A count is neither

The survey sorts the site’s quantities by the power a scaling raises them to. Twelve rows, four values, every one an integer to fifteen figures.

There is a class of quantity it cannot sort, and it is the class the subject’s foundation is built on.

What the probe requires #

The scaling probe calls a quantity a shape when two things hold: scaling every length leaves it unchanged, and perturbing one length changes it.

The second condition is not decoration. Without it a constant would be classified as a shape, and asserting that a constant is scale-invariant is a sentence with no content. So each row is perturbed twice — everything together, and then one length by five per cent — and a row is reported only if the first perturbation moves it by nothing and the second moves it by something.

Every one of the twelve rows passes both. Their responses to a five per cent change in one length run from 0.5% to 13%.

A mobility fails the second. Change any length by five per cent and the mobility is what it was. Change it by fifty per cent and it is still what it was, right up until the mechanism stops assembling at all.

An integer is constant on an open set #

That is the property, and it is worth stating precisely because it is what distinguishes a count from a shape.

A shape is a continuous function of the parameters that happens to be invariant along one direction. Freudenstein’s K₁ moves smoothly when the crank changes and does not move when everything scales; it is a function with a level set.

A mobility is an integer-valued function. Integer-valued functions of continuous parameters are either discontinuous or locally constant, and this one is locally constant: on an open neighbourhood of any generic four-bar the mobility is 1, full stop, and it changes only at parameter values where the mechanism’s structure changes.

A quantity that is constant on an open set has no exponent, because log of it against log of anything is a horizontal line whose slope is zero for the same reason a constant’s is. The probe would report zero and the control column stops it.

The three kinds #

So the site’s quantities fall into three classes rather than two, and the third has been in plain sight since the foundation.

Sizes. Exponent 1, −1 or 2. A length, a curvature, an area. Need a size measurement.

Shapes. Exponent 0, and responsive to proportion. An angle, a ratio, an invariant. Recoverable from angles alone.

Counts. No exponent. A mobility, a number of assembly modes, a number of links, a chromatic number, a number of precision points, a Grashof class. Recoverable from anything, because they cannot be wrong by a little.

That third class is large on this site and it is where a great deal of the subject’s confidence lives. Grübler’s count, the sixteen eight-link chains, the two assembly modes of a four-bar, the forty solutions of a platform’s forward problem: none of them is a measurement in the sense the rest of this field uses.

The counts this site carries #

Worth listing, because the class is larger than it sounds and the site’s confidence rests on a good deal of it.

Mobility, by formula and by rank, on every mechanism here. The number of links and joints, which is where a formula’s inputs come from. The number of assembly modes — two for a four-bar, four for a Watt six-bar, up to forty for a Gough platform. The number of distinct chains with a given link count: one at four, two at six, sixteen at eight, two hundred and thirty at ten. A chromatic number, which says how many planes a machine’s links need. The number of precision points a synthesis may prescribe. A Grashof class, which is a label rather than a number and behaves identically.

Every one of those is an integer, every one is locally constant in the parameters, and none of them has an error bar anywhere on this site.

That is not a criticism — an integer with an error bar is usually a confusion — and it does mean a large part of what the site asserts is of a kind this field’s instruments cannot examine. A calibration recovers a count for free and can say nothing about how close it came to being a different count.

What a count is for a measurement #

The consequence for this field is the useful part and it cuts both ways.

A count is trivially identifiable. Nothing about scale, conditioning, noise or pose selection touches it. A protractor recovers a four-bar’s mobility, its Grashof class and its number of assembly modes exactly, from three readings, with any instrument whatever, because a count that is locally constant is determined by any observation that pins the parameters into the right open set.

And a count is brittle rather than uncertain. It has no error bar because it cannot be slightly wrong; what it can be is wrong, discontinuously, if the parameters cross a boundary. A four-bar whose Grashof margin is 0.002 is one measurement error away from being classified as a different kind of machine.

