The two that fold into a surface

Conway’s notation is built on a physical operation. Take a wallpaper pattern, and fold it up along its own symmetries: bring every point onto the one representative of its orbit, and what is left is a small object carrying the whole group in its shape. A rotation centre becomes a cone point — a spike where the surface closes up around a corner — and a mirror line becomes a boundary, an edge the surface simply stops at.

Those marks are the information. Conway’s symbols are a list of them, and the accounting that prices them is the classification.

Which raises a question the notation never asks. Is there a pattern whose fold has no marks at all?

The answer is yes, twice, and the two answers are the only two compact flat surfaces that exist.

What puts a mark on the fold #

A mark appears where a point comes back to itself. If some operation of the group carries a point exactly onto itself, then that point has a shorter orbit than its neighbours — the group cannot separate it from its own images — and the fold has to close up around it. Every mark on a folded pattern is the shadow of a fixed point.

So the question is entirely about operations, and it has four cases.

• A rotation fixes its centre. One point, held exactly still.
• A mirror fixes a whole line of them.
• A translation fixes nothing: every point moves by the same vector, and no vector but zero is zero.
• A glide fixes nothing either, and it is the interesting case. A glide reflects across a line and then slides along it. A point on the line is reflected onto itself and then moved; a point off the line lands on the far side. Nothing survives.

The fourth case is the one that makes the question interesting rather than trivial. A glide is not a mirror with a translation attached in the sense of being either of them — it is the odd fourth motion, a single operation that no choice of origin turns into a reflection, and its refusal to hold anything still is what makes the two surfaces below possible at all. A classification with three kinds of plane motion would have no free groups in it beyond the trivial one.

p1 has nothing but translations. pg has translations and glides. Everything else in the list has a rotation or a mirror somewhere in it, and the fold acquires a mark.

That is the whole argument, and it is short enough to be suspicious. It is worth checking the one case where it could go wrong.

A fixed point is a short orbit #

There is a second way to say the same thing, and it is the one that connects the fold to arithmetic the collection already uses everywhere.

The orbit of a point under a plane group is the set of places the group sends it. For almost every point that orbit has one member per operation of the group: p4m has eight operations per cell, so a general point has eight images in each cell and its orbit is as long as the group is large. The exceptions are the points some operation leaves alone — the special positions — whose orbits are shorter, by exactly the order of the subgroup fixing them.

So “no operation fixes a point” and “every orbit is as long as the group” are the same statement. A group acting freely is one whose orbits are all full length, and the fold of such a group is a surface because there is nowhere the orbit collapses. Its area is the cell’s divided by the number of operations, exactly and with no correction: a whole cell for p1, half a cell for pg.

That is worth setting beside the asymmetric unit in space, where the corresponding statement carries a caveat — the unit is always a little larger than the cell divided by the group’s order, and the excess is the special positions counted whole. The excess is zero exactly for the groups in this essay. There are two of them out of seventeen, which is a fair measure of how special the free case is.

pgg is not a third #

Four of the seventeen carry a glide: pg, pgg, p4g and cm. Only one of them acts freely, and the other three fail for three different reasons — which is why the glide by itself is not the condition.

A word about how those four are counted, because the table below disagrees with the sentence above and both are right. The operations are listed modulo the lattice, so each group appears with one representative per coset of its translations rather than with every operation it has. cm’s glide is its mirror composed with the centring translation, so modulo the lattice it is the mirror and no separate glide appears; p4g’s mirror and its glides are genuinely different cosets and both appear. A glide that vanishes on reduction is still a glide of the group, and it is still not what decides the question.

pgg has glides in two directions, and composing two perpendicular glides gives a half turn. The group’s own closure produces a rotation nobody supplied — which is what a closure does throughout this subject — and a half turn fixes its centre. So pgg’s fold has cone points on it and is not a surface, in spite of having no mirror anywhere.

The rotation in pgg’s row is the one to look at, because nobody put it there. A group is closed under composition, so writing down two perpendicular glides is writing down their product whether or not anybody names it, and the product of two perpendicular glides is a half turn about the point where their axes cross. That is what a closure does throughout this subject: the operations supplied are generators, the group is what they generate, and a fixed point can arrive from a product of operations none of which has one.

cm has a mirror, plainly, even though it also has glides. p4g has four-fold centres. Each fails on something present rather than something derived, where pgg fails on something the closure supplied. Only pg gets through, and it gets through because a strip of glides in one direction produces nothing but translations when it closes: compose two parallel glides and the reflections cancel, leaving a translation by the sum of the two slides, and a translation holds nothing still.

