A reference on a sheet with no corner

A fold needs something to align. That is the whole of why folding is a construction method rather than a way of creasing paper approximately: a fold is exact when it brings one marked thing exactly onto another, and inexact otherwise.

The question this raises and which is usually passed over is what a fresh sheet has to align.

What a square starts with #

Four edges and four corners, and that is all.

The edges are lines. The corners are points, and they are points only because two edges meet there — a corner is not a separate feature, it is an intersection.

From those, the whole apparatus follows. Bring one corner onto another and the crease is a diagonal. Bring one edge onto the opposite edge and the crease is a midline. Every reference point in the collection is built from those first folds, and the tree of them grows from four points and four lines.

What a disc starts with #

A disc of paper has one edge and no corners.

Its boundary is a smooth circle with no distinguished point on it anywhere: every point of it looks like every other, and there is nothing an axiom could take as input.

So a construction on a disc cannot begin. Bringing the edge onto itself is satisfied by every diameter; bringing a point onto the edge needs a point, and there is none.

That is a fact about circular sheets which folders know practically — one starts by folding a disc in half, arbitrarily, and everything afterwards is measured from that arbitrary first fold.

What a cylinder starts with #

Two circles of edge, no corners, and a seam that is not a mark.

The seam is where the sheet was joined. Physically it is a line of tape and a real feature of the object. Mathematically it is a line of ordinary paper: the identification joins the two edges at points that are now interior, there is no crease there, and nothing distinguishes it from any other line running the length of the tube.

So a cylinder is in the disc’s position twice over: two smooth boundary circles, nothing marked, and no first fold that is not arbitrary.

Trying it with a disc #

The claim that a circular sheet has nothing to start from is worth testing, because it sounds like an exaggeration.

Cut a circle of paper. Now make an exact fold — one whose crease could be described to somebody else precisely enough that they could reproduce it.

Every fold available brings the edge onto itself, and every diameter does that equally well. There is no way to say this diameter rather than that one, because the sheet has no feature to refer to.

Make one anyway. Now the sheet has a crease, and the crease meets the boundary at two points, and those two points are marks. From there everything works: bring one onto the other and get a perpendicular; bisect; divide.

So the first fold is free and every subsequent fold is exact. That is a perfectly workable situation and it is different in kind from a square, where the first fold is exact too.

The difference shows up when two people do it. Two people folding a square into eighths produce identical objects. Two people folding a disc into eighths produce objects that are identical up to a rotation neither can specify.

Where a mark can come from instead #

There are three ways to give a cornerless sheet something to start from and all three are used.

An arbitrary first fold, as above. Free, and it makes everything afterwards relative.

A printed mark. The sheet arrives with something on it: a registration mark, a printed pattern, a hole. That is how manufactured sheet material is handled, and it turns the construction problem into an alignment problem.

A physical feature. A seam that is a fold rather than a join, a fitting attached at one point, an edge that is not quite smooth. Any of those breaks the symmetry.

The third is worth noting because it is what actually happens with a manufactured tube: the seam is a stiff line and everything gets measured from it, in practice, even though mathematically it is nothing. So the physical object has a mark the mathematical one does not, and the folder uses it.

That is a mild version of a situation this collection has met before — the sheet decides which points exist — and here the physical sheet decides and the mathematical one declines to.

Why a corner is worth more than an edge #

The essay has been treating corners as the valuable thing and edges as secondary, and the ranking deserves an argument.

An edge is a line. Bringing a line onto a line is one axiom, and it produces a crease — but a line has infinitely many points and no distinguished one, so aligning two lines usually leaves a family of solutions rather than one.

A corner is a point. Bringing a point onto a point is one axiom and it produces exactly one crease. Bringing a point onto a line produces one or two. Points pin things down and lines do not.

So a sheet’s constructive richness is measured by its points, and a square’s four are the seed of everything.

