The allowance is spent at the end

A tolerance is a direction settled one half of this. The closure of a general quadrilateral mesh is four equations in twenty lengths, so its solutions form a surface sixteen directions wide; an error along that surface is free at any size, an error across it is fatal at any size, and the fifth of a millimetre an earlier rung reported was the allowance in one direction out of twenty.

Every number in that account was measured at one fold angle. Which is where a tolerance is normally quoted, and is the thing nobody had thought to question: a mesh is not a position, it is a motion — the whole sheet follows from one crease, and the closure is an equation in the fold angle. There is no reason in advance why the sixteen free directions at a third of a radian should be the sixteen free directions at two and a half.

Two questions, then, and they turn out to have different answers. Does the surface move? And does the price of leaving it change?

The surface does not move #

The first answer is a flat no, and it is worth stating how flat.

The test does not compare bases, which are arbitrary. It takes the direction the closure’s residual grows fastest along at one fold angle — a specific vector in the space of twenty lengths — and asks how much of it lies inside the equations’ row space at a different angle. A share of one means the surface has not moved at all. Anything less means a part cut to a tolerance measured at one configuration is being cut to the wrong tolerance for the rest of the motion.

Over every pair of angles from 0.3 to 2.5 radians, the stiff direction found at one lies inside the row space at the other to 1.000000000000. And the complementary check runs the same way: a direction that is free at one angle has 5.5 × 10⁻⁸ of itself inside the row space at any other, which is zero to the precision the Jacobian is computed at.

So the set of meshes that close is one set. It does not depend on where in the fold the question is put, and a mesh cut differently in a free direction is a mesh that folds all the way rather than one that happens to close once.

And the price rises all the way #

The second answer is the essay’s, and it is not a subtlety.

Fix a budget — a worst residual of 0.02 radians, which is about a degree and is generous for something meant to be a mechanism. Ask how far the mesh may be cut wrong along the closure’s own worst direction before it spends that budget, on a sheet 150 mm across.

A factor of 12.9 between the two ends. The 0.211 mm at 0.60 radians is the number this site published, and it is now readable as one row of a table rather than as a property of the mesh.

The direction of the effect is the awkward one. A deployable is nearly flat when it is deployed and packed when it is packed, and the packed end is where the allowance is smallest. The mesh is most forgiving where it is doing least.

The shape of the curve is a quantity the site already has a name for #

The table is a curve and the curve is very nearly a known function.

Divide the closure’s sensitivity at each fold angle by tan(ρ/2) — the half-angle tangent — and what comes back is 6.007 at 0.30, 6.001 at 0.45, 5.991 at 0.60, 5.972 at 0.80, 5.946 at 1.00 and 5.909 at 1.20. Constant to within 1.7 per cent over the range a builder uses. It then leaves: 5.677 at 1.80, 5.405 at 2.10, 4.845 at 2.40.

The divisor is not arbitrary and it is not new here. tan(ρ/2) is the fold-angle multiplier, the quantity that relates one crease’s fold angle to another’s at a degree-four vertex, and it is the natural coordinate for a rigid motion for exactly the reason it shows up here: the closure is a composition of rotations, and the half-angle tangent is what turns a rotation into an algebraic object.

So the practical statement is one number and a known function. Measure the allowance once, anywhere in the middle of the fold, and the allowance anywhere else is that number times cot(ρ/2) over cot of the angle it was measured at — accurate to a couple of per cent until the mesh is very nearly closed, and increasingly optimistic after that.

Where the fit gives out, and why that end matters #

The departure at the closed end is not noise and it should not be smoothed over, because it is the end an engineer cares about.

Near ρ = 2.4 the sensitivity is 19 per cent below what the multiplier predicts, and falling further. What is happening is that the mesh is approaching a configuration where panels are nearly parallel, the closure’s Jacobian is changing in a second way that is not the half-angle scaling, and the growth in sensitivity slows.

