Crystallography & Symmetry
patterns derived from their groups
What a lattice forbids

The same group means the same pattern

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

Assumes Why there is a list at all and What is left when the order is forgotten.

17 min read 10 figures Symmetry is decidableExactly this manyThe same arithmetic, renamed

The seventeen are usually presented as seventeen objects: this pattern has a four-fold centre and that one has not, this one has mirrors and that one only glides. Nothing in that presentation says they are seventeen groups.

The distinction is not pedantic. Two patterns that look entirely different could, in principle, have symmetry groups that are isomorphic as abstract groups — the same multiplication table, wearing different clothes. If that happened, the classification would be a classification of drawings, and the number seventeen would be an artefact of how the drawings were sorted rather than a fact about algebra.

It does not happen, and the reason is a theorem rather than an accident.

p6: the map comes back. p6 written on two bases related by an integer matrix of determinant one, and about two origins. The two descriptions share no coordinate; they are the same group. The matrix and the origin shift were then recovered from the two operation sets alone — which is what Bieberbach's theorem promises, carried out as a search over the integer matrices and the origins the lattice permits, and checked by applying what was found.
Fig. 1 The same group written on two bases and about two origins. Every coordinate differs and nothing about the group does. The question this essay is about is whether that is the only way two crystallographic groups can be abstractly the same.

What the theorem says

Bieberbach proved three results about crystallographic groups in 1911 and 1912, in the course of answering Hilbert’s eighteenth problem. The second of them is the one here:

An isomorphism between two crystallographic groups of the same dimension is realised by an affine change of coordinates.

That is, if two such groups have the same abstract structure, there is an invertible linear map and a translation carrying one onto the other as sets of motions. Abstract sameness forces geometric sameness, up to a stretch and a shear.

The consequence for this collection is immediate. The seventeen are seventeen abstract groups; the two hundred and thirty are two hundred and thirty; and every count made anywhere here is a count of algebra rather than of pictures. The classification is not a taxonomy — it is a complete list of isomorphism types, and nothing depends on how the plates were drawn.

Where the work is done: the translations are characteristic

The theorem’s engine is Bieberbach’s first result, and it is the one that does the real work.

The translations of a crystallographic group form a lattice of full rank and finite index. More than that, they are characteristic: they are the unique maximal abelian normal subgroup of finite index, so no isomorphism can send them anywhere else. Any map preserving the group structure must carry the translations of one group onto the translations of the other.

Once that is granted the affine map builds itself.

The isomorphism restricted to the translations is an isomorphism of ℤ² onto ℤ², so it is an integer matrix of determinant ±1 — that is the linear part. Matching the image of one more element fixes the translation part. A homomorphism of abstract groups turns into a matrix and a vector, and the reason it can is that the lattice is not something the group is sitting on but something it contains.

14 fingerprints, and 17 with the class. Every plane group described by things an isomorphism cannot change: how many operations it has modulo translations, how many of them are rotations of each order, how many are reflections and how many glides. That separates fourteen of the seventeen; the three pairs picked out need one more invariant, and the arithmetic class supplies it, because the translations are characteristic and so the lattice the point group acts on is carried along by any isomorphism. Together the two say that no two of the seventeen are the same group.
Fig. 2 Two invariants of a plane group, both of which an isomorphism must preserve because the translations are characteristic: the census of its operations by kind, and the arithmetic class — the point group together with the lattice it acts on, up to a change of basis. Neither separates the seventeen alone; together they do, and the pairs picked out are the ones the first misses.

The constructive half, and what it costs

The theorem promises a map exists. Finding it is a different matter, and doing so is what makes the statement a computation rather than an assertion.

The procedure here goes the other way round. A group is written on a different basis and about a different origin — conjugated by an integer matrix of determinant one, then shifted — which produces the same abstract group with every coordinate changed. The conjugator is then recovered from the two operation sets alone: the integer matrices with small entries are tried, the origins the lattice permits are tried, and any candidate found is applied and compared exactly before it is reported.

204 of 204 bases put back. Every one of the seventeen groups written on four bases and about three origins, and the change of coordinates recovered from the operation sets in every case. The recovery is a search, so a failure would mean either that no affine map exists — which would contradict the theorem — or that the search was too narrow; both are worth distinguishing, and the check that separates them is that every map found is applied and compared before it is counted.
Fig. 3 Every one of the seventeen written on four bases and about three origins, and the change of coordinates recovered in every case. A failure would mean either that no affine map exists — contradicting the theorem — or that the search was too narrow, and the two are worth telling apart: every map found is applied and its operation set compared, so a wrong answer cannot be reported, only a missing one.

