The error that is repeated

This site’s practice field has a standing argument about stacks of dimensions. Add four lengths, each with a tolerance, and the worst case is the sum of the tolerances while the statistical estimate is the square root of the sum of their squares — and on four terms the two differ by a factor of two, which is the usual reason a designer reaches for the second. Where the errors accumulate through a mechanism rather than along a line the same choice has to be made again.

A network of repeated units is that argument with thirty-two terms instead of four, and with a reason to believe the terms are not independent.

The measurement #

A lazy tong of $n$ units spans $2nL\cos\theta$, exactly. Cut one unit’s bars so that its opening angle is a hundredth of a radian away from the drawing and the span is out by that unit’s share: 0.0117 in units of the bar length, at an opening of 0.62 radians, and it is 0.0117 whether the tong has two units or thirty-two.

Cut every unit that way and the errors add.

0.0234 at two units, 0.0468 at four, 0.0936 at eight, 0.187 at sixteen, 0.374 at thirty-two. The ratio of the second column to the first is 2.000000, 4.000000, 8.000000, 16.000000 and 32.000000 — the unit count, to six figures, because the mechanism is linear in the openings and an error goes through the same multiplication the motion does.

The third column is what independence would give: 0.0165, 0.0234, 0.0331, 0.0468 and 0.0662. At thirty-two units the two differ by $\sqrt{32} = 5.66$.

What one unit’s error is worth #

Before the accumulation, the single-unit case, because it is the baseline everything else is a multiple of and it is worth having exactly.

A tong unit spans $2L\cos\theta$. Differentiating, an error $\delta\theta$ in the opening angle gives a span error of $2L\sin\theta \cdot \delta\theta$ — so the sensitivity is not a constant: it vanishes at $\theta = 0$, where the unit is fully extended, and is greatest at a right angle, where it is fully closed.

At the opening used throughout, 0.62 radians, $2\sin(0.62) = 1.1616$, so a hundredth of a radian gives 0.0117 of a bar length. That is the number in the first column of the table, and it is a number about one unit measured at one opening.

Two consequences follow that are worth separating from the accumulation argument. A tong is most sensitive to its own dimensions where it is most closed, so an assembly specified at full extension is being specified at its best case. And the sensitivity is a derivative, so it is the right quantity for a small error and the wrong one for a large one — the same distinction the practice field draws between a sensitivity and a swept band, and here the errors are small enough that the derivative is exact to six figures.

Which column applies #

That is not a question about the mechanism. It is a question about how the parts were made, and for a repeated unit the answer usually goes the wrong way.

The statistical estimate assumes the errors are independent draws from a distribution centred on nought. Thirty-two units cut on one setting of one machine are not that. They are thirty-two copies of whatever that setting was wrong by, with the same sign and very nearly the same magnitude — one draw, used thirty-two times.

The same is true of a mould, a die, a jig, a program, a tool that wears in one direction, a drawing with an arithmetic slip in it. Every one of those produces correlated errors, and every one of them is the normal way to make many copies of one part. The independence assumption is doing all the work in the statistical estimate, and repetition is exactly the circumstance in which it is least defensible.

There is a version of that in the practice field already: a stack-up stops working when the terms are not independent, and the diagnosis there is the same. What is new here is that the terms are not merely correlated by accident — they are copies by construction.

What it does to a specification #

The practical shape is a tolerance that gets harder with size.

A designer specifies a tolerance on the finished span. On a one-unit mechanism that translates straight into a tolerance on the part. On a tong of $n$ units it must be divided by $n$, because the part’s error is multiplied by $n$ on the way back.

A tong of thirty-two units held to a span tolerance of one millimetre needs its units to a thirty-second of a millimetre. The same tong of two units needs a half. Linearly harder with size, which is the worse of the two available behaviours; had the errors been independent it would be harder as the square root, and thirty-two units would need a fifth of a millimetre rather than a thirty-second.

That is also the answer to a natural hope. A tong is symmetric, so it is tempting to think an error one way somewhere will be cancelled by an error the other way somewhere else. It will — if the errors are independent, which is precisely what the third column is. The mechanism itself has no way of cancelling anything: the span is a plain sum of the units’ contributions and nothing feeds one unit’s error back to another. Compare Peaucellier’s linkage, where the straightness is an algebraic identity that survives a whole family of link lengths. A sum of thirty-two wrong numbers is wrong.

