The symmetry a gluing adds

A square patch cut out of a tessellation has whatever symmetry its outline allows: some reflections, perhaps a quarter turn, and that is all. It is a finite object with a finite symmetry group.

Glue its opposite edges and it acquires translations — the shifts by one cell, in both directions, which on the patch ran off the edge and on the glued sheet come back round.

That is more symmetry rather than less, which is the opposite of what closing a shape usually does, and it changes what a lettering has to satisfy.

What a lettering has to respect #

A lettering assigns a mountain or a valley to every crease. On a glued sheet the creases are the classes of the identification: a crease running off one edge and back on at the other is one crease and takes one letter.

So a lettering of the glued sheet is automatically invariant under the translations. It cannot fail to be, because the object it is a lettering of does not distinguish a crease from its translate.

A lettering of the patch is not. It has separate letters for the two pieces of a divided crease, they need not agree, and if they do not the lettering does not extend to the plane.

What that buys #

A lettering of the glued sheet is one answer for every patch at once.

Write it onto a patch of one cell, four cells, nine cells: the same rule, and the letters agree wherever two patches overlap, because they came from a statement about the pattern rather than from a search on a square.

A lettering found on a patch is an answer for that patch, and nothing more. It may extend and it need not, and the search that found it was not asked.

What a periodic lettering is, exactly #

The phrase does a lot of work above and deserves a definition.

A lettering of the plane is periodic with respect to a lattice when translating it by any lattice vector gives back the same lettering: every crease and its translate have the same letter.

A lettering of a glued cell is such a thing, expressed compactly. The cell’s creases are the classes of the pattern’s creases under the lattice, so assigning a letter to each class assigns one to every crease of the plane, consistently.

That is why the two are the same object described two ways, and why searching one is searching the other.

The alternative — a lettering of the plane that is not periodic — is a perfectly good thing to want and is not what a glued cell can express. A plane pattern may have letterings with a larger period, or with no period at all, and none of those is visible to a search on one cell.

So the gluing constrains in two ways at once. It requires invariance, which is what makes an answer general, and it only sees invariant answers, which is a genuine restriction on what can be found.

Larger periods, which are also available #

The restriction has an escape and it is worth knowing.

A lettering with period two cells rather than one is not found by searching a one-cell sheet, and it is found by searching a two-cell sheet: gluing a rectangle two periods across identifies creases two cells apart rather than one, so a lettering of that sheet has period two.

So the family of searches — one cell, four cells, nine cells, sixteen — is a family of questions about letterings of increasing period, and each finds everything the smaller ones do plus more.

That is the reason the collection searches cells at several sizes rather than at the smallest, and it is why the costs grow so fast: a four-period cell has sixteen times the panels and asks for letterings of period four, which is a much larger space.

It also means a negative answer at one size says nothing about the next. A pattern with no lettering of period one may perfectly well have one of period two, and the grid at odd sizes is refused for a different reason entirely that does not go away by taking a bigger cell — the parity of an odd multiple is odd.

The two kinds of symmetry, and what each is for #

It is worth separating them properly, since the essay’s whole point is that they behave differently.

Translations are the symmetries the gluing supplies. Every lettering of the glued sheet respects them, necessarily, because the object identifies a crease with its translate. They are what makes a lettering general.

Point symmetries — reflections, rotations — are symmetries of the drawing that a lettering may or may not respect. Some patterns’ letterings cannot respect them, because the symmetry swaps two creases the vertex conditions force to differ.

The difference is in how each acts on the creases. A translation acts freely: no crease is fixed, and a letter can be assigned consistently by choosing one per orbit. A reflection has fixed creases — the ones lying on its axis — and a fixed crease has to be its own image, which is only possible if its letter is its own opposite, which no letter is.

So the split is not arbitrary. A symmetry with no fixed creases can always be respected; one with fixed creases sometimes cannot.

Where this leaves the searches #

The consequence for how this collection’s results should be read is short and it applies to a good deal of what has been published.

A lettering found on a patch is a fact about that patch. It may not extend, it says nothing about the pattern, and the search was not asked whether it does.

