Thickness round a closed loop

Paper has thickness and the ideal model does not, and the gap is where a large part of the engineering lives. A crease pattern says two panels meet along a line; two panels of real material meeting along a line would have to interpenetrate, so the hinge is moved.

Every technique for doing so is a way of offsetting the hinge from the ideal crease by an amount related to the material’s thickness — and each of them buys the travel by giving something else up.

The offsets accumulate. On a flat sheet they accumulate outward and end at the boundary. On a closed sheet they have to come back.

What an offset is #

Take one fold between two panels of thickness $t$. If the hinge sits on the ideal crease line, the panels collide before they are anywhere near closed. So the hinge is moved: onto one surface, or into a chamfered notch, or into a gap bridged by a membrane.

Whichever technique is used, the physical hinge is displaced from the ideal line by something of order the thickness — and the displacement has a direction, decided by which side the fold goes.

Why they accumulate #

Walk across a panel of the pattern to the next, then to the next, and each crossing displaces the physical structure from the ideal one by that crease’s offset.

Cross five creases and the accumulated displacement is the sum of five offsets, each with its own sign depending on which way that crease folds.

On a flat sheet that is fine: the accumulation grows outward from wherever the walk started, and the sheet’s edge is where it stops. A designer notices it as the finished object being slightly larger than the pattern says, which is the allowance every folded assembly carries.

And why a closed sheet is different #

On a closed sheet a walk can come back to where it started.

Go once round a tube, crossing every crease that runs along it, and the accumulated offset has to be zero — because the panel arrived at is the panel departed from, and it cannot be displaced from itself.

That is a condition on the offsets: their signed sum round every non-shrinkable loop must vanish. It is not a condition at any crease and it is not satisfied automatically.

The four techniques, and what each offsets #

Setting them out makes the accumulation concrete, since each displaces the structure by a different amount.

Solid panels, hinge on the crease. Offset nought, and the panels do not move at all: they collide immediately. The control case, and it is why a technique is needed.

A chamfer. Material removed at the crease so the two panels can rotate into the notch. The hinge sits at the bottom of the notch, offset from the ideal line by something less than the thickness, and the panel is thinnest exactly where it is worked hardest.

A surface hinge. The hinge on one face of the panel rather than in the middle. Offset is the full half-thickness, and the direction is fixed by which face — so the technique writes the mountain-or-valley assignment into the hardware and the pattern can no longer be folded the other way.

A membrane. A gap between panels bridged by flexible material. Offset is half the gap, which is a design parameter and is usually larger than the thickness.

Four techniques, four offsets, and a real structure often uses more than one — a chamfer where the folds are shallow and a membrane where they are deep. That is the case where the sum round a loop stops cancelling.

Why the sum is signed #

The accumulation is signed because the offsets point, and the pointing is what makes cancellation possible.

A surface hinge on the mountain side displaces the structure one way; the same hinge on the valley side displaces it the other. So a walk crossing a mountain and then a valley picks up two offsets of opposite sign, and if they are equal in magnitude the walk ends where it began.

Which means an alternating pattern with a uniform technique and a uniform thickness closes automatically, at every loop, for free.

That is a strong statement and it explains why the condition has never been noticed. The ordinary case — one material, one thickness, one technique, alternating letters — satisfies it identically, and the residual is exactly nought rather than small.

The condition bites when any of those four is broken, and breaking one of them is an ordinary engineering decision rather than an exotic one.

What breaks the cancellation #

Four ways, each of them a thing a real design does.

Mixed techniques. Chamfers in one region and membranes in another, chosen because the fold angles differ. The offsets have different magnitudes and no longer cancel in pairs.

Varying thickness. A panel stiffened locally, or a laminate whose layers stop. The offset scales with thickness and the pairs stop matching.

Unequal letter counts. A loop crossing three mountains and one valley — perfectly possible, and required at some vertices — accumulates two offsets’ worth rather than nought.