The other failure, which is not an inaccuracy #

The tong’s error shows up as a wrong length. That is a mechanism working slightly incorrectly, which is the ordinary tolerance situation and the one this field has spent eleven rungs on.

An assembly whose constraints repeat one another fails differently, and the difference is a change of kind.

A deployable ring of eight pairs works because its elements’ kink angle is exactly $180° - 360°/8$. Get it wrong by a degree and the ring does not open badly — it does not go together. The four dependencies among its constraints vanish, the rank rises to full, and the count of nought becomes the truth.

The same for a folded sheet. A pattern’s shape has to satisfy one condition for each dependency among its constraints, which is $(n-2)^2$ on a Miura sheet: one at nine panels and a hundred at a hundred and forty-four. A drawing that misses them by a manufacturing tolerance is off the foldable surface by that tolerance, and the residual is what a fold would have to absorb.

$5.9 \times 10^{-6}$ at a fold of 0.02 radians on a grid displaced by a tenth of a panel; $1.6 \times 10^{-3}$ at half a radian. There is no fold, at any angle, and no amount of solving produces one.

How wrong a pattern may be #

The folded sheet allows a more direct version of the question than the ring does, because its conditions can be counted and its residual can be measured against the size of the error that caused it.

Displace the interior vertices of a three-by-three Miura pattern by a tenth of a panel and the best closure it can reach at a fold of 0.3 radians is $5.6 \times 10^{-4}$. That is not a small number in this field’s terms: the same routine reaches $10^{-16}$ on the undisplaced pattern.

What the residual is measuring is the amount of closure error a fold would have to absorb, and there is nothing in a rigid-panel mechanism to absorb it with. Real card absorbs it by bending, which is why a badly drawn pattern can still be folded by hand and why the failure is invisible until somebody tries to make the same thing out of panels and hinges.

The scaling of that with the error is worth stating carefully because it is the part a tolerance argument needs. The residual grows monotonically with the fold — $5.9 \times 10^{-6}$ at 0.02 radians, $1.6 \times 10^{-3}$ at 0.5 — so a pattern that is off the surface is worst at full fold, which is generally where a deployable is asked to work. A tolerance that looks acceptable at a small fold is not a tolerance on the mechanism.

Two questions, and which one to ask #

So an assembly of repeated units has two tolerance questions and they are answered differently.

How far off is the output? That is the ordinary one, and for a network it is the ordinary answer multiplied by the unit count — with worst case as the honest model rather than the pessimistic one.

Will it assemble at all? That question does not arise for a four-bar, whose four lengths always give some mechanism, and it is the governing question for anything whose count is wrong. It has a sharp form: the assembly’s dependencies are conditions on its dimensions, and a dimension outside tolerance is a condition unsatisfied.

Reading the redundant column as a tolerance risk is the practical form of everything the network field measures. Nought means the assembly will go together whatever its parts turn out to be, and it is why a lazy tong is easy to make. Four means four relations have to hold. A hundred means the drawing has to be computed rather than drawn.

Where the site has been here before #

This is the practice field’s twelfth rung and it is worth locating against the eleven before it, because the change is not one of degree.

A length is a range opened the field by giving every dimension a tolerance and asking what the output does. The four lengths do not matter equally measured which dimension the answer is most sensitive to. Seven lengths and a hundred corners took the worst case seriously by enumerating them. In every one of those the mechanism exists and the question is how far off it is.

What a network adds is a class of assembly for which the prior question — is there a mechanism — is not automatic. Nothing in the first eleven rungs needed it, because a four-bar with any four lengths that close is a four-bar. A deployable ring with the wrong kink is not a worse ring; it is a set of parts.

That is why the redundancy count deserves to be read as a tolerance risk rather than only as an arithmetic curiosity. It is the count of relations the parts have to satisfy, and it is available from the same rank that produces the mobility.

What a real assembly does about it #

Two things, and both are recognisable from real mechanisms rather than being deduced here.