So the right way to report a count is with the distance to the boundary rather than with a tolerance. Not Grashof class: crank rocker ± nothing, but crank rocker, with a margin of 0.4 units on s + l ≤ p + q, and the margin is a length with exponent one and an ordinary error bar.

Those state the split at one configuration, which is where it is easiest to see and weakest as evidence. A count that survives a scaling has to survive it at every position the machine can reach, so the honest version of the claim is drawn over a whole turn rather than at a chosen angle.

An integer with an error bar is a confusion #

Worth defending the practice this essay is refining, because it is easy to read the recommendation as put error bars on counts and that would be wrong.

A mobility of 1 ± 0.3 is not a statement anybody can act on. Mobility is not a quantity that takes the value 1.3; the interval says nothing about the machine and it invites a reader to imagine a mechanism with a third of an extra freedom, which does not exist.

What a margin says is different in kind. It does not say the count might be another value; it says how far the parameters are from a boundary at which the count would be another value. That is a length or a ratio, it has an ordinary error bar of its own, and it answers the question a reader actually has: could a small manufacturing error make this a different kind of machine.

The uncertainty lives in the parameters and the count is a function of them, so the honest propagation is to report the parameters’ uncertainty and the distance to the boundary, not to smear the integer.

That is the same discipline the rank routine already applies and it generalises to every count on the site.

The margin is the measurement #

That reframing is worth pressing, because it recovers a count into the field’s own vocabulary.

Every count on this site is the sign of something, or a rank, or a solution tally, and behind each of them is a continuous quantity that decided it.

Grashof’s class is the sign of s + l − p − q. That difference is a length, exponent one, and the site computes it. Its magnitude says how far the machine is from being a different kind of machine.

Mobility by rank is the number of singular values of the constraint Jacobian above a cut. The ratio between the smallest kept and the largest discarded is a dimensionless margin, and this site has reported it since the foundation precisely because a rank without a gap is a decision rather than a measurement.

A count of assembly modes is a count of real roots, and behind it is the discriminant, whose vanishing is where two roots merge.

In every case there is a continuous quantity with an exponent, and reporting it alongside the count turns an integer into a measurement with a margin. The count is the answer and the margin is the evidence.

Two counts that a measurement cannot settle #

The claim that a count is trivially identifiable needs one qualification, and it is a real one.

A count is identifiable when it is a function of the parameters and the parameters are in a known open set. Mobility, Grashof class and link count are all of those.

Some counts are not functions of the parameters at all. Which of two assemblies a machine is in is one: the parameters permit both and the machine is in one of them, and no measurement of the input and output angles distinguishes them. Which of three cognate linkages traced a curve is another: three different parameter vectors, all consistent with the data.

Those are counts of alternatives rather than counts of a machine’s structure, and they behave completely differently. A structural count is free; a count of alternatives is a statement that a measurement has several answers and that the additional information needed is of another kind.

Both are integers with no error bars and only one of them is settled by measuring. Filing them together — which the word count invites — loses the distinction that decides whether more measuring helps.

Where counts and shapes part company #

One case where the distinction is not academic, and it is one this site has already met.

Grübler’s formula declares a working mechanism immobile: three parallel bars, five links, six pins, count zero, Jacobian rank one. Two counts, disagreeing, both integers, neither with an error bar.

That disagreement is resolved by the margin: the rank’s smallest kept singular value is comfortably above the cut and the gap is large, so the rank is a measurement and the formula is a prediction that is wrong. Without the margin the two counts are simply two assertions.

A count’s authority comes from the continuous quantity underneath it, and where two counts disagree the one with a measured margin wins. That is a rule this site arrived at empirically in its earliest essays and it is what this essay’s classification says in general.

Three kinds of count, by where the integer comes from #

Sorting the counts themselves turns out to be worth doing, because the three kinds behave differently under measurement.

Counts of structure. Links, joints, loops. These are properties of a graph and have no parameters at all, so no measurement of a machine’s motion touches them and none needs to: anybody can look at the machine and count. The topology field’s enumerations are of this kind.