An asymmetric hinge. Any hardware hinge whose axis is not on the panel’s midplane, which is most of them.

So the cancellation is the default and it is a fragile default. Any of the four takes the residual from exactly nought to something of order a thickness, and a tube with twenty creases round it can accumulate several thicknesses.

What several thicknesses does #

Worth quantifying, since the seam carries the residual is vague.

A structure of one-millimetre panels with twenty creases round a loop, of which the offsets fail to cancel by a tenth of a thickness apiece, accumulates two millimetres of residual — which for a tube of a hundred millimetres circumference is two per cent.

Two per cent is a large misfit for a mechanism. The seam has to close a two-millimetre gap, or the panels have to be redrawn two per cent narrower, or the structure is assembled pre-stressed.

The third is what usually happens, because the misfit is discovered at assembly rather than at design, and by then the panels exist.

That is the practical cost of the condition and it is the sort of thing that reads in a project report as tolerance stack-up at the closure seam — which is exactly what it is, arrived at from the folding rather than from the tolerancing.

The general shape, once more #

The condition is another instance of the pattern this whole phase has been about.

On a sheet with a boundary, an accumulated quantity has somewhere to go: it grows outward and the edge absorbs it. On a closed sheet it has to return to its starting value, which is a condition.

The layer order has the same shape: a stack accumulates upward and a rim is where it stops, and a closed sheet’s order climbs instead. So does the parity, where the accumulated quantity is a sign. So does the closure of the folded motions, where it is an isometry.

Four accumulated quantities, four conditions, all of them vacuous on a disc and all independent on a closed sheet. The thickness offset is the only one of the four that is about the material rather than about the ideal pattern, which is why it turns up in a different literature and under a different name.

The parity, and the thickness, and how they differ #

The two conditions on a closed sheet are worth setting side by side because they look alike and are not.

The parity is a signed sum modulo two: crossing a crease exchanges which face shows, and the count round the loop has to be even. It is discrete, exact, and about the ideal pattern.

The offset is a signed sum of real numbers: crossing a crease displaces the structure by that crease’s offset, and the sum round the loop has to be nought. It is continuous, approximate, and about the material.

So a tube has to satisfy both, they are independent, and only one of them is a property of the drawing.

How the condition is usually met #

Not by solving it. By symmetry.

A corrugation rolled into a tube usually has its creases alternating — mountain, valley, mountain, valley — so the offsets alternate in sign and cancel in pairs. Round a loop crossing an even number of creases, half of them displace one way and half the other, and the sum is nought.

That is the same evenness the parity condition requires, and it is why the two are so easy to conflate: on the ordinary patterns, satisfying the parity also cancels the offsets.

Where the two come apart is when the offsets are not equal. Different techniques on different creases, or a thickness that varies, or a chamfer on some folds and a membrane on others: then the signs still alternate and the magnitudes do not, and the sum round the loop is a small nonzero number.

What a nonzero sum does #

The structure does not fail; it strains.

A tube whose offsets do not close is a tube whose two ends of the loop are displaced from each other by the residual. Joining them anyway pre-loads the structure: the panels are slightly out of position, the material takes up the difference, and the object is stressed even at rest.

For a paper model that is invisible. For a metal deployable it is a stress concentration at the seam, which is where such things fail.

So the practical form of the condition is: make the offsets round the loop cancel, or the seam carries the residual. That is a familiar shape of requirement in mechanism design and it is not usually stated for folded structures, because folded structures are usually analysed as ideal sheets.

What a designer should do #

Three things, and the first two are free.

Count the letters round the loop. If the mountains and valleys crossed are equal in number and the technique and thickness are uniform, the offsets cancel and there is nothing to do.

Check for mixed techniques. If the design uses more than one accommodation, or the thickness varies, sum the offsets round each loop with signs and see what is left.

