position takes a number of orientations fixed by a group index, and its occupancy is the reciprocal. Many different disorders share one occupancy — eleven at a single kind of tetragonal site — and what separates them is which of the site's operations the molecule keeps, which the averaged structure records and the occupancy does not.

restriction · Local symmetry What lies between a group and its copies at index 25. For each group, the lattices carried to themselves by its point group that contain a copy of the group at index 25, arranged by index, with lines for containment; each lattice is labelled by the Gaussian or Eisenstein integer that generates it. Copies are the bottom row, drawn large when nothing lies between them and the whole lattice. p4: 3 copies at index 25, 0 maximal, with invariant lattices between of index 5; p4m: 1 copies at index 25, 1 maximal, with invariant lattices between of index none.

The primes a cell can grow by

A plane group contains copies of itself in bigger cells, and the International Tables list the ones that are maximal — the copies nothing else sits between. For p4 they come at 2, at 5, 13, 17 and 29 twice each, and at 9 and 49 once, and the list is the list of primes of the Gaussian integers. Put mirrors on the pattern and the copies at 5 and 13 vanish while 25 becomes maximal: a mirror cannot keep one factor of a prime without the other.

space-groups · Isomorphic subgroups Subgroups, the classes a group sorts them into, and the sets its normaliser does. For every plane group, the number of subgroups of index two and of index three, the number of conjugacy classes those fall into under the group's own operations, and the number of sets they fall into under its Euclidean normaliser. Over the seventeen there are 74 subgroups of index two in 74 classes and 56 sets, and 82 of index three in 36 classes and 32 sets. 9 of the thirty-four rows have fewer sets than classes, which is where the tables' "equivalent" entries come from. Counts of subgroups and of classes agree with an independent count from transitive actions on n points.

Three of them, and they are equivalent

The subgroup tables print a count and sometimes a word beside it. Three subgroups of one type may be three copies the group itself shuffles, or three the group holds firmly apart and only a change of description exchanges. p3 has three copies of itself at index three, no operation of p3 moves any of them, and one shift by a third of a cell exchanges all three.

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 P4₁: the lattices, as an ideal across and a multiple along. Every sublattice the point group of P4₁ carries to itself, indexed by the norm of the ideal it uses across the axis and by the multiple it takes along it. The entry is the space group that sits on it: the parent's own type in one colour, a different type in the other, and a dash where no group with the parent's point group survives at all. A dot marks a lattice that is maximal — one whose step is a single prime, across or along, with nothing between it and the whole. The rows and columns are two divisibility orders and the table is their product, which is the shape the plane's answer predicted.

An ideal across and a prime along

In the plane a copy of a group inside itself grows by a prime ideal, and the maximal indices are the norms of the primes of a ring. In space with one principal axis there are two directions to grow in, and the question the plane left was whether the two constraints multiply. They do not — and the place they fail is an index the plane calls maximal, because the step in between carries the group's mirror image.

space-groups · Isomorphic subgroups A row of coefficients for every group. For eight of the seventeen, the number of sublattices of each index that the group's point group carries to itself. A nought means the group has no copy of itself at that index at all. p1 and p2 preserve every sublattice, so their rows are the counts of sublattices themselves — 1, 3, 4, 7, 6, 12 — which is the sum of the divisors. The rows thin out as the point group grows, and p4m and p6m have almost nothing in them. Every row is a sequence a Dirichlet series can be built on, and the next figure is what that series factors into.

A row written as a product

Every group's copies of itself sit at a row of indices, and every row so far has been read one entry at a time. Counting all of them at once turns a row into a Dirichlet series, and every one of the seventeen rows factors into a product over the primes — which is the statement that a copy is a chain of maximal steps, written as arithmetic. The plainest group of all has the most famous series in mathematics.

space-groups · Isomorphic subgroups One lattice, two copies of pm. pm on a lattice doubled across its mirrors. The mirrors of the parent are every vertical line; a copy of pm on the doubled lattice has mirrors every other line, and there are two ways to choose which — the solid set or the dashed set. Both are copies of pm with the same lattice and the same point group, and no translation of the parent carries one onto the other, because the translation that would is exactly the one the doubling removed. So a count of invariant lattices is not a count of subgroups, and the gap is visible in the smallest case there is.

A lattice is not a subgroup

Every count of copies so far has counted lattices, and the International Tables count subgroups. One invariant lattice can carry several copies of a group that nothing in the parent carries onto one another — pm's doubled lattice carries two, with its mirrors on the even lines or the odd ones — and how many is a cohomology computation, a first where the classification of the seventeen used a second.

space-groups · Isomorphic 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.

SublatticeSubgroupCosetMaximal subgroupIsomorphic subgroupGaussian integerHolohedryKlassengleicheNormaliserSuperstructureConjugationDivisor sum

All concepts