Nowhere to put the error

A crease put down a fraction of a degree out of place does not stay a fraction of a degree out of place. It is carried by every fold after it, and the sheet beyond it ends up somewhere it was not meant to be.

On paper this is almost never visible, and the reason is not that the error is small. It is that the paper has somewhere to put it.

What a surface does that a plane cannot #

The idealised sheet of this subject is a surface with no thickness that folds along lines. Both halves of that are false of paper, and the list of what else is false is longer than the description. But one property of paper is no idealisation at all, and is the whole of what follows: paper is allowed to be a surface rather than a plane.

A panel is not. A rigid panel is a plane region and stays a plane region through the entire motion — that is the definition of the object, not an approximation to it, and it is what makes a panelled pattern a different kind of thing from a folded sheet. Nothing about it may deviate anywhere.

Take a sheet whose crease is a third of a degree from where the drawing put it. The paper beyond it is misaligned with the paper it is supposed to lie against — and the sheet takes on a slight double curvature over several square centimetres, and the two edges come into line. The flatness given up is a fraction of a millimetre spread over a hand’s width, beneath the resolution of anybody looking at it. The surface has adopted a shape that is not flat, which it was always entitled to do.

This is a geometric statement and should be kept as one. It says nothing about how stiff paper is or how much force the folder applied — those questions belong to a subject this site does not enter, and the moment a clearance is asked how rigid the resulting joint is, the argument has left geometry. The claim is only that the shapes available to a sheet outnumber the shapes available to a plane, and that a shape absorbing a small misalignment is among the first and not the second.

A panel has an account of exactly zero. So the error does not disappear at the panel and it does not disappear on the way; it travels to the only place in the assembly that is permitted to move, which is the hinge.

The error that has to be absorbed #

How large that error is, and how it grows, is the rung below this one, quoted here rather than re-derived. A folded position is a composition of reflections, a reflection in a line that is out by an angle turns everything beyond it by twice that angle, and the two ways of being out do not grow alike.

A bias of 0.4° at every crease puts the far end 0.056 panel widths out after eight creases, 0.112 after sixteen, 0.168 after twenty-four, 0.223 after thirty-two and 0.335 after forty-eight — a straight line through the origin, which is what a per-crease constant looks like. The same size of error scattered independently gives 0.019, 0.023, 0.032, 0.039 and 0.052: the same numbers divided by roughly the square root of the count rather than by the count.

The consequence for a specification is immediate. Suppose an assembly of thirty-two creases has to land its far end inside a tenth of a panel width. Both laws are exactly linear in the error, so the requirement inverts directly: the systematic component must be held under 0.18° and the scattered component may be as large as 1.0°. Those are two specifications differing by a factor of nearly six, and a drawing writes them with the same words.

On paper both the fan and the blur are recoverable: a folder squares the stack up with two hands and the sheet takes the difference as curvature. In a panelled assembly the fan is the final shape, and the only place any of it can be taken up is at the joints.

Clearance is a length before it is a specification #

So the joint has to be built with room in it, and the room is a length in the same units the drift is measured in. The cleanest case is the hinge that is nothing but room: two panels not touching at all, joined by a membrane across the space between them. The gap is a number the designer chooses, and what it buys is found by driving the panels together until their outlines meet.

Sweeping the gap gives the whole relation. At a panel thickness of 22 per cent of the panel length, a gap of 0.08 reaches 40.0°, 0.16 reaches 72.1°, 0.30 reaches 107.5°, 0.45 reaches 127.9°, 0.62 reaches 140.9° and 0.90 reaches 152.5°. The room and the fold angle are the same quantity read two ways.

Two things follow. The clearance is not free: every hundredth of a panel length of gap was paper in the drawing and is empty in the thing, and across a tessellation that is a serious fraction of the sheet — added directly to the proportion already doing nothing.

The second is a limit rather than a cost. The curve of travel against gap rises and flattens: 0.62 buys 140.9°, and half as much again, at 0.90, buys only 152.5°. It never reaches 180°. A membrane joint between two panels of any finite depth cannot fold flat at any finite gap — it can only approach flat, more and more expensively. The subject’s central object, a sheet that lies flat, is unavailable to this arrangement by construction rather than by imprecision.