Counts derived from parameters. Mobility by rank, Grashof class, the number of real assembly modes. These are functions of continuous parameters, locally constant, with a boundary somewhere — and a margin to that boundary that can and should be reported.

Counts of alternatives. How many machines fit the data. Not a function of the parameters, not settled by measuring more, and the subject of the two ambiguity essays in this field.

The first is free and uninteresting to a metrologist. The second is free and interesting, because its margin is a measurement. The third is not free at all and is where a calibration’s real difficulties live.

Only the middle kind has an exponent-shaped question to ask about it, and even there the answer is about the margin rather than about the count.

The mobility of a scaled machine #

For completeness, the trivial statement the probe would have made had it been allowed to.

Scale a mechanism and its mobility is unchanged. Scale it by a thousand or by a thousandth and every constraint is still the same constraint, the Jacobian’s entries scale but its rank does not, and the formula’s inputs are counts of links and joints which have no size at all.

So a mobility is scale-invariant, and saying so is true and uninformative. The probe declines to say it because the same sentence is true of the number seven.

That is the whole of what the control column is protecting against, and it is worth having as a general caution about invariance arguments: an invariance is only evidence when the quantity varies. A proof that a quantity is unchanged under a group action says nothing until it is also shown to be changed by something.

A rank is a count with a margin already attached #

The best-behaved count on this site is worth holding up as the model for the rest, because it already does what this essay recommends.

Mobility measured by the rank of the constraint Jacobian is an integer, and the routine that computes it also returns a gap: the ratio of the smallest accepted pivot to the largest rejected one. A large gap means the rank is unambiguous; a gap near one means the matrix is near-singular and the answer depends on the tolerance.

That is exactly a count with the evidence for it. The integer is the answer, the gap is how far the answer is from being a different answer, and a reader can see whether to believe it.

The mobility machinery here has reported it that way from the start, for a reason that had nothing to do with this field: an overconstrained mechanism’s rank decision is genuinely close, and reporting it without the gap would be reporting a convention. The habit was already right and this essay is a name for it.

What the rest of the site’s counts lack is the equivalent. A Grashof class is reported without its margin, an assembly-mode count without the discriminant, a chain enumeration without any statement of how it would change under a perturbation — though for that last one the answer is that it would not, since a chain count is combinatorial and has no parameters at all.

Why the class was not obvious #

A closing observation about why this took a survey to notice, since the fact that a mobility is an integer is not news.

The site has always treated its counts differently from its measurements without saying so. A mobility is printed as 1; a transmission angle is printed as 42.7° with the configuration it was computed at. Nobody would put an error bar on the first and everybody expects one on the second, and the difference was carried entirely by convention.

What the survey does is force the question. Its probe has to decide, for each quantity, whether it is a shape or a size, and a mobility answers neither — so the probe either mis-files it or declines, and the declining is what makes the third class visible.

An instrument that has to classify everything finds the things that do not classify, which is a general reason to build instruments that are exhaustive rather than selective. The same thing happened when the ancillary check asked whether every generator was listed and found fifty-two that were not.

The classification is not a discovery about mobilities. It is a discovery about what the site’s own vocabulary had been doing silently for twenty-five fields.

What this adds #

Three things, all small and all useful.

A third class in the survey, so that a reader asking what happens to this number under a scaling has somewhere to put a count rather than mis-filing it as a shape.

A reporting rule. A count should carry the margin of the continuous quantity that decided it. That is one extra number, the site usually computes it already, and it is the difference between an integer and an integer with evidence.

And a caution about the probe itself. The scaling survey is a good instrument for continuous quantities and it is silent on the discrete ones, which are a large part of what this site asserts. Knowing which questions an instrument cannot answer is the same discipline as knowing which combinations a matrix cannot recover, one level up.

The three together are a habit rather than a result, and the habit is the thing to carry. Before believing a number, ask what kind of number it is: whether it moves when the machine is scaled, whether it moves when the machine’s proportions change, and whether it can move at all. Three questions, all answerable by two perturbations, and the answers decide what a measurement of it could ever mean.