Which also explains why a hexagon is not obviously better than a square despite having more corners: six seeds rather than four, but the binary structure of a square’s diagonals and midlines is what makes division by folding so clean, and a hexagon’s threefold structure does not have the same arithmetic behind it.

What the collection has measured #

The reference work here is quantitative and it is all about squares, which is worth being explicit about since the essay is a scope note on it.

How many reference points are reachable at each fold depth, how close the reachable set comes to an arbitrary target, which sheet proportions make which points available: each is a measurement over a square’s coordinates, and the coordinates come from the corners.

None of it transfers to a cornerless sheet, not because the computation would be hard but because the question does not have a well-formed statement there. How far is the nearest reference from this point needs the point to be specifiable, and on a cylinder half of its specification is a convention.

So the honest position is that the whole reference-point apparatus is about polygonal sheets, that this has never been said, and that saying it costs a clause.

The first arbitrary fold #

The practical response, on any sheet with no corners, is to make one fold arbitrarily and construct from it.

That is what a folder does with a circular sheet, and it works: the arbitrary fold produces two points where its crease meets the boundary, and from two points the whole apparatus starts.

What is lost is not the ability to construct but the canonicity of the result. On a square, the midpoint of the left edge names a specific point of the sheet. On a cylinder, every construction is relative to an arbitrary origin, and two folders working on identical tubes produce constructions that are the same up to a rotation nobody can pin down.

That may or may not matter. For dividing a tube into thirds it does not; for specifying where a feature goes relative to something else on the tube it does, and the something else has to be marked by hand.

The measurement that becomes impossible #

There is one thing a construction on a square does that a cylinder cannot do at all.

Reference points are counted and their reach measured: given a target point, how many folds are needed to reach within a given distance of it, and how the reachable set fills the sheet. Those measurements are relative to the sheet’s own coordinates, which come from its corners.

On a cylinder there are no such coordinates in one direction. A point is at a definite height and at an arbitrary angle, so half of every coordinate is a convention, and any measurement of reach in that direction is measuring relative to the first arbitrary fold.

That is not a small thing. It means the entire quantitative apparatus of reference points is about sheets with corners, and on a closed sheet the questions have to be reformulated before they can be asked.

The symmetry that causes the trouble #

There is a clean way to say why a cornerless sheet has no canonical points, and it is worth having because it is the same reason in every case.

A symmetry of a sheet that moves a point to a different point means the two points cannot be told apart by anything intrinsic to the sheet. If every point of the boundary can be carried to every other by a symmetry, then no point of the boundary is distinguishable, and no construction can name one.

A square’s symmetry group has eight elements. It moves each corner to three other corners and nowhere else, so the corners form an orbit of four and a construction can name a corner, and having named one, everything else follows.

A disc’s symmetry group is infinite: every rotation. Its boundary is one orbit, so no boundary point is nameable, and there is nothing for a construction to grasp.

A cylinder’s is infinite too, in the closed direction. Rotations round the tube carry every point to every other at the same height, so a height is nameable and an angle is not.

That is the whole account, and it says something slightly more general: a sheet with continuous symmetry has no canonical points, and the amount of symmetry is exactly the amount of arbitrariness a construction inherits.

What a first fold buys #

Making one arbitrary fold is, in that language, breaking the symmetry.

A disc with one crease in it has a symmetry group of two rather than infinity — the reflection in that crease — so its boundary now has two orbits of points and both are nameable. Everything afterwards is exact.

A cylinder with one lengthwise crease has the same. One arbitrary act, and the rest of the construction is as good as on a square.

That is a reassuring way to end, because it says the loss is a single bit of information rather than a structural inability. A cornerless sheet is a sheet needing one arbitrary decision before it becomes a constructive object, and the decision is free.

The situation only becomes awkward when the result has to be canonical — when two people or two machines have to produce the same thing on the same sheet, and there is nothing to agree on.

What is left #

A cylinder is not useless as a construction surface and it is much poorer than a square, and the honest inventory is short.

