Which of the seven survive

There are exactly seven axioms, and the completeness of the list is one of the tidiest results in the subject: every way of specifying a single fold by aligning combinations of points and lines is one of the seven, and the enumeration is exhaustive over the combinations.

The enumeration is about points and lines on a sheet, and it says nothing about which sheet.

What a fold is, in the axioms #

Each axiom says: here are some marked things, and there is a fold bringing one onto another.

The first two are about points. A fold through two points; a fold bringing one point onto another. Both are perfectly well defined on any sheet where two points can be joined by a line.

The third brings a line onto a line. The fourth drops a perpendicular from a point to a line. The fifth and sixth bring points onto lines, one and two at a time. The seventh brings a point onto a line while making the crease perpendicular to another.

Five of the seven name a line, and naming a line is where the sheet enters.

What reflecting assumes #

A fold reflects one side of the crease onto the other. That sentence has a hypothesis in it: the crease has two sides.

On a disc of paper it does. Any line drawn across a disc separates it into two pieces, and folding takes one piece onto the other.

On a cylinder it need not. A line running once round the tube separates it into two pieces; a line running along the tube, from one open end to the other, does not — cut along it and the tube unrolls into a rectangle in one piece.

So on a closed sheet, fold along this line is not always a well-formed instruction. There is no one side to bring onto the other.

Which axioms are affected #

The split is by whether the axiom’s output is a fold along a specified line, and all seven are, so the question is really which of them can name a line that does not separate.

Axiom one — the fold through two points — can. Two points on a cylinder can be joined by a line running along it, which does not separate.

Axiom two — bringing a point onto a point — produces the perpendicular bisector, which on a cylinder may or may not separate depending on where the two points are.

Axioms three to seven all take a line as input, and if the given line does not separate, the reflection they ask for does not exist.

So the honest statement is that the seven are seven ways of specifying a fold when the fold’s line separates the sheet, which on a disc is always and elsewhere is not.

Which lines separate a cylinder #

The condition is easy to state for the sheets this collection can build, and having it settles which axioms are usable where.

On a disc, every line separates. Nothing to check.

On a cylinder, a line separates when it is a loop that can be shrunk, or a line running from one open end to the other and back — that is, when it does not go once round. A line running round the tube separates it into two shorter tubes; a line running along it does not, because cutting along the tube gives a rectangle in one piece.

Curiously, a line running along the tube from one end to the other does separate it — cutting there opens the tube into a rectangle, which is one piece, so it does not separate. Whereas a loop round the tube leaves two pieces.

So the rule for a cylinder is: a line separates when it is a closed loop going round, and does not when it runs the length of the tube. That is the opposite of the naive guess, and it is worth checking against a paper tube, which takes ten seconds and a pair of scissors.

On a torus, neither kind of loop separates. Cut along either and the torus opens into a cylinder, in one piece; cut along both and it opens into a rectangle. So on a torus, fold along this line is never a well-formed instruction for a non-shrinkable line, and the only usable lines are the ones that bound a disc-shaped region.

The paper test #

The claim about which lines separate is worth verifying, since the answer is counter-intuitive.

Take a paper tube — a strip taped into a loop. Cut round it, all the way, parallel to the ends: two tubes. It separated.

Take another and cut along its length, from one open end to the other: one rectangle. It did not separate.

That is the whole of it, and the second result is the one people expect to give two pieces. It gives one because the cut, having gone the length of the tube, arrives back where it started from the other side of the seam, and the paper on both sides of the cut is connected round the other way.

So the fold that would be specified by fold along the length of the tube has no side to fold onto, and the axiom asking for it is asking for nothing.

What can still be constructed #

The picture is not entirely negative, and it is worth saying what a cylinder does admit.

Folds along shrinkable lines are fine. Anything happening within a disc-shaped region of the cylinder is exactly as it is on a flat sheet, and all seven axioms apply there without qualification.

So a construction confined to a patch of the tube works. What fails is anything using the tube’s own global structure — a fold along its length, a reference built from going once round — and those are precisely the operations that would exploit the closure.

Which means a designer on a cylinder can use the whole apparatus locally and gains nothing from the sheet being closed. The closure costs the operations it forbids and buys no new ones, which is a fair summary of a closed sheet as a construction surface.

A related loss: which side is which #

There is a second thing an axiom needs and it is not the separation.

Bring this point onto that line has two answers in general, and choosing between them is done by which side of the crease things end up on. On a disc that is unambiguous.

On a sheet with one side — a Möbius band — it is not, for the same reason mountain and valley are not globally defined there. An instruction distinguishing two solutions by which side they land on has nothing to distinguish them by.

So the Möbius band loses even the axioms whose lines separate, because the disambiguation fails rather than the reflection.

That is a further step down and it is complete: on a sheet with one side, the axioms specify folds up to a choice that cannot be made, which is not a specification.

The sheet with no marks at all #

There is a second and more basic difficulty on a closed sheet and it comes before any axiom.

Every construction starts from something marked. On a square that is the four edges and the four corners: two edges give a corner, a corner is a point, and the whole apparatus of reference points is built from them.

A cylinder has two circles of edge and no points at all. A circle has no corners, no distinguished position, and nothing on it that an axiom could take as input.

So even the axioms that survive have nothing to be applied to, which is a difficulty of its own and is arguably the bigger one.

What the axioms are for #

It is worth remembering why the list matters, since the essay is about its scope.

The seven axioms are what makes folding a construction method: a way of producing exactly specified points and lines from a starting configuration, comparable with straightedge and compass. The sixth axiom is what lets a fold solve a cubic, and that is why paper trisects an angle and Euclid’s tools cannot.

