Euler counts the gluing

A construction that says these two edges of the paper are the same edge has to be checked, and the obvious checks are weak.

Counting the panels afterwards and finding fewer than before proves nothing: any wrong identification also merges panels. Counting the letters and finding the count falls proves nothing either. Even checking that the counts add across the two pairs of edges — which they do, and which is a real result — passes a construction with a hole in it, as it turns out.

What catches it is a quantity that is not a count of anything the construction is trying to produce.

The number #

For any sheet built out of vertices, creases and panels, form

$$\chi = V - E + F.$$

On a disc of paper it comes to one. On a cylinder and on a torus it comes to nought. Those are facts about the shape of the sheet rather than about the drawing on it: any drawing at all on a disc gives one, and any drawing at all on a torus gives nought.

That is what makes the number useful. It does not depend on the pattern, the tiling, the number of cells, the crease angles or anything else the construction varies, so a computed value that does not match is a defect in the construction and cannot be anything else.

Why it comes out that way #

The arithmetic is short enough to follow, and following it explains why the corner matters later.

Start with the cut rectangle. Its interior contains $V$ vertices, its creases are cut into $E$ pieces by the rectangle’s edges, and its panels number $F$. For a disc, $V - E + F = 1$; that is the ordinary statement about a planar graph drawn in a disc, with the outer region not counted.

Now glue one pair of edges. Each crease the pair divided becomes one crease rather than two, so $E$ falls by the number of such creases. Each panel touching one of those edges merges with its partner, so $F$ falls by the number of such panels. And $V$ does not change, because the edges were placed to miss every vertex.

The two falls are not equal, and their difference is exactly one: the panels merge one more time than the creases do. On the plain grid’s cell the creases fall from twelve to ten and the panels from nine to six, which is two against three; on the Miura’s, four against five. So $\chi$ falls by one, from one to nought, and the sheet is a cylinder.

Glue the second pair and the falls are now equal, so $\chi$ does not move. The reason they are equal is the corner: the four pieces of paper at the corners of the rectangle are all one panel of the pattern, and the second gluing merges four into one rather than merging two pairs. That is one panel-merge fewer than the first gluing had, and losing it is exactly what stops $\chi$ going to minus one.

So the corner is not a detail of the bookkeeping. It is the single merge that distinguishes a torus from something that is not a surface at all, and if it fails to happen the number says so.

The corner, and the defect it hid #

The rectangle’s corner is not placed at a round number. It is searched for: the two coordinates are chosen independently so that each edge falls in the largest gap between the drawing’s own features in that direction, which keeps every edge clear of every vertex.

That search is doing the right thing and it cannot see one case. A crease can run exactly through a corner of the rectangle without coming anywhere near a vertex — the corner is the meeting of two edges, and a line can pass through it while staying in the middle of the gap in both coordinates separately.

A corner is where four edges of the cell meet. A crease piece ending there has no partner on any one of them: the matching rule looks for a piece arriving at the opposite edge at the same height, and at a corner there is no single opposite edge to look at.

The symptom was one unmatched crease piece per glued pair, which is small enough to look like rounding, and a torus whose Euler number came out at minus one.

Nothing else noticed. The panel counts were sensible. The letter counts were sensible and additive. The vertex count was unchanged, as it should be. The search ran and returned an answer. Only the topological invariant was wrong, and it was wrong by exactly the number of merges the corner had failed to make.

Reading the number the other way #

Everything above uses $\chi$ as a check on a construction whose answer is known. It can also be used the other way round, and doing so is a good way to see what it is measuring.

Suppose somebody hands over a drawing and an identification rule and declines to say what surface results. Counting $V - E + F$ gives one number, and counting how many circles of boundary edge the sheet retains gives another, and between them the surface is determined: characteristic one with one boundary circle is a disc; nought with two is a cylinder; nought with none is a torus or a Klein bottle, and which of those it is depends on whether any identification carries a flip.

