Ten ways for space to be flat

Two of the seventeen plane groups hold no point still, and folding the plane along one of them gives a surface with nothing marked on it: a torus, or a Klein bottle. Both are flat, in the strong sense that every small piece of them is a piece of ordinary plane.

The same question in three dimensions has a longer answer and a far better argument, because the answer cannot be reached by inspection. Which of the two hundred and thirty space groups hold no point still? There are thirteen, they give ten shapes rather than thirteen, and getting there requires almost no searching at all.

What flat means here #

“Flat” is doing precise work in that sentence and is worth pinning down before the count.

The fold is by isometries — motions that preserve distances — so the result inherits its geometry from ordinary space. Every small ball inside one of the ten shapes is an ordinary ball of ordinary space; every triangle drawn small enough has angles summing to two right angles; parallel lines started close stay close. The shape is finite in extent and has no boundary and no edge, and it still has no curvature anywhere.

That is the property this collection has met twice before from the other side. Conway’s accounting prices a folded wallpaper pattern and requires the total to come to exactly two, and past two the list does not stop — spend more and the geometry is hyperbolic and the list is infinite. Spend exactly the amount that leaves nothing curved, and the answers are finite in number. The seventeen are the two-dimensional case of that with marks allowed; the two that fold into surfaces are the case with no marks; and the ten below are the same question one dimension up.

Which operations hold a point still #

The test is the one the plane used, restated for the operations space has and the plane does not.

An operation is a linear part M and a translation t, and it fixes a point when M x + t = x has a solution. Split t into the part that lies along whatever M leaves alone — the intrinsic translation, the average of t around one cycle of M — and the rest, which is a statement about where somebody put the origin. Then:

an operation moves every point exactly when its intrinsic translation is not zero.

A rotation has intrinsic translation zero and holds its axis still; a screw does not and holds nothing. A mirror holds its plane still; a glide does not. An inversion holds one point still and there is nothing to be done about it, because its linear part leaves no direction alone, so its intrinsic part is zero whatever t is.

The last row is the one that cannot be repaired and it is worth reading separately. An inversion’s linear part leaves no direction alone, so the averaging projection is the zero map and the intrinsic translation vanishes whatever translation is attached — and the figure checks that by trying all 1,728 translations on a grid of twelfths and finding not one that escapes. That is the difference between an operation that happens to hold a point still and one that cannot be made to let go, and it is why the first cut of the argument below throws away every centrosymmetric class before looking at a single translation.

That last observation does most of the work before any searching begins.

Sixteen of the thirty-two, for nothing #

If a class contains an operation whose linear part fixes only the origin, then that operation has intrinsic translation zero for every possible t, so it always holds a point still, so no choice of translations can save the class.

Which operations are those? An inversion, a 4̅, a 3̅, a 6̅ — every operation whose matrix has no eigenvalue of one, which is a determinant to evaluate and nothing more.

Sixteen classes go, and no translation has been chosen. It is worth noticing which ones: every centrosymmetric class, which is more than a third of the thirty-two, and every class with a rotoinversion in it. A crystal that folds into a flat shape has no centre of symmetry, and that is the first thing the argument says.

Eight more, for one line of group theory #

Sixteen classes remain and eight of them are also impossible. The search that shows it never runs.

A subgroup of a group that acts freely acts freely. The reason is that a subgroup’s elements are elements, and an element that moves every point moves every point whether or not other elements are present. So if a class has been shown to have no free extension, no class containing it can have one either.

Five classes are searched and come back with nothing: 32, 3m, 422, 4mm and 23. Three more then need no search: 622 contains 32, 6mm contains 3m, and 432 contains 422. That is a third of the surviving candidates decided by one sentence.

The direction of the inheritance is the part to get right, because it runs the opposite way from the one intuition offers. A subgroup of a free group is free, so failure travels upward: a class that contains a failing class fails, and a class contained in a succeeding class succeeds. It does not follow that a class contained in a failing one fails — 2 sits inside 222 and inside 422, and 222 succeeds while 422 does not. So the argument prunes only in one direction, and every one of the eight survivors still has to be searched for itself.

The saving is nevertheless real and it is worth measuring rather than admiring. Sixteen of the thirty-two classes go on a determinant and three more on the containment sentence, so nineteen are decided before a single translation is chosen. Thirteen are searched — the five below that return nothing and the eight that yield the list — which is what makes an exhaustive walk over translation parts affordable at all.

The search that is left #

Eight classes are searched properly: 1, 2, m, 222, mm2, 3, 4 and 6, each on every centring its system’s lattice permits. For each, the generators’ translation parts run over a grid whose denominator is the class’s own exponent — which is where an intrinsic translation’s denominator has to lie — the group is closed, and the closure is tested.

Two things keep that search small. Each generator is tested on its own before any closure is attempted, since a generator with a fixed point cannot sit inside a group without one. And the closure is capped one element above the order the class must have, so an inconsistent choice of translations is abandoned after a handful of compositions rather than filling up a group of fifteen hundred.