That is the whole of why one direction of glide is safe and two are not. A second direction gives a product whose linear part is a rotation rather than a translation, and a rotation of the plane always has a centre. So pg is not merely the group that happens to escape; it is the only shape of glide group that can.

The count is complete because the list is. There is no seventeenth plane group waiting to be found, so there is no third free one either. That is the shape of nearly every count on this site — the answer is a corollary of a classification that closed — and it is worth naming here because the corresponding statement about surfaces looks as though it ought to need a surface-theoretic proof of its own. It does not. Two of seventeen, decided by looking at operations.

In Conway’s accounting that is the same statement about money. A boundary costs a whole unit, a cross-cap costs a whole unit, a cone point of order n costs (n − 1)/n and a corner of order n costs half of that; every wallpaper group’s features add to exactly two. A fold with no marks has bought nothing but handles, and a handle costs two — so a group acting freely is one that spends its entire budget on a single handle and has nothing left for a cone point or a boundary. There are two ways to spend two that way, and they are the two below.

The torus, which is the easy one #

For p1 the fold is the cell with its edges joined. A point walking off the right edge reappears on the left, one lattice vector back; a point walking off the top reappears at the bottom. That is a torus, and the construction is the one every video game with a wrapping screen uses.

Nothing is held still, so the surface has no marks. And the surface is flat in a strong sense: it is made from a piece of the plane by gluing, so every small patch of it is a patch of ordinary plane, with angles adding up as they always did. A torus made this way has no curvature anywhere, which is not what the doughnut sitting on a table looks like — the doughnut is a picture of the same gluing that had to be bent to fit into space, and the bending is a property of the picture rather than of the surface.

The Klein bottle, which is the good one #

For pg the vertical edges are joined differently. The glide slides half a cell and reflects, so a point leaving the right edge comes back through the left edge upside down.

The result is the Klein bottle: a surface with only one side, on which a shape can be carried around a loop and come back as its own mirror image. It is flat in exactly the sense the torus is — every patch is a patch of plane — and it cannot be built in three-dimensional space without passing through itself, which is again a fact about the picture rather than about the surface.

Handedness is what pg is about, and this is the clearest statement of it the collection has. The reason the motif on every plate here is a comma rather than a dot is that a comma has a hand and a dot does not; the reason pg is a different group from p1 is that its glide reverses that hand. Folded up, the two facts become one fact: the surface is one-sided.

Two numbers that say the same thing #

The site has computed something about pg before, in a context with no surfaces in it at all. Abelianising a plane group throws away the order of the letters in every word and leaves a small abelian group behind, computed by a Smith normal form and mentioning no geometry whatever.

For a group acting freely, that abelianisation is the first homology of the surface it folds into — the count of independent loops, with the loops that come back reversed showing up as factors of finite order. p1 gives ℤ², which is the torus: two independent loops, neither of finite order. pg gives ℤ ⊕ ℤ₂, which is the Klein bottle: one ordinary loop, and one that cannot be undone by going round it twice.

The qualification matters and is not decoration. cm’s abelianisation is also ℤ ⊕ ℤ₂ and cm’s fold is not a Klein bottle — it has a mirror boundary on it. An abelianisation is a fact about the group; it is a fact about the surface only when there is a surface, which is exactly when the action is free. This is the collection’s standing caution about invariants met in a new place: a number that agrees in two cases is not a proof that the two cases are alike.

Where the two surfaces were named #

The two shapes have separate histories and neither of them starts in crystallography.

The torus is old and needs no naming. Klein’s bottle is from 1882 and was introduced as a surface, not as a wallpaper: an example of a closed surface with no inside — a thing that a curve can traverse and return from mirrored. That the plane group pg produces it was not the point of either construction, and the connection between “the seventeen” and “the flat surfaces” was made from the other end.

That connection is a theorem of Bieberbach’s, from 1911, and the pieces of it are already in this collection under other names. His first result is that every crystallographic group contains a lattice of translations of finite index — which is the argument averaging over a finite group makes here. His third is that in each dimension there are finitely many such groups, which is the reason there is a list at all. What is added by the present question is his consequence for the torsion-free case: those groups are exactly the fundamental groups of compact flat manifolds, and classifying them classifies the shapes.

In two dimensions the classification is small enough to check by eye and the answer is the two above. The next dimension is where the arithmetic starts doing work nobody can do by inspection.

Why exactly two, from the surfaces’ side #

The count of two came out of asking every operation whether it holds a point still. There is a completely independent route to the same number, it uses no plane groups at all, and the agreement is worth having.