Two hundred and four distortions, two hundred and four recoveries. The search is bounded — matrices with entries beyond two are not tried — so what has been shown is that no case needed one, rather than that none ever could.

What a change of basis is allowed to be

The recovery searches over integer matrices of determinant ±1 and no others, and the restriction is not a convenience.

A change of basis has to take the lattice to the lattice. Written in the old basis, the new basis vectors are integer combinations of the old, and the inverse change must be integral too — which forces the determinant to be ±1. Anything else either loses lattice points or gains them.

The consequence is visible the moment a forbidden change is attempted. Stretching one axis by two and leaving the other alone is a perfectly good linear map of the plane; applied to the operations of p4 it turns the four-fold rotation into a matrix with a half in it, which is not an operation of any lattice at all. The construction refuses it rather than producing a subtly wrong group, and the refusal is the same one the whole collection’s operation constructor makes: an operation whose matrix is not integral does not map the lattice to itself.

That refusal is what makes the recovered map trustworthy. A search over all invertible matrices would find something for every pair of groups, because any two lattices are related by some linear map — and the something would not be an isomorphism of the groups.

p4m: the map comes back. p4m written on two bases related by an integer matrix of determinant one, and about two origins. The two descriptions share no coordinate; they are the same group. The matrix and the origin shift were then recovered from the two operation sets alone — which is what Bieberbach's theorem promises, carried out as a search over the integer matrices and the origins the lattice permits, and checked by applying what was found.
Fig. 4 p4m with the two axes exchanged and the origin moved three twelfths along a. Both changes are legitimate — the matrix has determinant one and the shift is a lattice fraction — and the group that comes out is the same group. Recovering which pair of changes was applied is the search this essay is about, and it succeeds here in the first few candidates because the exchange of axes is among the smallest matrices tried.

What would break the theorem, and does

The hypothesis in crystallographic is discreteness, and it is not decoration. Dropping it makes the conclusion false, and the counterexample is small enough to draw.

The group generated by 1 and √2 under addition is isomorphic to ℤ², abstractly: it is free abelian on two generators, because √2 is irrational and no whole-number combination of the two vanishes. As a group of motions of the line it has nothing in common with a lattice at all — its points come arbitrarily close together, and no affine map of the line carries one onto the other, since an affine map takes a discrete set to a discrete set.

Isomorphic groups, and nothing else in common. Two subgroups of the line, both isomorphic to ℤ² as abstract groups. The first is a lattice; the second is generated by 1 and √2 and comes arbitrarily close to every point of the line, so no affine map carries one onto the other. Bieberbach's theorem says an isomorphism of crystallographic groups is realised by an affine map, and this is what the word crystallographic is doing: without discreteness the conclusion is simply false.
Fig. 5 Two subgroups of the line, both isomorphic to ℤ² as abstract groups, drawn to the same scale. The first is a lattice; the second is generated by 1 and √2 and is dense. Bieberbach’s theorem applies to the first situation and says nothing about the second, and the word doing that work is discrete.

So the theorem is a statement about crystallographic groups specifically, and it is worth knowing which hypothesis is load-bearing. It is not finiteness of the point group, which follows; it is not the dimension; it is that the translations are a lattice rather than merely an abelian subgroup.

What separates the seventeen, computed

Bieberbach’s theorem says isomorphic groups are affinely conjugate, hence of the same type. The other direction — that no two of the seventeen are isomorphic — needs invariants, and two suffice.

The operation census. How many operations a group has modulo translations, and how many of them are rotations of each order, mirrors and glides. Every one of those numbers is preserved by an isomorphism, because the translations are characteristic and the kinds are decided by determinants and fixed points. Measured across the seventeen it takes fourteen distinct values: eleven groups are pinned by it alone and the remaining six fall into three pairs.

The mirror-against-glide split is doing real work in that count, and it is easy to leave out. A coset of determinant −1 is a mirror if some operation in it fixes a line, which needs the component of its translation along its own axis to vanish, and adding a lattice vector can make that happen. Counting the cosets without asking would put p4m and p4g at four reflection cosets each and make them a fourth stuck pair — and unlike the other three they share an arithmetic class, so nothing below would rescue them. p4m has four mirror cosets; p4g has one and three glides.

The arithmetic class. The point group together with the lattice it acts on. An isomorphism carries the translations onto the translations and the action along with them, so the pair is an invariant. It separates the three remaining pairs — pm from cm, pmm from cmm, and p3m1 from p31m — each of which has the same operation census on a different lattice.

Together they distinguish all seventeen, which is the computed half of the statement that the classification is a classification of groups.