Slots and adjusters. Giving one joint per unit a slot instead of a hole adds a freedom per unit and destroys the dependencies deliberately: the assembly then goes together at the cost of being a looser mechanism, which is a clearance treated as a link applied on purpose. That is why deployable structures are so often full of adjustment.

One part, many times. The other route is to accept the correlated error and design so that it does not matter — to make the assembly’s behaviour depend on quantities that are the same for every unit rather than on differences between them. A ring’s kink angle is one number on one part; if every element is cut on the same fixture the angle is wrong by the same amount everywhere, and a symmetric error in a symmetric assembly changes what the mechanism is rather than whether it exists. That is a much better failure than the alternative and it is available only because the units are identical.

The count of terms, and what it does to the choice #

There is one more thing the repetition does to the argument, and it cuts against the statistical estimate a second time.

The factor between worst case and the square-root estimate is $\sqrt{n}$. On four terms that is 2; on thirty-two it is 5.66; on a hundred and forty-four panels’ worth of dimensions it would be 12. So the two models diverge exactly as the assembly grows — which means the decision between them is least consequential where it is best justified and most consequential where it is least.

A designer choosing the statistical estimate on a four-term stack is making a modest bet on independence. The same designer choosing it on a thirty-two-unit assembly of identical parts is making a large bet on an assumption that repetition specifically undermines.

The honest position is the one this field takes on its ledger: report both, and say which model the numbers came from. The worst case is arithmetic and needs no assumption; the statistical estimate needs a distribution nobody has measured, and the difference between them is a number rather than a matter of judgement.

There is a third position and it is worth naming because it is what a careful designer actually does. Measure the parts. Thirty-two units cut on one setting have an error that is nearly a constant, and a constant can be found by measuring one of them — at which point the accumulated error is not a tolerance question at all but a known offset, and the assembly can be built to allow for it. That is available precisely because the units are identical, and it is the only good thing repetition does for a tolerance.

A tenth of a correlation doubles the answer #

The two columns are the two extremes — perfectly correlated and perfectly independent — and a real process is neither. What happens in between is worth computing, because the answer is not halfway and the asymmetry decides which model a designer should default to.

With $n$ units each carrying an error of standard deviation $\sigma$, and a correlation $\rho$ between any two of them, the span error’s standard deviation is $\sigma\sqrt{n + n(n-1)\rho}$. At $\rho = 0$ that is $\sigma\sqrt{n}$, the comforting column; at $\rho = 1$ it is $\sigma n$, the honest one. In between the correlated term carries a factor of $n(n-1)$ against the independent term’s $n$, which is what makes the middle so unbalanced.

Put thirty-two units and a correlation of a tenth into it: $\sqrt{32 + 32 \times 31 \times 0.1} = \sqrt{131.2} = 11.45$, against $\sqrt{32} = 5.66$ for independence. A correlation most people would describe as negligible has doubled the accumulated error, and it has done so because the covariance terms outnumber the variance terms by $n - 1$ to one.

The threshold falls out of the same expression. The correlated term dominates once $\rho > 1/(n-1)$, which at thirty-two units is 0.032. So on an assembly of this size, a correlation of three per cent is where the statistical estimate stops being approximately right, and anything above it is closer to the worst case than to the square root.

That settles which model to default to, and the answer is not the one the practice field’s four-term stacks suggest. On a short stack, independence is a modest bet: at four terms the threshold is a third, and a process would have to be strongly correlated to break it. On a long one the threshold collapses as $1/n$, so the longer the assembly the more nearly the worst case applies, whatever the process is doing.

Which turns the essay’s argument into something a designer can act on without knowing $\rho$. The correlation does not have to be measured, only bounded: a process that cannot be shown to have $\rho$ below $1/(n-1)$ should be costed at the worst case. And for thirty-two units cut on one setting of one machine, $\rho$ is not near 0.03 — it is near one, which is where this essay began.

What is not measured here #

Force, as everywhere in this field. Whether a slot slips, whether an adjuster holds, and what a redundant constraint does to the load in a member are all outside a field whose quantities are lengths and ranks.

And a distribution. The third column above is what independence would give; it is arithmetic on an assumption rather than a measurement of anything. What is measured is the first two columns, and the point of putting the third beside them is to show how much the assumption is worth — a factor of 5.66 at thirty-two units, in the direction that flatters the design.