The gap a target angle costs #

The closed form inverts, and the inverse is the expression a designer actually needs: given a fold angle to reach, how much room does it take?

Solving $\theta = 180° - 2\arctan(t/g)$ for the gap gives

$$g = t\,\tan\!\big(\theta/2\big)$$

At a panel thickness of 0.22 and a target of 72.1° that is $0.22 \times \tan 36.05° = 0.160$, and at 140.9° it is $0.22 \times \tan 70.45° = 0.620$ — both of the essay’s own numbers, recovered from a tangent.

Which prices the last few degrees #

Written that way the limit stops being a curve that flattens and becomes an expense that diverges, and the arithmetic is brutal.

Reaching 170° takes $g = t\tan 85° = 11.4\,t$. Reaching 175° takes $22.9\,t$. Reaching 179° takes $114.6\,t$.

At the essay’s own thickness of 0.22 of a panel length, 175° needs a gap of five panel lengths — five times the width of the panels being joined. The assembly at that point is two small plates a long way apart held by a membrane, and it is not a fold.

Even a gap equal to the entire panel, $g = L = 1$, leaves the joint twenty-five degrees short of flat.

So the last ten degrees cost more room than the first hundred and seventy, and the shortfall from flat is $2\arctan(t/g) \approx 2t/g$, which halves only when the gap doubles.

What that leaves a designer #

Two readings, and the second is the one that decides a design.

Thinner panels help in exact proportion: at $t = 0.05$ a gap of one panel length reaches 174.3° rather than 155.2°, because the shortfall is proportional to $t$. The ratio $t/g$ is the only thing the joint knows, so a panel five times thinner buys the same angle at a fifth of the gap.

And there is a threshold worth naming. A gap equal to the panel thickness gives exactly 90°; a gap of ten thicknesses gives 168.6°; a gap of a hundred gives 178.9°. Every factor of ten in the gap buys about a factor of ten off the shortfall and nothing more — so a designer choosing a membrane joint is choosing, at the outset, which decade of the approach to flat to pay for.

The other places to put the room #

The gap is one answer among several, and the useful way to sort them is by what each one gives away rather than by what the cross-section looks like. That taxonomy is a rung of its own and is not repeated here. What matters for tolerance is narrower: each technique puts the room somewhere different, and where it goes decides what can absorb the error.

That is the control, and the reason the question exists. Everything else is a way of removing material or moving the axis until the joint moves.

The chamfer’s room is inside the material, and that matters for error in a way the gap does not: a tapered panel still reaches the crease line, so the two halves locate each other by their own edges. The membrane gives that up, and where its panels sit is decided by whatever the membrane is fixed to.

The surface hinge is the one arrangement here that reaches flat, and the price is stated in the figure. Its axis is on one face, so it closes toward that face and cannot close away from it: the mountain-and-valley assignment is no longer something a folder chooses, but is welded into the hardware. The two things a builder might want most — a joint that folds flat, and a joint that can go either way — are not available together. Either the axis is off the mid-plane, in which case the assignment is fixed, or it is on the mid-plane, in which case flat is a limit and never a value.

Which condition was checked, and how #

The travel numbers are the load-bearing measurements here, so how they are obtained matters. Contact is tested by separating axis on the actual outlines of the two panels, and the travel is found by bisection on the first angle at which the test reports a meeting. Nothing is judged by eye. A test slightly too generous would report travel where two solids overlap, and a joint credited with a fold angle it cannot reach is a joint that jams in the assembly rather than on the page.

The membrane sweep was then checked against something the drawing never evaluates. The panels’ inner corners sit at a known offset from the axis, and the first contact between them is a plain trigonometric statement: the travel should be exactly 180° − 2·arctan(t/g) for panel thickness t and gap g. That closed form gives 39.97°, 72.05°, 107.49°, 127.89°, 140.93° and 152.53° at the six gaps above, against 40.0°, 72.1°, 107.5°, 127.9°, 140.9° and 152.5° from the contact test. The two agree everywhere to the precision the label carries and share no arithmetic — one drives polygons together and the other evaluates an arctangent. A disagreement would have meant the picture and the number were about different objects.