Thirteen groups. P1, the group with nothing but translations. P2₁, P3₁, P3₂, P4₁, P4₃, P6₁, P6₅ — a single screw axis, at every order the restriction permits and at every pitch that leaves no rotation behind. P2₁2₁2₁, three mutually perpendicular screws. Pc and Cc, a glide plane on a primitive and on a centred cell. Pca2₁ and Pna2₁, a screw and two glides.

The absences are as informative as the list. P4₂ is not there and P4₁ is: a four-fold screw of pitch a half squares to a two-fold rotation, which holds a line still, while a screw of pitch a quarter squares to one of pitch a half in a doubled cell and holds nothing. P6₂, P6₃ and P6₄ are not there for the same reason. The rule is that the screw’s pitch and the axis’s order must share no factor.

The trap, which cost this derivation a factor of six #

There is a way of asking the question that is wrong, gives a plausible answer, and passes every check that does not know better. This site asked it that way first.

Composing an operation with a lattice translation gives another element of the same group, and its intrinsic translation is not the same one. The translation’s component along the operation’s own axis is added — and for a two-fold, that component is half a lattice vector, because averaging a vector with its own reflection halves it. So an operation written down as a screw can have, one cell over, a partner that is an honest rotation.

Asked with the weaker test — is each operation, as written, a screw? — the class 32 reports seventy-two groups acting freely. There is no flat three-dimensional shape with a symmetry group of that kind, and the answer is wrong by exactly the amount that a test of representatives differs from a test of cosets.

The right question is asked of the whole coset and answered in integers: with P the averaging projection onto whatever M leaves alone, some element of the coset holds a point still exactly when P t lies in the lattice P Λ. That is a membership test in an integer span, it is decided by a Hermite reduction with no tolerance anywhere in it, and it is what makes the count above thirteen rather than eighty-five.

What the eight surviving holonomies are #

The eight classes that yield anything are 1, 2, m, 222, mm2, 3, 4 and 6 — and read as abstract groups they are the trivial group, three copies of the group of order two, the group of order four with no element of order four, and the cyclic groups of orders three, four and six. Nothing non-abelian survives.

That is not an assumption anywhere in the search and it is the clearest single statement the derivation makes. The five classes that were searched and returned nothing — 32, 3m, 422, 4mm, 23 — are precisely the smallest non-abelian classes whose every element has an axis, and the three that were never searched at all inherit their failure. A flat three-dimensional shape has an abelian holonomy, and it comes out of an exhaustive walk over translation parts rather than out of a theorem quoted for the purpose.

P2₁2₁2₁ is worth a second look because it is the one an experimentalist meets constantly. It is among the commonest space groups in the structural literature, for a reason that is exactly this essay’s subject read backwards: a molecule with a hand cannot sit in a group containing a mirror or an inversion, and among the groups it can sit in, the ones without rotation axes leave the molecule free to pack without having to lie on anything. The group whose fold is a flat shape and the group a protein crystallises in are the same group, arrived at from opposite directions.

Thirteen groups, ten shapes #

P3₁ and P3₂ are the same group built with the screw turning the other way. As space groups they are different, because no rotation of space carries one onto the other; as shapes, they are the same shape seen in a mirror, and a shape and its mirror image are one shape.

This collection has counted that difference before. Two hundred and thirty or two hundred and nineteen is the same arithmetic on the whole classification: eleven pairs of space groups are mirror images that no motion relates, and whether they are counted once or twice depends on which question is being asked. Three of those eleven pairs are in this list, and they are exactly the three that make thirteen into ten.

Six of the ten are orientable and four are not. The four one-sided ones are the groups with a glide in them — Pc, Cc, Pca2₁ and Pna2₁ — which is the three-dimensional form of the observation that pg’s fold is a Klein bottle. A glide reverses handedness, so a shape folded along one has a path that brings a hand back as the other hand.

Two counts are being compared there and it is worth keeping them apart, because they disagree and both are printed. Nine of the thirteen groups are orientable and six of the ten shapes are. The gap is not an error: all three of the merging pairs are screws, screws reverse nothing, and merging two orientable groups gives one orientable shape. So the count of orientable groups falls by three when the pairs merge and the count of one-sided ones does not fall at all — four groups with a glide, four one-sided shapes, and no pair among them.

That the glide is the whole of the difference is a statement worth testing rather than repeating. An operation reverses handedness exactly when its linear part has determinant −1, and the only such operations available to a group acting freely are the glides, since a mirror holds its plane still and is excluded before the search begins. So a free group is orientable exactly when it contains no glide, and the four one-sided shapes are the four groups that do — which is the same sentence as the plane’s, where pg is the only free group with a glide and the Klein bottle is the only one-sided fold.

The theorem that makes the count possible #

The search above enumerates candidates and checks them, and behind it sits a result that says the enumeration is the right thing to do at all. It is worth naming, because without it neither the thirteen nor the ten would be a classification.