Fourteen censuses, seventeen groups. The two invariants that between them tell the seventeen apart, with the rows the first cannot separate marked. The census counts a group's operations modulo translations, sorted by kind: rotations by their order, and the reflection cosets split into mirrors and glides according to whether any member of the coset fixes a line. That split is what separates p4m from p4g, which share an arithmetic class and would otherwise be a fourth stuck pair. The census takes fourteen values across the seventeen, so three pairs collide — pm with cm, pmm with cmm, p3m1 with p31m — and each of the three is a pair of groups with the same operations on different lattices. The arithmetic class settles all three, and the figure refuses to draw unless the two together take seventeen distinct values. An invariant that agrees proves nothing; only one that differs proves anything, which is why the pair is needed and why the classification's other direction has to come from Bieberbach's theorem rather than from any table.
Fig. 6 The two invariants side by side, with the rows the first cannot separate marked. The census takes fourteen values across the seventeen, so three pairs collide, and each of the three is a pair of groups with the same operations sitting on different lattices — which is exactly what the arithmetic class reads. The split between mirror cosets and glide cosets is what keeps p4m and p4g apart; they share an arithmetic class and would otherwise be a fourth stuck pair.

Why this is harder than it looks

The natural objection is that the theorem must be obvious: surely a group determines its own action.

It does not, and general group theory is full of counterexamples. The same abstract group can act on wildly different spaces, and two subgroups of a Lie group can be isomorphic and not conjugate. The free group on two generators embeds in the rotations of space in continuum-many inequivalent ways. Even for lattices, ℤ² sits inside the plane as a lattice and inside the line as a dense set, as above.

What makes crystallographic groups special is the combination: an abelian normal subgroup that is both maximal and of finite index, forced by discreteness plus cocompactness. That combination pins the group’s action to the group, and it is why Bieberbach’s three theorems are stated together — the first supplies the lattice, the second uses it to rigidify the map, and the third bounds the number of groups in each dimension.

p4g in 4 letters and 8 relations. The presentation of p4g, derived from the group's own operations. The two translations commute; each conjugation relation is read off a column of a matrix; and the point group's relations are corrected by the translation they actually come back as, which is what makes this group an extension rather than a semidirect product. Every relator is evaluated where the group lives and must be the identity, and coset enumeration on the letters alone returns 8, which is the order of the point group.
Fig. 7 A plane group with the plane taken away: four letters and the words in them required to equal nothing. Two such presentations can look entirely different and describe the same group, and deciding whether they do is the word problem’s harder cousin. Bieberbach’s theorem says that for these groups, once the question is settled, the geometry follows.

The third theorem, and why the counts are finite

Bieberbach’s third result is the one this field is named for. In each dimension there are finitely many crystallographic groups up to affine conjugacy. Seventeen in the plane, two hundred and nineteen in space up to affine equivalence, 4,783 in four dimensions.

That is the answer to Hilbert’s eighteenth problem in the form he asked it, and its proof rests on the same lattice: the point group is a finite group of integer matrices, there are finitely many of those up to conjugacy, and each admits finitely many extensions.

So the three theorems are one argument used three ways, and everything in this collection that ends in a number depends on the first of them.

18 extension classes, 17 groups. Each of the thirteen arithmetic classes with the number of ways translations may be attached to it — its cohomology — the shape of that group, and how many distinct plane groups the classes come to once the changes of basis that are mere relabellings are quotiented out. The two columns differ in exactly one row, 2mmp, where four extension classes are three groups because two of them are the same group with the axes swapped. No lattice is drawn anywhere in this computation.
Fig. 8 The extension count, which is the third theorem’s machinery in the plane: thirteen arithmetic classes, each admitting finitely many ways of attaching translations, adding to seventeen. Finiteness at each step is what makes the total finite, and the first step — that there are thirteen classes — is the one Bieberbach’s argument supplies.

What is not being claimed

Three limits, and the second is the sharpest.

The theorem is not proved here. What is computed is the conjugator, in every case this collection can construct, together with the failures that occur when the hypotheses are dropped. A construction is not a proof, and the search that finds it is bounded.

Affine, not Euclidean. As with the choice of cell, the map realising an isomorphism is affine — it may stretch and shear. Two patterns with isomorphic groups need not be congruent, or even similar: a rectangular pmm and a differently proportioned pmm are related by a stretch and are the same group. The classification identifies things a photograph would not, and the word affine is where that identification lives.