A lettering found on a glued cell of $n$ periods is a fact about the pattern: a rule with period $n$, valid on every patch of every size, checkable by writing it back.

The collection has both, and the second has been available only recently. Where an essay says the pattern admits this lettering, the question is which of the two was searched, and the answer for anything before the gluing existed is the first.

That is not a defect in the earlier work, which said what it measured. It is a change in what can be asked.

The cost #

The invariance is a constraint, and constraints cost.

A patch of a two-period square twist cell has forty free letters. The glued sheet has thirty-two. The eight that go are exactly the ones whose two halves could have disagreed, and requiring them to agree is requiring the lettering to be invariant under the translation.

So a periodic lettering is harder to find than a patch lettering, and it says much more when found. That is the ordinary trade between a constrained search and a general answer.

Why more symmetry, and not less #

The direction is worth pausing on because it is counter-intuitive.

Closing a shape usually loses symmetry: a square has eight symmetries and a square with one corner cut off has one. Gluing a patch’s edges gains them, and the reason is that gluing is not a change to the shape’s outline. It is a change to which points are the same point, and identifying points can only make more maps of the object to itself, never fewer.

More precisely: the patch’s symmetries are the ones preserving its outline, and the glued sheet has no outline, so nothing is preserved and everything that acts on the pattern acts on it.

What the letters still cannot keep #

There is a companion result and it goes the other way, so the two should be set beside each other.

A pattern’s symmetry is not always a symmetry of its letterings: a drawing invariant under a reflection can have no lettering invariant under it, because the reflection swaps two creases whose letters are forced to differ.

That remains true on a glued sheet and it is now about a larger group. The translations are symmetries of the drawing and the letterings are invariant under them, by construction. The reflections and rotations are symmetries of the drawing and the letterings need not be, exactly as before.

So the group splits in two. The translations are respected necessarily; everything else is respected or not, as before.

The practical version #

For anybody searching a tessellation for a lettering, the choice is between two questions.

Search a patch. Cheap, many answers, and none of them says anything about the pattern. That is what this collection did for a long time.

Search the glued sheet. Dearer, fewer answers, and each is a rule for the whole plane that can be written onto any patch of any size.

The second is what the gluing is for, and the extra symmetry is the reason it works: the object’s own translations do the constraining, and a solver working on it cannot produce a non-extending answer even by accident.

The numbers behind the trade #

Since the essay claims a constrained search is dearer and says more, both halves deserve figures.

Dearer. A two-period square twist cell has forty free letters cut out and thirty-two glued, and the searches cost thirteen nodes and nine. At three periods, eighty-four letters and seventy-two, and thirty nodes against six hundred and twenty-five. The constrained search is not merely dearer; past a threshold it is dearer by orders of magnitude.

Says more. A patch lettering is one answer for one square. A periodic lettering, checked, holds on nine patches at three sizes on four tilings, up to fifteen hundred creases — and would hold at any size, since it is a rule rather than an answer.

So the trade is steep in one direction and unbounded in the other, which is a reasonable description of the difference between an instance and a theorem.

It also explains why the collection did patches first. A patch search is affordable at sizes a glued search is not, and for a long time the choice was between a cheap answer about a square and no answer at all.

Free actions and fixed creases #

The mechanism separating the two kinds of symmetry deserves one more paragraph, because it is general and it is not specific to paper.

A group acts freely on a set when no element other than the identity fixes anything. A lattice of translations acts freely on the creases of a tessellation: no translation leaves any crease where it was.

A group acting freely admits an invariant assignment of anything, always: pick one crease per orbit, choose its value, and copy the choice round the orbit. Nothing can conflict, because nothing is asked twice.

A group with fixed points does not. A reflection fixes the creases on its axis, and an invariant assignment has to give such a crease a value equal to its own image — which for mountain and valley means equal to its own opposite, and there is no such letter.

So free action is the condition under which a symmetry can always be respected, and it is exactly the condition the translations satisfy and the reflections do not. That is the whole of why the group splits, and it would split the same way for any two-valued quantity on any object.

What a designer would notice #

The essay is filed under design and its consequences for a designer are indirect, so they are worth stating.