Bieberbach’s theorems, proved in 1911 and 1912 in answer to the eighteenth of Hilbert’s problems, say three things about a group of isometries acting on n-dimensional space with a compact quotient.

First, such a group contains a lattice of translations of finite index. That is the statement the whole subject rests on and the one it is easiest to assume: it says a crystallographic group has a lattice, rather than being defined to have one, and it is what makes the point group finite and the enumeration a search over finitely many things.

Second, two such groups that are isomorphic as abstract groups are conjugate by an affine transformation. This is the one that decides the arithmetic on this page. It says the classification of these folds is a classification of abstract groups — so two shapes are the same shape exactly when their groups are isomorphic — and it explains why the eleven enantiomorphic pairs behave the way they do here: P3₁ and P3₂ are isomorphic, so they give one shape, and the two are separated only when the conjugating map is required to preserve handedness.

Third, there are finitely many in each dimension. That is what licenses the phrase and no eleventh.

So the thirteen and the ten are two counts under two equivalences, and Bieberbach says exactly which. Thirteen is the count up to proper affine conjugacy, which is what a crystallographer means by a space group; ten is the count up to affine conjugacy outright, which is what a geometer means by a manifold. Neither number is more correct, and a claim that omits which equivalence is meant is not yet a claim.

Where the classification came from, and where it stops #

The ten were worked out by Werner Hantzsche and Hermann Wendt in 1935, a little over twenty years after Bieberbach’s theorems made the question well posed, and the paper is a search of much the same shape as the one above — the candidate holonomies, then the extensions each admits.

One of the ten carries their names. The Hantzsche–Wendt manifold is the fold along P2₁2₁2₁, and it is the odd one out of the list in a way worth recording: it is orientable, it has finite first homology, and it is the only one of the ten whose first Betti number is zero. Every other flat three-manifold has a circle’s worth of freedom in it somewhere; this one has none, which is the topological face of the fact that P2₁2₁2₁ has three mutually perpendicular screws and no direction left over.

The next dimension is a real classification and not a small one. The compact flat four-manifolds were enumerated later and there are seventy-four of them, twenty-seven orientable — quoted here rather than derived, since the search needs the four-dimensional space groups, of which there are 4,783. The pattern of growth is the interesting part: two, ten, seventy-four. Bieberbach guarantees each of those is finite and says nothing about how large, and no formula is known that produces the sequence.

What this does and does not say about crystals #

No crystal is one of these shapes. A crystal is a pattern in space; a flat manifold is what space itself becomes when it is folded along the pattern’s group. The groups above are perfectly ordinary space groups with perfectly ordinary crystals in them — P2₁2₁2₁ is among the commonest groups in the structural literature, because a molecule with a hand can sit in it — and nothing about those crystals is unusual.

What is unusual is the fold. A structure in P2₁2₁2₁ has an asymmetric unit that is exactly a quarter of the cell, with no correction anywhere for special positions, because there are none: every point of that cell has four images and not one of them coincides with itself. That is a statement a crystallographer uses daily and it is the same statement as the one about shapes.

Six of the ten are shapes a chiral crystal could make. The orientable six — P1, P2₁, P2₁2₁2₁, P3₁, P4₁ and P6₁ — contain no operation of negative determinant at all, so each of them is a group in which a single-handed molecule can crystallise. The four one-sided ones each contain a glide, and a glide reverses hands. So the orientability of the shape and the chirality of what could live in the crystal are the same fact, stated once about the fold and once about the contents.

The ten are also a list of candidate universes, and this is the one place where the subject leaves crystallography entirely. If space is finite and has no curvature, its shape is one of these ten, because the argument above used nothing but flatness and finiteness. Which of them — or whether space is finite at all — is a measurement rather than a theorem, made on the largest scale anybody measures anything, and nothing in this collection bears on it. What the classification supplies is the list the measurement would have to choose from, which is the same service it supplies to a crystallographer holding a diffraction pattern.

And the ten are all of them. There are ten compact flat three-dimensional shapes and no eleventh, for the same reason there are seventeen plane groups and no eighteenth: the classification closed. What this essay adds to that closure is which entries of the list of two hundred and thirty it applies to, found by a determinant, a sentence about subgroups, and a search over eight classes.

The next dimension is not done here and is worth naming as a limit rather than as an omission. There are 74 compact flat four-dimensional shapes and 1,060 in five dimensions, and the counts stop being reachable by the kind of argument above almost immediately: the classes to filter run into the thousands, and the subgroup argument prunes a smaller share of them each time. This collection stops where its own machinery reaches, which is three.

One last framing, because it says where this rung sits. Everything here is a statement about the two hundred and thirty read as abstract groups rather than as symmetries of anything, and the crystallographic restriction is doing its usual work underneath: the eight surviving holonomies have orders 1, 2, 3, 4 and 6 and no other, which is the same list, arrived at without ever asking about a crystal.