And handedness is not preserved. An affine map may have negative determinant. In the plane this changes nothing, because every plane group is affinely equivalent to its own mirror image. In space it does not, and eleven pairs of space groups are related by an isomorphism whose affine map reverses handedness — which is exactly why two hundred and thirty and two hundred and nineteen are both correct.

pgg: the map comes back. pgg written on two bases related by an integer matrix of determinant one, and about two origins. The two descriptions share no coordinate; they are the same group. The matrix and the origin shift were then recovered from the two operation sets alone — which is what Bieberbach's theorem promises, carried out as a search over the integer matrices and the origins the lattice permits, and checked by applying what was found.
Fig. 9 pgg on a sheared basis, with the origin left alone. The shear is a legitimate change of description — the lattice goes to a lattice, and the operations stay integer matrices — and the group is unchanged. A stretch of one axis alone would not be: it takes the operations out of the integers, and the construction refuses it rather than producing a subtly wrong group.

The invariants, and what they are not

It is worth being clear about the direction each invariant runs.

An invariant that differs between two groups proves they are not isomorphic. An invariant that agrees proves nothing at all — the groups may still differ in some way the invariant cannot see. So a table of invariants is only ever evidence in one direction, and the seventeen being pairwise non-isomorphic rests on the combination of two invariants that between them differ for every pair.

The other direction is supplied by the classification itself, and this is where Bieberbach’s theorem earns its place in the argument. Two groups of the same type are affinely conjugate by construction, hence isomorphic; two groups of different types are separated by the invariants; and the theorem is what closes the loop by saying that isomorphic groups must be of the same type. Without it, the invariants would establish only that seventeen drawings are different.

A third invariant is available and is strictly weaker, which is worth having in view because it is the one a reader coming from abstract group theory reaches for first. The abelianisation throws away the order of the letters in every word of a presentation and leaves a small abelian group, computed from the relator matrix by an integer normal form with no lattice anywhere in it. It separates pm from pg — ℤ ⊕ ℤ₂ ⊕ ℤ₂ against ℤ ⊕ ℤ₂ — and it separates p3m1 from p31m, which is one of the three pairs the census cannot reach. But it returns only twelve different groups from the seventeen: p2, pmg, cmm and p4m all abelianise to ℤ₂ ⊕ ℤ₂ ⊕ ℤ₂, and pgg, p4 and p4g all to ℤ₂ ⊕ ℤ₄.

So the three invariants fail in different places and none of them is redundant. The census cannot see a lattice; the arithmetic class cannot see a translation part; the abelianisation cannot see how a point operation moves a translation, which is most of what a plane group is. Any two of them happen to suffice here, and that is a fact about seventeen small groups rather than a principle — in three dimensions the same three leave pairs standing that need a fourth.

And every one of them depends on being able to decide whether two products of a group’s generators are the same element, which for these groups has a procedure and in general does not: a normal form, computed by rewriting, is what makes an abstract comparison possible at all.

Who proved it, and what it was for

Ludwig Bieberbach published the two papers in 1911 and 1912. The problem was Hilbert’s eighteenth, asked in 1900, and its first part was whether the number of crystallographic groups is finite in every dimension — a question that had an affirmative answer in three dimensions from Fedorov and Schoenflies by enumeration, and no argument covering all dimensions at once.

Bieberbach’s proof replaced enumeration with structure. Rather than listing the groups, it established what any such group must contain, and finiteness followed. That is the shape of most good answers to “how many” in this subject, and it is why the essays here that end in a number so often turn on a structural fact rather than on a search.

The theorems were reproved several times in the following century — Frobenius in 1911 gave an independent argument, and Zassenhaus in 1948 turned the structure into the algorithm that produced the four-dimensional count. Nothing in the modern treatment is easier than Bieberbach’s, which is unusual for a result of that age.

One more place the same rigidity appears

The theorem has a shape that recurs, and naming it makes the result less surprising.

A rigidity theorem says that a weak kind of sameness forces a strong kind. Mostow’s rigidity says that a hyperbolic manifold’s fundamental group determines its geometry in dimension three and above. Bieberbach’s says that a crystallographic group’s abstract structure determines its action up to an affine map. In both cases the mechanism is a subgroup that cannot be moved — here the lattice, there the whole fundamental group’s action on a boundary.

The plane is the case where rigidity nearly fails, and it is worth knowing why it does not. Flat tori are not rigid at all: a torus has a moduli space, a whole two-dimensional family of shapes, and two tori of different shapes are not isometric. What Bieberbach’s theorem says is that they are nevertheless affinely equivalent, and that the classification quotients by exactly that freedom. The moduli are real and the classification is coarser than they are, which is a choice about what “the same” means and is worth making explicitly rather than by accident.