The closed form is also what settles the limit claim, as a fact rather than an impression of a curve. An arctangent of a positive quantity is positive, so 180° − 2·arctan(t/g) is strictly less than 180° for every finite gap and every panel of positive thickness. The flat state is not merely expensive; it is absent.

The drift figures carry their own refusals: the systematic law is checked as a drift per crease that must stay constant across the upper half of the range, and the random law as a drift per root crease with a wider allowance, since a mean over forty strips is noisy. A systematic drift not proportional to the count would mean the reflection mechanism is not the one operating, and the figure refuses to draw rather than report it.

Paper pays the same bill and calls it a crease #

There is a symmetry here that is easy to miss, and it changes what the clearance is.

A crease on paper is not a line either. It is a small arc, and the paper wrapped round that arc is longer than the distance the stack advances by — room, provided at the joint, priced by the joint, and increasing with the number of joints.

So paper does not fold without clearance and never did. The panelled assembly and the paper sheet are charged the same line item at the same place, and the difference is the currency. The panel pays in fold angle and in a pattern that is no longer quite the drawing. The paper pays in area, quietly, out of a sheet whose size was never the point — which is why it goes unnoticed until the crease count gets large and a fine tessellation comes out measurably short.

The two bills grow the same way as well. The paper spent on creases is a constant per crease, so it accumulates linearly, exactly as a biased angular error does. Neither is diluted by dividing the sheet more finely; both are made worse by it.

Where the cross-section stops #

The pictures above are cross-sections of one hinge, and three things are beyond them.

They cannot show a vertex. Four hinges meet at an interior vertex of a rigid pattern, their clearances are not independent, and a gap chosen to give one hinge its travel may take that travel from a hinge beside it. A cut through a single joint is silent about that, so everything above is a lower bound on the difficulty rather than the difficulty.

They cannot show the hinge. In every drawing here the axis is a point and the panels are the only solids present. A real joint is a component with its own thickness, its own play and its own manufacturing error — one more contribution to the total rather than a correction to it.

They cannot show whether the assembly moves. That question has its own vocabulary and its own owner; this site measures where solids meet and stops, and what moves is quoted from elsewhere rather than argued here.

Three idealisations do work in every number above, each false in the direction that makes the real answer worse. The panels are exactly flat, and a manufactured panel is not. The hinge axis is exactly where it is said to be, and putting it consistently a little off is the systematic error the drift figures price. Contact is treated as the only thing that stops a fold, which is a geometric limit and the last one anything reaches. Something more forgiving can be arranged instead, by refusing to let the panels be a surface at all.

Who noticed it, and when #

The engineering side came first, from people who had a structure to deliver rather than a theorem to prove. Koryo Miura’s solar-array work in the 1980s handled thickness with membrane hinges and generous clearances, and the generosity is the point: those clearances were sized to swallow accumulated error, and nobody at the time was calling that a geometric quantity.

Hoberman’s patents in the early 1990s carry the idea of moving the crease off the mid-plane, arrived at while designing deployable structures. Tomohiro Tachi’s offset-panel construction in 2011 is the first general treatment: given a rigid-foldable pattern, it produces thick panels folding along the same path, by giving up the requirement that the panels form a surface. Each is a different answer to where the room goes, and each was found by somebody building something.

What arrived later is the reading offered here — that the room is a consequence rather than a concession, forced by the difference between a surface and a plane and sized by a crease count. A drawing specifying an angular accuracy per crease and a clearance per joint has specified two numbers that are not independent, and their relation turns on something the drawing rarely records: whether the process is biased or merely imprecise.

Where the ladder goes next #

The immediate direction is the one the cross-section could not reach: the vertex, where several joints share their room and the clearances have to be solved together rather than chosen one at a time.

The other direction goes back to the sheet. Almost no pattern satisfies the folding conditions exactly, and no folded sheet satisfies them exactly either, so every working model in the subject is a pattern that fails the theorems being folded by a material that forgives it. The size of that forgiveness is what this rung has been measuring. On paper it is a curvature nobody sees; in anything that gets built it is a slot with a width, drawn on a drawing, and priced twice — once in the area it takes out of the sheet, and once in the fold angle it fails to reach.