That is the whole classification for the sheets this collection can build, and it is four integers. It is worth having in that form because it says exactly how much information the counts carry: enough to name the surface, and nothing whatever about the drawing on it.

Why a wrong count is worse than a crash #

A construction that produces the wrong surface does not fail loudly. That is the property that makes the check necessary rather than merely tidy.

The identification with a missing merge produced a sheet with a panel that should have been joined to another and was not. Everything downstream then ran perfectly: the vertex tables were built, the search explored them, a lettering was found, and a verdict was returned. The verdict was about an object that is not the object anybody intended, and there is nothing about it that looks wrong.

The general shape of that failure is familiar in this collection and has been recorded several times: a checker that passes a sheet with no vertices, a test imported without the hypothesis that says which sheets it is about, a lettering found on a patch that says nothing about the pattern. Each is a computation returning a well-formed answer to a question nobody asked.

The defence in every case is the same: hold a quantity whose value is fixed from outside, and compare.

What the search would have said #

It is worth being concrete about how far the broken construction got, because the answer is: all the way.

The Yoshimura’s one-period cell, glued into a torus with the corner defect, reported two vertices, seven creases and four panels. Those are perfectly plausible numbers — small, in the right proportion to each other, and consistent with the letter counts on both cylinders. The search over letterings ran on them and terminated.

$2 - 7 + 4 = -1$. There is no surface with that characteristic and no boundary.

After the repair the same cell reports two vertices, six creases and four panels, which gives nought. Six is also the number two vertices of degree six force: the degrees sum to twelve and every crease has two ends, so there are six creases, and that is an independent confirmation that the repaired count is the right one.

Two vertices, six creases, four panels: the smallest object this construction has ever produced, and the one it was getting wrong.

Why this is the right kind of check #

There is a general principle here about what makes a check worth having, and this is a clean instance of it.

A check is useful in proportion to how independent it is of what it is checking. Counting panels checks a construction that produces panels, using the construction’s own idea of what a panel is; the two share a failure mode. Euler’s number is computed from the same three counts, but the value it must have comes from somewhere else entirely — from the shape of the sheet, which the construction did not choose and cannot influence.

So a wrong value cannot be explained away. There is no reading of the drawing under which a torus has $\chi = -1$.

The same logic runs through the rest of this collection’s checks. A crease pattern is verified against four theorems before it is drawn; a folded state is checked by composing reflections round every loop; two routes to one panel are required to agree. Each is a quantity whose correct value is known from outside.

What the number cannot see #

Being independent of the drawing is what makes it a good check and also what limits it.

$\chi$ does not know whether the pattern folds. Two drawings on the same sheet have the same Euler number whether one of them is flat-foldable and the other is nonsense.

It does not distinguish the two cylinders. Gluing across and gluing along give the same value, and they are genuinely different sheets with different letter counts and very different search costs.

It does not distinguish a cylinder from a torus, which is the most surprising of the three: both come to nought, because the characteristic depends on the surface and a cylinder and a torus happen to share theirs. What separates them is the boundary — a cylinder has two circles of edge and a torus none — and the check that sees that is simply counting the boundary edges the cell retains.

So the number belongs to a family of checks rather than standing alone, and the family’s members answer different questions: whether the identification is structurally sound, whether it preserves the side the paper shows, and whether the drawing on it folds.

The other counts that were checked instead #

Before the topological check existed, three things were being checked about every gluing, and each of them passed on the broken cell. Listing them says what kinds of check are weak.

Two routes to one panel must agree. The construction propagates a lattice offset as it merges panels, and if two different chains of merges reach the same panel with different offsets, that is a contradiction and it is counted. On the broken cell there was no such contradiction, because the piece that failed to merge simply never got a second route.

Every crease piece on an edge must find a partner. This one did fire — one unmatched piece per glued pair — and it was reported as a small integer among a dozen other small integers, in a place where a nonzero value had no consequence attached to it. A diagnostic nobody has attached a meaning to is not a check.