Two consequences, and the first runs the reassuring way. The extrapolation is conservative exactly where the tolerance is tightest — a designer extrapolating from mid-fold is told the part must be better than it actually has to be, and errs toward precision rather than away from it. And a measurement taken at the packed end is still the one to take, because it is the binding case and the case the model does not cover.

The one number, written as an allowance #

The multiplier account is stated as a sensitivity divided by a tangent, which is the right way to see the mechanism and the wrong way to use it. Written directly in the quantity a workshop is given, it is one product.

Multiply each row of the table by the tangent of half its fold angle. At 0.30 radians, 0.425 mm times 0.1511 is 0.0642. At 0.60, 0.211 times 0.3093 is 0.0653. At 1.20, 0.098 times 0.6841 is 0.0671. Three rows spanning a factor of four in allowance, agreeing on the product to within four per cent.

So the rule is: allowance times tan(ρ/2) is about 0.065 millimetres, on this mesh at this budget, and the allowance anywhere in the buildable range is that constant divided by the tangent. No sensitivity has to be computed and no derivative has to be taken; a workshop given one number and a fold angle can produce the tolerance for any other fold angle with a calculator.

At the closed end the product drifts up — 0.085 at 2.40 radians — which is the same departure the multiplier figure shows, seen in the units that matter. It drifts in the safe direction: the true allowance there is thirty per cent larger than the constant predicts, so a part cut to the extrapolated figure passes.

Which is why the ratio and not the millimetres is the result #

That constant is a property of one mesh, one budget and one direction, and quoting it as though it were general would repeat exactly the mistake this essay is about.

What is general is the shape. The allowance goes as the cotangent of half the fold angle over the range a mechanism is built for; the constant of proportionality is whatever a single measurement on the mesh in hand says it is; and the two ends of a full motion differ by a factor of about thirteen.

That separation is what makes the measurement transferable. A designer with a different mesh, a different budget or a different sheet size does not need this essay’s numbers — they need one measurement and the function, and the function does not depend on any of the three. A designer with this essay’s numbers and a different mesh has nothing, which is the position anybody quoting a single-configuration tolerance has always been in without knowing it.

It also says what a specification should contain. Not a millimetre figure, and not a millimetre figure with a fold angle attached, but a millimetre figure, a fold angle, and the note that it scales as the cotangent — three items, none of which costs anything to write down, and which between them turn a number that is right once into a number that is right throughout.

What the two answers say together #

Put the two halves side by side and the statement is short.

The geometry of the solution set is fixed and its scale is not. Which twenty-length directions a mesh may be cut differently in is decided once, by the pattern; how much it may be cut differently by is decided by which part of the fold is being asked about.

That separation is useful because the two are used for different things. A designer choosing where to spend precision needs the directions, and the directions do not move — a part measured against the wrong direction is wrong at every angle, so a bad choice is uniformly bad. A manufacturer setting a number needs the scale, and the scale changes by an order of magnitude, so a number quoted without a configuration is quoted without most of the information in it.

A tolerance on a rigidly folding mesh is a direction and a moment. Neither alone is a specification.

Against the mesh that folds for a structural reason #

The comparison this anchor keeps returning to is worth making again with the new measurement, because it comes out the same way and it comes out more strongly.

A Miura folds because one crease family runs straight through every vertex and the whole sheet repeats one vertex. That is a property of its structure, and a length cannot destroy a structure: change a Miura’s panel width, height and shift by five per cent each and the result is another Miura, which closes exactly.

So a Miura’s allowance is not 0.425 mm at one end and 0.033 mm at the other. It is unbounded at both, in every direction that keeps the structure, and zero in the directions that break it. There is no curve to draw because there is no scale to measure.

That is the sharpest statement of what this anchor has been circling: a mesh that folds because an equation holds has a tolerance, and a mesh that folds because a structure holds does not have one at all. The first is a number that changes through the motion; the second is a yes-or-no about whether the structure survived being made.