Decide where the residual goes. Into the seam as a designed gap, into the panel widths as a correction, or into the structure as pre-load. All three are used and only the third is a decision made by default.

None of that requires the computation this collection has not done. It requires knowing the condition exists, which is the essay’s contribution.

An analogy from another kind of assembly #

The situation has a familiar name in mechanical engineering and borrowing it makes the condition less exotic.

A chain of parts each with a tolerance accumulates: the last part’s position is uncertain by the sum of the individual uncertainties. That is a tolerance stack, it is standard, and it is analysed in every assembly.

A chain that closes on itself — a ring of parts, a linkage that returns to its start — has a stack that must come back to zero, and the residual is a closure error. Mechanisms that fail to close are either forced together, which pre-loads them, or redesigned.

That is exactly the situation here, with the thickness offsets playing the part of the tolerances and the loop round the tube playing the part of the closed chain.

The difference is that a tolerance stack is about manufacturing variation, which is random and unavoidable, while an offset stack is about a designed displacement, which is systematic and can be made to cancel. So the folded case is the easier one: the residual can be driven to exactly zero by choosing the technique, where a tolerance stack can only be bounded.

Which is worth knowing, because it means the condition is a design opportunity rather than a limitation. A tube whose offsets cancel exactly is a tube that assembles without force, and getting there is a matter of arithmetic on the drawing.

The condition, stated once #

For a closed sheet with real panels:

Assign each crease an offset — a signed displacement of the physical hinge from the ideal crease line, whose magnitude comes from the accommodation technique and the material thickness, and whose sign comes from the crease’s letter.

Then for every loop that cannot be shrunk to a point, the signed offsets crossed by that loop must sum to nought.

On a sheet with a boundary the condition is vacuous, since every loop bounds and the accumulation has an edge to end at. On a closed sheet it is one equation per loop, it is about lengths rather than parities, and it is satisfied identically by the ordinary case of one technique, one thickness and alternating letters.

That is why nobody has written it down, and it is why it will be met the first time somebody builds a folded tube with two different hinges in it.

What would be computed #

For completeness, since the essay ends in an owed item.

Give each crease of a glued cell an offset: a signed length, determined by the accommodation technique, the material thickness and the crease’s letter. Then sum round each independent loop and report the residual.

The loops are the ones the gluing supplies — one per glued pair — and they are already identified, since they are the same loops the parity is computed on. The signs come from the letters, which a lettering supplies. The magnitudes come from a table of techniques, which the collection already has.

So the computation is a sum over a loop with data that all exists, and the reason it has not been done is that the collection’s glued sheets are ideal and its thickness work is on single folds, and nothing has joined the two.

That is a small piece of work and it would produce a number per pattern per technique, which is the sort of table a designer could use.

Where this sits among the thickness results #

The collection’s thickness work has been about a single fold: how far it can close before the panels touch, and what each accommodation technique buys.

That is the right first question and it is entirely local. What it does not address is how the accommodations interact across a pattern, which on a flat sheet is an accumulation with somewhere to go and on a closed one is a condition.

Thickness has a sign is the nearest existing result, and it is the observation that an offset points — which is exactly the ingredient the accumulation needs. This essay is that observation applied round a loop.

So the sequence is: a fold has a travel, an offset has a sign, and on a closed sheet the signed offsets have to sum to nothing. Three steps, of which the first two are measured here and the third is not.

What is measured here #

Nothing about thickness on a closed sheet, and the essay should be read as an argument rather than a measurement.

This collection’s thickness work is a cross-section through one fold, driven until the panels touch, with the travel each technique buys measured on the actual outlines. That is a local computation and it is exact.

The accumulation round a loop has not been computed, because it needs a glued sheet with thicknesses on it and the collection’s glued sheets are ideal. Building one means giving each crease an offset and summing round the loops, which is arithmetic and has not been done.

Recorded as owed, with the observation that the condition exists and that symmetry is why nobody has met it.