The counts must add across the two pairs of edges. They did. Additivity is a real property and it is a property of the letters, which were being counted correctly; the merge that went missing was a panel merge, and the panel counts are the ones that do not add anyway.

So of the three, one was blind, one was a number with no threshold, and one was measuring the wrong column. What was needed was a quantity with a known value, and none of the three had one.

A note on what is being counted #

One point of care, since $V$, $E$ and $F$ have to mean the right things for the number to come out right.

$V$ counts the interior vertices of the cell — points where creases meet. It does not count the points where a crease crosses the cell’s edge, because on a glued sheet those are not vertices at all: two crease pieces meeting in line at an ordinary point of the paper.

$E$ counts the creases after identification, so a crease that ran off one edge and back on at the other is one crease. It does not count the cell’s own boundary edges, which are not creases and, once glued, are not boundary either.

$F$ counts the panels after identification, likewise.

Get any of those wrong and the number comes out wrong for a reason that has nothing to do with the construction — which is a real hazard, and the reason the counts are taken from the quotient rather than from the drawing.

What it costs #

Nothing worth mentioning, which is most of the argument for having it.

The three counts are already produced by the construction. The vertices come out of the vertex tables the search needs; the creases out of the identification; the panels out of the face walk that has to happen anyway. Adding them with alternating signs and comparing against a constant is arithmetic on three integers.

Against that, the defect it found had been present in every Yoshimura cell built, and would have propagated into every measurement made on one. The cost of not having the check is not the cost of a wrong number: it is the cost of a wrong number that looks right, in a table of numbers that are right, in an essay whose argument does not depend on it — which is the failure mode this collection has met before and expects to meet again.

Where the number came from #

The invariant is far older than anything in this subject and it arrived here without any of its usual apparatus, which is worth a sentence.

Euler wrote it down for convex polyhedra: vertices minus edges plus faces is two, for a cube, a tetrahedron, a dodecahedron and everything else of that kind. The generalisation — that the number depends only on the surface and not on how it is divided up — took another century and is one of the founding results of topology.

None of that machinery is used above. What is used is the value of the number for four particular surfaces, taken as known, and three counts the construction was producing anyway. That is the ordinary way an invariant earns its keep outside its home subject: not as a theory to be developed, but as a constant that a computation has to hit.

It is also why the check is honest about its limits. It says the surface is right and it says nothing else, because that is all the number was ever about.

The check as a habit #

There is a habit behind this that is worth naming, because it is the one that produced the repair rather than a lucky glance.

Every construction in this collection is asked to produce something whose value is known independently, and the two are compared. A generator that draws a crease pattern is asked whether the pattern satisfies four theorems it did not choose. A folded state is asked whether two routes to one panel agree. A twist tessellation is asked whether its distances are consistent, by a computation that shares no code with the one that placed them.

The gluing had three such comparisons and none of them was against a constant. Adding one that was — a number that has to be nought and no argument about the drawing can change that — took an afternoon and turned up a defect present in every cell of one family.

What makes the habit pay is that the checks are cheap and the failures are quiet. A construction that crashes is free to find. A construction that returns a well-formed wrong answer costs whatever is built on top of it, and the only defence is to have asked it something it could not have got right by accident.

The version that generalises #

For a sheet built by identifying edges of a polygon, the characteristic can be read off the identification rather than computed from the drawing, and having both is what makes the comparison a check rather than a definition.

A rectangle with nothing identified is a disc: $\chi = 1$. Identify one pair of opposite edges preserving direction and it is a cylinder: $\chi = 0$. Identify both pairs and it is a torus: $\chi = 0$. Identify one pair with a flip and it is a Möbius band: $\chi = 0$. Identify both with one flip and it is a Klein bottle: $\chi = 0$.

Four of those five come to nought, which is why the number is a weak invariant and a strong check. It rules out a great deal without distinguishing much, and ruling out is what a check is for.