Compact surfaces without boundary are classified, and the classification is short: a sphere, a connected sum of some number of tori, or one of those with cross-caps attached. Each has an Euler characteristic, and the characteristic decides which geometry the surface can carry — positive for spherical, zero for flat, negative for hyperbolic.

So the flat compact surfaces are exactly those of characteristic zero, and there are precisely two of them: the torus, which is orientable, and the Klein bottle, which is not. The sphere has characteristic two and the projective plane one, both positive, so neither can be flat. Everything else — two handles or more, three cross-caps or more — has negative characteristic and is hyperbolic.

That is the same trichotomy the orbifold accounting runs on, arriving from topology rather than from a price list, and the two answers must agree because they are answers to the same question: a flat compact surface is the plane folded by a group acting freely, and a group acting freely is one whose quotient has no marks.

Two by two arguments sharing nothing. One counts fixed points among seventeen groups’ operations; the other classifies surfaces and reads off which have characteristic zero. Neither knows about the other, and both say two.

The picture of a Klein bottle is not the object #

One caution belongs with the second surface, because the familiar image of it is misleading in a way that matters for what has been claimed here.

A Klein bottle cannot sit in three-dimensional space without passing through itself. The usual drawing — a bottle whose neck re-enters its own side — is an immersion: a map that is locally faithful and globally not, with a circle of self-intersection that is not part of the surface.

The object folded out of pg has no such crossing. It is a surface with a flat metric, every point of it locally indistinguishable from a piece of plane, and the self-intersection is an artefact of trying to draw it in a space one dimension too small. The fold is intrinsic — it is the plane with points identified — and nothing in that construction produces a crossing.

That is worth saying because the flatness is the whole point. A surface that genuinely crossed itself would have a place where the local picture is not a piece of plane, and the claim that pg folds into a flat surface would fail there. It does not fail, and the drawing is what is wrong, in the same way that a map of the Earth is distorted without the Earth being.

The same answer on a strip #

The seven friezes are the same classification with one direction of repeat instead of two, and the same test gives the same shape of answer: two free groups out of seven, one orientable and one not. Their folds are the cylinder and the Möbius band — which are the two flat surfaces that are complete in one direction and infinite in the other, and which stand to the torus and Klein bottle exactly as a frieze stands to a wallpaper.

The Möbius band is the strip’s own version of the handedness argument. Its glide reverses a mark just as pg’s does, and walking a comma once round the band brings it back reversed — which is the sentence usually offered about a paper Möbius band with a pencil line on it, here derived from a group rather than demonstrated with scissors.

Four flat surfaces, then, from the plane and the strip: cylinder, Möbius band, torus, Klein bottle. With the plane itself that is five, and there are no others. The classification of flat two-dimensional shapes is a corollary of the classification of plane groups, obtained by asking which of them hold nothing still.

What the fold does not decide #

The same statement runs the other way and is worth having in that form, because it is the one a crystallographer meets. A group’s special positions are exactly the marks on its fold. p4m has a two-dimensional cell full of them — the mirror lines, the four-fold centres where two mirrors cross, the two-fold centres between them — and every one of those becomes a boundary or a corner when the plane is folded up. A group with no special position anywhere has nothing to become a mark, and folds into a surface. So the table of Wyckoff positions and the orbifold symbol are two spellings of one list, and a group whose table has only a general position is a group whose symbol has only a handle in it.

A surface is not a pattern. The fold throws away everything about the motif and keeps only the group, so two wallpapers with entirely different designs and the same group fold into the same surface. Nothing here is a statement about what a pattern looks like.

The fold keeps the group and not the lattice. p1 acts on an oblique lattice and on a square one, and both fold into a torus — a different torus each time, longer in one direction or shorter, but a torus. Which torus is the shape question the space of lattices answers, and it is a two-dimensional continuum of answers rather than one. So “p1 folds into a torus” is a statement about which surface, not about which of the tori, and the second question needs the whole region to answer.

And a flat surface is not a shape in space. Neither the torus nor the Klein bottle above is being embedded anywhere. The claim is that they are made by gluing a piece of the plane, so that every neighbourhood is a neighbourhood of the plane and angles behave as they do on paper. The doughnut on the table has curvature — positive on the outside, negative on the inside — and is a different object wearing the same name.

The last of those refusals is not needed in the plane and is needed at once in space, and it is the reason this test has to be stated about cosets rather than about operations. In two dimensions an operation and its lattice translates agree about whether they hold something still. In three they do not, and the same question asked of the two hundred and thirty has an answer that a careless version of this argument gets wrong by a factor of six.