Available: everything within a disc-shaped patch of the tube, exactly as on a flat sheet, once an origin has been chosen. Every axiom, every reference tree, every division.

Available with a convention: anything relative to a first arbitrary fold, which is most practical work.

Not available: any construction whose result is meant to be a canonical point of the sheet, and any measurement of reach in the closed direction.

Not available at all: folds along lines that do not separate the sheet, which is a separate difficulty and applies to lines running the length of the tube.

The seam, once more #

The seam deserves a final look because it is the one thing on a cylinder that looks like a mark and is not.

Mathematically it is nowhere. The identification joins two boundary edges point by point, the joined points are interior points of the sheet, and there is no crease, no feature and no way for any axiom to refer to it. A cylinder built by gluing and a cylinder that was always a cylinder are the same object, and the second has no seam at all.

Physically it is very much somewhere. Tape, adhesive, a weld, a fold-over: stiffer than the paper, thicker, visible, and it is where a folder measures from because it is the only thing to measure from.

So the physical object has a mark the mathematical object lacks, and the mark is an artefact of manufacture rather than a feature of the sheet.

That is worth carrying because it means a construction on a real tube is possible, in practice, and is not a construction on a cylinder — it is a construction on a cylinder with a mark on it, which is a different and richer object. The mathematics describes the poorer one, correctly, and the poorer one is not what anybody has in their hands.

Four sheets, ranked by what they start with #

A short table, since the essay’s content is a comparison.

A square. Four edges, four corners. Four points to start from, and the binary structure of its diagonals and midlines behind them. The richest constructive sheet in ordinary use.

A rectangle. The same four points and no diagonal symmetry, so slightly less.

A hexagon. Six points, a threefold arithmetic, and division by folding that is less clean than the square’s.

A disc. One edge, no points. One arbitrary fold and then everything.

A cylinder. Two edges, no points, no canonical angle. One arbitrary fold in the closed direction, and heights are canonical while angles are not.

A torus. No edges at all. Nothing anywhere, in either direction, and two arbitrary choices before anything can be said.

Read down and what falls is the number of marks, and with it the number of arbitrary decisions a construction has to make before it becomes exact.

What a machine does #

There is a version of this problem in manufacturing and its solution is instructive.

A machine folding sheet material does not use the sheet’s corners. It uses registration: the sheet is placed against stops, or it carries printed marks that a camera finds, and every fold is positioned relative to the machine’s coordinates rather than to the sheet’s.

That removes the whole problem. A cornerless sheet is no harder than a square one, because neither is being used as a coordinate system.

And it is why the reference-point apparatus is a hand-folding apparatus. Its whole value is that a person with no measuring instrument can produce exact points, using the sheet as its own ruler. A machine has a ruler and does not need the sheet to be one.

So the correction in this essay applies to constructions by hand and not to manufacture, which is a narrow scope and is the scope the reference-point work has always had. Worth stating, because it is easy to read a limitation on a sheet as a limitation on the objects made from it, and here it is not.

The clause, again #

The apparatus of exact folding — the axioms, the reference points, the reach measurements, the constructibility results — is an apparatus for sheets with corners.

Every result in it is right and every one of them starts from marks that only a polygonal boundary supplies. A smooth boundary supplies edges without points; a closed sheet supplies fewer edges and no points; and a construction on either begins with an arbitrary choice that a square does not require.

Which is the same clause the rest of this phase has been adding, arriving in the one corner of the subject where the sheet is not a crease pattern at all.

The sentence #

A construction needs a marked point to start from, and a fresh sheet’s marks are where its edges meet each other.

A square has four such points. A disc has none. A cylinder has none, and its seam — which looks like a mark and is a real feature of the physical object — is an ordinary line of paper as far as any axiom is concerned.

So the apparatus of exact folding is an apparatus for polygonal sheets, and the polygon is doing more work in it than anybody has had reason to say.