So the list is the foundation of the constructibility results, and those results are about what numbers can be reached. Every one of them is a statement about a flat sheet with a corner to start from, and the scope correction above does not touch any of them: the numbers a fold reaches are the numbers a fold reaches on a square, which is the question anybody asks.

What the correction touches is any attempt to carry the construction apparatus onto a closed sheet, which nobody has attempted and which now has a stated reason not to work.

An enumeration that is complete over the wrong thing #

There is a general observation here about completeness results, and it is worth having because this subject contains several.

A completeness result says: here is a list, and nothing is missing from it. What it always means is: nothing is missing from the class of things enumerated over.

The axioms enumerate over combinations of alignments — how many points, how many lines, brought onto what. That enumeration is exhaustive and the proof is a case analysis over small numbers.

What it does not enumerate over is sheets, because sheets were not a variable. So the result is complete over its class and its class does not include the variation that turns out to matter.

That is not a defect in the result; it is what every completeness result is like. The useful habit is to ask, on meeting one, what the class was — and to notice that a variable held constant throughout the enumeration is invisible in the conclusion.

The same question applied to the rule families of a corrugation or to the thirty-two rules of the waterbomb gives the same answer: complete over assignments, on a flat patch.

The seventh, and why it is the odd one #

A closing note on the list itself, since the seventh axiom has always sat awkwardly.

The seventh — bring a point onto a line while making the crease perpendicular to another line — was found after the other six and by a different person, and it is the one most often left out of statements of the list. It is genuinely independent of the first six, so the list is six or seven depending on when it was written.

It takes two lines as input, which makes it the most exposed of the seven to the condition above: both lines have to separate for the operation to be well formed, and on a closed sheet the chance of two arbitrary lines both separating is lower than for one.

That is a small observation and it fits the pattern of the axiom’s history: the one that was hardest to find is the one with the most hypotheses, and both facts have the same cause, which is that it constrains the most.

Why nobody constructs on a tube #

The scope correction is about an activity nobody performs, and the reasons are worth naming so that the essay is not read as a criticism of anybody.

Paper is flat. A construction is performed by folding a sheet, and a sheet is flat. Taping it closed first makes every fold harder and every unfolding impossible.

Constructions want references. The whole method is building points from points, and a closed sheet has none to begin with — so a construction on one would have to start from marks somebody put there, which defeats the purpose.

The applications do not need it. Constructions are used to divide a sheet, place creases exactly and reach specific numbers. All of that happens on the flat pattern, before any joining, and the joined object inherits whatever was constructed.

So the flat sheet is the construction surface for the same reason it is the design surface, and there is no pressure to change it.

Which makes this a scope note rather than an obstacle: the axioms are complete for what they are used for, and the missing clause matters only to somebody asking whether the apparatus extends. The answer is that it does not, and that the reason is the sheet rather than the axioms.

What is measured here #

Nothing, and the essay should be read accordingly.

The claims are geometric and topological: which lines separate which sheets, what a reflection presupposes, which axioms take a line as input. Each is elementary and none is a computation this collection runs.

What is measured, elsewhere in this phase, is the combinatorics of glued sheets — free letters, panels, Euler’s number, verdicts and search costs — and none of that touches the axioms, which are about constructions rather than about crease patterns.

So this essay is a scope note on a result rather than a result, and it is placed here because the scope note has never been written and because the objects that violate the assumption now exist.

The three things a sheet has to supply #

Pulling the essay together, a construction needs three things from its sheet and a square gives all three.

Marked points to start from. A corner is the intersection of two edges and is a point. A square has four; a disc has none; a cylinder has none.

Lines that separate. Every axiom that takes a line assumes the fold reflects one side onto the other. A disc’s lines all separate; a closed sheet’s do not all.

A consistent notion of side. Distinguishing the two solutions an axiom often has requires knowing which side of the crease a thing lands on. Any orientable sheet supplies it; a Möbius band does not.

A square supplies all three, which is why the whole apparatus was built on one and why nobody has had occasion to name any of the three.

Each of the other sheets fails at least one, and the failures are in that order: a disc of paper fails the first, a cylinder fails the first and second, a Möbius band fails all three.

What the completeness result actually says #

Worth restating precisely, because the result is correct and the essay is not a refutation of it.

The seven are complete in the sense that every combination of alignments — bringing points onto points, points onto lines, lines onto lines, in every number — either has no solution, or is one of the seven, or is a consequence of them. The enumeration is over combinations and it is exhaustive.

What it does not enumerate is sheets. It assumes throughout that a fold along a line is a reflection of one part onto another, which is a property of the sheet rather than of the alignment.

So the correction is the same one this phase has made everywhere: the result is exact on a disc, the hypothesis is unstated, and it is unstated because nothing violated it.

Whether the seven could be extended #

An obvious question, and the answer here is no rather than not yet.

On a closed sheet the operation fold along this line is not always available, so the axioms lose cases rather than gaining them. There is no new alignment that becomes possible; there are alignments that stop specifying anything.

Whether a useful notion of fold exists on a sheet where lines need not separate is a real question — one could ask for an isometry of the sheet fixing a line, which on a cylinder is a reflection when the line separates and something else when it does not — and nothing here settles it.

The clause #

The seven axioms are the complete list of ways one fold can be specified on a sheet where every line separates, which is a disc.

Elsewhere, the ones that take a line as input are conditional on the line separating, and the ones that produce a line have to be checked. And on a sheet with a smooth boundary there is nothing marked to start from, so the list is complete and has nothing to be applied to.

That is a narrow correction to a result that is otherwise exactly right, and it is the same clause the rest of this phase has been adding: on a disc.