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.
Fig. 10 The shape space of plane lattices, where every point is a lattice and the five types are its interior, its edges and its corners. Two lattices at different points of this region are not congruent; their groups may still be isomorphic, and are, whenever the two lattices are of the same type. The moduli are what the affine equivalence in Bieberbach’s theorem throws away.

The problem this makes decidable

Bieberbach’s theorem is a statement about existence, and it has a consequence about computation that is worth naming because the corresponding question outside this subject has no answer at all.

Dehn’s third problem asks, of two finitely presented groups, whether they are isomorphic. In general it is undecidable — no algorithm settles it, by the same encoding of a Turing machine that makes the word problem undecidable for general presentations. That is Adian and Rabin’s result, and it is one of the standard demonstrations that group theory in general is not a computable subject.

For crystallographic groups it is decidable, and the theorem is why. An isomorphism must carry translations to translations, so its linear part is an integer matrix of determinant ±1 conjugating one point group to the other, and its translation part is determined up to an origin shift. Both live in finite sets, so the search is finite: enumerate the candidate matrices, test each, and either produce a conjugator or report that none exists.

That is the algorithm the recovery in this essay runs, and it is what a program does when handed a list of operations and asked to name the group. The naming is not a lookup against a table of symbols; it is a decision procedure, and it terminates because the theorem bounds what an isomorphism can be.

So two of Dehn’s three problems have answers for the plane groups, by two unrelated arguments: the word problem by a normal form, and the isomorphism problem by Bieberbach. The conjugacy problem is the third, and for the same structural reason it is decidable too — by a single test of whether one vector lies in a lattice.

What the theorem does not say about a picture

The title’s claim runs one way, and it is worth stating the direction it does not run, because a reader can reasonably read it both ways.

Two patterns with isomorphic groups are the same type of pattern. That is the theorem: the groups are affinely conjugate, so the classification is a classification of groups rather than of drawings, and there is no eighteenth plane group hiding as an unrecognised presentation of one of the seventeen.

Two patterns with isomorphic groups need not look remotely alike. A wall of hexagons and a field of asymmetric commas can both have p6, and nothing about one is recoverable from the other. The group is a statement about which motions preserve the pattern and about nothing else — the motif is free, the cell’s proportions are free within the lattice type, and the drawing is free entirely.

That is the distinction between a group and a pattern, and the whole of this collection’s practice depends on keeping it. A figure here is a pattern generated from a group and a motif; the group is what is claimed and checked, and the motif is chosen to make the claim visible. Change the motif and every figure changes and no assertion does.

Where the ladder goes next

Downwards, into what an isomorphism is allowed to move. The same pattern, described twice is the question of which coordinate changes leave a group alone rather than carry it somewhere — the normaliser rather than the conjugator — and the two are the same arithmetic asked from opposite ends.

Sideways, into the failure case. Discrete, or dense, and nothing between is the theorem’s hypothesis examined on its own: a classification of the subgroups of the plane in which the lattices are one case out of five, and the other four are what every count in this collection has quietly been avoiding.

Where this goes next

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. p3m1 and p31m, told apart without a picture. The two groups this site returns to most often: same point group, same lattice, same number of operations, and distinguished in every other essay here by where their mirrors sit relative to the lattice — which is a fact about the plane. Abelianised, they are ℤ2 and ℤ6, which are not isomorphic. That difference is a fact about the groups: no change of basis, no redrawing and no relabelling can carry one to the other, and the argument never mentions a mirror line. What is left when the order is forgotten Abelianising a group throws away the order of the letters in every word and leaves a small abelian group behind. It is computed by a Smith normal form, it never mentions the plane, and it separates p3m1 from p31m — which a picture can only illustrate. The origins of p2 that change nothing. One cell of p2 with its pattern, and every point marked to which the origin may be moved without a single operation of the group changing its translation part. There are 4 of them per cell, and the count does not change when the search grid is refined, so it is a fact about the group rather than about the grid. Two coordinate lists differing by one of these vectors describe the identical arrangement, which is why no structure's coordinates are ever unique. The same pattern, described twice Two coordinate lists for one structure can disagree in every number and describe exactly the same arrangement, because a group does not fix its own origin. The operations that may be applied to a description without changing what it describes are its normaliser, and they can be found by looking at pictures rather than at matrices.

What this makes readable

Essays that name this one as a prerequisite.

Named alongside this one

Essays reaching for the same objects. Nobody chose these; they are what the concept index makes visible.

What links here

Every essay whose body links to this one.

The objects this essay names

Each one links to every other essay that touches it.

Arithmetic crystal classChange of basisClassificationDiscretenessGroup extensionLattice translationNormal subgroup