Why the closed end is the tight one #

The scaling has a mechanism and it is worth spelling out, because a reader who has the mechanism can predict the direction without the table.

The closure round a face of a quadrilateral mesh is a composition of four rotations, one per crease, and it has to come back to the identity. Differentiate that with respect to a panel length and the derivative is a rotation composed with the rate at which the fold angles respond to the length — and that rate is what the vertex’s own gearing sets. Near the flat sheet the fold angles are all small and the rotations are all nearly the identity, so a length can move a long way before the composition notices. Near the closed state they are not.

That is why the half-angle tangent is the right divisor rather than the angle itself. A rotation by ρ enters the algebra as tan(ρ/2), which is zero at flat and unbounded at fully folded, and the sensitivity inherits it directly.

It also explains why the effect has never been reported. Anybody measuring a mesh measures it at a comfortable fold angle, which is somewhere near the middle, and the middle is where the curve is least interesting: over 0.6 to 1.2 radians the allowance falls only by a factor of two, which reads as scatter rather than as a law.

What is being held fixed #

Three things, and stating them is what keeps the numbers about geometry.

No material property appears anywhere. No modulus, no stiffness, no yield. The residual is a mismatch in radians between the composition of rotations round a face and the identity, and the allowance is how far the lengths may move before that mismatch passes a stated number. A steel mesh and a paper mesh with the same lengths have the same allowance in this sense.

No panel has thickness. Thickness is a separate constraint with its own arithmetic, and mixing the two would produce a number that is neither.

And the budget is stated rather than derived. 0.02 radians is a choice; a different budget scales every number in the table by the same factor, because the residual is very nearly linear in the step over this range. What does not scale is the ratio between the two ends, which is the finding.

The measurement, and the two ways it could have lied #

The allowance is found by stepping. Take the mesh’s twenty lengths, move them along a stated direction by increasing amounts, recompute the worst closure residual at each step, and find where it crosses the budget by interpolating between the two steps that bracket it.

The obvious failure is a one-sided derivative read as a two-sided one. The residual is a magnitude, so it has a corner at zero: a central difference across a solution reads the two arms of a V and reports no slope at all. The sensitivity here is therefore computed one-sided, which is the same care the earlier rung had to take and for the same reason.

The other is a step that leaves the branch. A mesh solved at one fold angle and evaluated at another has to be evaluated on the same motion, and a solver asked for each angle independently will hop between configurations. Every residual here comes from the closure equations evaluated directly at the given angle rather than from a re-solve, so there is no branch to lose.

The same question asked of the other error #

There is a second thing that goes wrong when a sheet is made rather than solved, and it is worth checking whether it has the same shape.

An error in a crease’s position is not an error in a panel length: it is a reflection in a line that is slightly off, which turns everything beyond it by twice as much. That error accumulates along a chain of creases and it arrives in the columns rather than in the row it was made in, and its growth with the crease count is a different question from anything above.

But its price through the motion is the same question, and the answer is the same. A misplaced crease and a mis-cut panel both enter the closure through the same Jacobian, so both are multiplied by the same tan(ρ/2), and both are cheapest at the flat sheet and dearest at the packed one.

So the statement generalises past its own measurement. Every error a maker can make is worth least where the mechanism is doing least, and a rigid origami’s qualification test should be run closed rather than open.

What a builder should do with it #

Three things follow, and none of them requires reading the mathematics.

Quote the tolerance at the packed state. It is the tightest, it is the one the extrapolation gets wrong, and a deployable that fails does so on the way in rather than on the way out, which is also where the sheet starts meeting itself.

Do not read a mid-fold measurement as a specification. The 0.211 mm this site published at 0.6 radians is five times the allowance the same mesh has at 2.4, and a workshop working to it would produce parts that deploy and do not pack.

And spend precision by direction before spending it by amount. The directions do not change through the motion, so a decision about which lengths to measure carefully is a decision that stays correct — which is the one piece of good news in the whole measurement.