A designer working with a tessellation wants a crease assignment that folds, and normally gets one by taking a patch, assigning letters, and checking. That answer is about the patch, and enlarging the patch may require a different assignment.

A periodic assignment does not have that problem. It is a rule — this kind of crease is a mountain, that kind is a valley — and it works at any size, which is what a designer wanting to make a large panel of the pattern actually needs.

So the practical form of the result is: assignments for tessellations should be found as rules rather than as answers, and the way to find a rule is to search a glued cell rather than a patch.

That is available now and it was not, and it is the sort of change that alters a workflow rather than a theorem.

And what a folder would notice #

Nothing, which is worth saying.

A folder given a patch with a lettering folds it. Whether the lettering extends to a larger patch is not a question that arises while folding, and a patch lettering that does not extend folds exactly as well as one that does.

The distinction matters when the same pattern has to be made at several sizes, or when a claim is being made about the pattern rather than about a sheet — which is a matter for whoever writes the claim rather than for whoever folds it.

That is a fair description of most of what this phase has found: the corrections are to what can be said, and the paper behaves the way it always did.

Where the collection’s letterings come from #

It is worth naming which of the collection’s results are patch letterings and which are periodic ones.

The lettering that folds nowhere and the lettering nobody could draw are about patches: answers for a square, found by searching one.

The lettering that was proved impossible is periodic — found on a glued cell, written back onto patches, and checked at three sizes on four tilings.

The difference is exactly the one this essay is about, and it is why the second is a statement about the pattern and the first two are statements about squares.

What symmetry is doing, in one sentence #

The gluing’s translations are what turn a search into a proof.

A search on a patch explores assignments of letters to the patch’s creases, and nothing in the search knows that some of those creases are the same crease of the pattern. A search on the glued sheet explores assignments to the pattern’s creases, and the identification does the work of saying which are which.

So the extra symmetry is not decoration on the object. It is the mechanism by which an answer becomes general, and it arrives by making the object smaller rather than by asking the search to be cleverer.

The check that makes it a rule #

A claim that a lettering is a rule for the plane has to be checked, and the check is worth describing because it is done with machinery that knows nothing about gluing.

Take a lettering found on a glued cell. Expand it onto ordinary patches — one cell, four cells, nine cells — by giving every crease the letter its class has. Then hand each patch to the collection’s ordinary checkers: the ones that read vertices and look for forced loops, which have never heard of an identification.

Every vertex condition passes, no loop is forced, on four tilings, up to one thousand five hundred and twelve creases and seven hundred and twenty vertices.

That is what turns this lettering satisfies the glued sheet’s conditions into this lettering works. The two are not obviously the same statement, since the glued sheet’s conditions are computed on a quotient and the patches’ are computed on the drawings, and having them agree at every size is the evidence.

It is also how an error in the glued sheet’s own consistency test was caught — the letterings that test rejected turned out to pass every check on ordinary patches, which is a contradiction that had to be resolved somewhere.

An analogy, and its limit #

The situation is familiar from crystallography and the analogy carries some way.

A crystal is described by a unit cell plus a lattice, and any property of the crystal is a property of the cell that respects the lattice. Computing on one cell with periodic conditions is the standard method, and computing on a finite cluster instead gives answers contaminated by the cluster’s surface.

That is exactly the situation here: a patch is a cluster, a glued cell is a cell with periodic conditions, and the surface contamination is the free letters the rim supplies.

Where the analogy stops is that a crystal’s cell computation is a continuous problem — energies, forces, band structures — and this one is combinatorial. A continuous periodic problem has a clean spectral theory and a combinatorial one has a search, so the methods share the framing and nothing else.

The framing is worth having anyway. It says what a patch measurement is: a cluster calculation, useful, and systematically biased by its surface in a direction that can be estimated.

Two symmetries, one of them new #

Stated once, since the essay is a distinction.

A patch has the symmetries of a square: reflections and rotations, at most eight of them, and its letterings may or may not respect any of them.

A glued sheet has those plus a lattice of translations, and its letterings respect the translations necessarily.

The second is not more symmetric in a vague sense. It is more symmetric in the sense that a specific infinite group acts on it, and the group is what turns an answer about a square into a rule for the plane.