The number on the box

Everything in this field is a mechanism most readers have touched. A car’s front suspension, the linkage that turns its wheels, the lift a warehouse uses, the hinge on a kitchen door, the latch on a folding table, the chain on a bicycle, the reduction in a robot’s elbow, the rocker over an engine’s valve.

They have one thing in common, and it is not their mechanism. It is that each arrives with a number. The suspension has a roll centre height. The steering has a percentage of Ackermann. The lift raises so many millimetres per millimetre of ram. The hinge opens to 95°. The chain is 53/11. The rocker is a 1.6. Somebody printed each of those on a specification sheet, and somebody else read it and made a decision.

This site can check them, and that is the whole of the field. Every one of those numbers is a claim about a mechanism whose geometry is known, and the eleven fields before this one have built the machinery to solve exactly that kind of geometry. Nothing new is needed. What is new is the comparison.

Five things a number can be #

The first surprise is that “is it right?” turns out to be the wrong question, because it presumes the mechanism has one of whatever is being quoted. Running the fourteen through gave five distinct answers, and the distinctions are not pedantic — they change what can be done with the figure.

Exact. The mechanism has the number, at every position, and no measurement can disagree. There are three of these, and the reason they are the three is the subject of the closing essay.

A mean. The mechanism has a function and the quoted number is its average over a cycle. A 53/11 chain drive turns the back wheel 4.818 times per turn of the pedals, to five decimal places, and does not have that ratio at any instant within the turn.

Bounded. The claim is false and the error has a stated bound that nothing cares about. A Watt’s linkage does not hold an axle still. It holds it to thirty-four microns over eighty millimetres of travel, which is a different sentence and a much more useful one.

An operating point. The number is the value of a varying quantity at one position, and it is honest exactly as long as the position is stated. A rocker ratio is this. So is a camber gain.

Quoted. The mechanism has no such quantity, at any position. The number names something that does not exist to be measured. Six of the fourteen are here, and this verdict is this site’s own — it went into the refutation index during the practice phase as a number where there is no number, on the strength of two examples. In this field it stops being a curiosity and becomes the most common case.

The three that survived #

Three rows are exact, and they have something in common that took the whole field to notice: all three are counts.

A compound epicyclic’s reduction is 1 / (1 − z₁z₄/z₂z₃), where the four z are tooth counts. The tooth counts are integers. The reduction is therefore a ratio of two integers — 2176/106 for the set drawn below — and it is exactly that ratio at every position of the drive, forever, on every unit that leaves the factory. There is nothing to measure and nothing to disagree with.

A scissor lift has one degree of freedom. Not approximately one; one. Mobility is a count, and the count that matters is the rank of a matrix, which is an integer.

A kinematic coupling is exactly constrained. Six contacts, six freedoms taken, none left and none doubled — again a rank, again an integer.

That is not a coincidence and it is not a consolation prize. A quantity that is a count cannot drift, cannot be a function of position, and cannot be quoted at an operating point, because it has no position-dependence to have an operating point in. Everything else in the field is a length, an angle or a ratio of two of those, and every one of those varies as the mechanism moves — because that is what a mechanism is.

What a quoted number looks like when it is swept #

The clearest of the six is the roll centre, because it is a construction rather than a part. Take a double wishbone: the wheel’s upright is the coupler of a four-bar, so at every instant it is turning about some point — the instantaneous centre — and the roll centre is where the line from that point to the tyre’s contact patch crosses the centreline of the car.

Every step of that is well defined and computable. What is not available is a height, because the answer moves.

The strut version is worse: over the same travel its roll centre runs from 121 mm to below the ground. The two suspensions are different mechanisms with different curves, and quoting either as a height compresses a curve into a point and then treats the point as a property.

This is exactly the ratio that is not a number, which the wrong field opened with in a gear train, arriving in a place where somebody has to sign a drawing.

The chain, which is the honest middle case #

A bicycle’s 53/11 is not a lie and it is not a quoted number either. It is a mean, and the mechanism underneath it is a good deal stranger than the ratio suggests.

A chain does not run on a circle. Its pins sit on a polygon, and the strand leaves along a tangent, so the effective radius of the sprocket varies within every single tooth — from R cos(π/n) up to R and back. On an eleven-tooth sprocket that is a four per cent swing in chain speed, once per tooth, eleven times per turn.

The mean being exactly the tooth ratio is not a modelling assumption; it follows from both sprockets having to pass the same pins. That is what makes “mean” a real verdict rather than a hedge: the quoted number is provably the average of the thing that varies, so it is the right number for a gear chart and the wrong number for anything that happens within a tenth of a turn.

Operating points, and the honest way to quote one #

A rocker arm over an engine valve is sold as a ratio: 1.6, or 1.65, or 1.5. What that names is the ratio of two moment arms measured at one position of the rocker — and the rocker swings twelve degrees while the valve opens, which moves both arms.

An operating-point number is honest if the operating point is stated and useless if it is not. The rocker’s is stated by convention — everybody in the trade knows a rocker ratio is quoted at zero lift — which is why the verdict here is point rather than quoted. The camber gain in the ledger is the same kind of number and is usually quoted with no operating point at all, which is how a rate that runs from −0.034°/mm to something quite different gets treated as a constant.

What the field does with this #

Each of the fourteen rows is an essay’s worth of mechanism, and the twelve essays after this one take them in turn. They are not arranged by machine. They are arranged by which of the site’s existing ladders they deepen, because that is what they are: a scissor lift is a mobility count with a roller in it, a steering trapezoid is a four-bar used as a function generator, a chain drive is a tooth question, a cabinet door is a coupler’s path with a body attached.

There is a second reason the field is arranged that way, and it is a caution rather than a plan. A field organised by machine would be a list, and a list has no end: there is always another mechanism somebody sells. A field organised by which claim it tests has a boundary, and the boundary is the last essay’s subject.

What this is not #

It is not an accusation. Nobody printing “roll centre 73 mm” on a drawing is being dishonest; they are quoting the value at the design ride height because that is what the convention is, and everybody who reads the drawing knows it. The failure this field is about is quieter: a number that starts as a convention gets used as a property, and then somebody computes with it.

It is also not a claim that the measurements here are what a real machine does. Every number in the right-hand column is a kinematic one. The wheel that scrubs 12 mm across the road here does so with no tyre, no compliance and no load; the chain that fluctuates by four per cent does so with no tension and no mass; the latch that unlocks after seventy microns of slider travel is not resisting anything. The boundary drawn two fields ago is exactly where it was.

What has changed is which side of that boundary the reader’s own machines sit on. Rather more of them than the site expected.

How the ledger is kept honest #

A table comparing claims with measurements is only as good as the second column, and the second column is the one that is easy to get quietly wrong. Three things keep it from being a list of assertions.

Nothing in it is typed. Every figure in the right-hand column is produced by a function in this repository at the moment the page is built. There is no data file of results. If the suspension’s geometry changes, the roll centre range in the ledger changes with it, and so does every sentence in every essay that quotes it, because those sentences quote the figure’s own caption or are checked against it.

Each row names the code that produced it. The from column — visible under each measurement — is the library and the function. That is not decoration: it is what makes a wrong number findable. A measurement whose provenance is “somebody computed this once” is indistinguishable from a measurement that was mistyped, and this site has published a number that was wrong by two thirds precisely because the provenance had been lost between one phase and the next.

Each measurement has a second route wherever a second route exists. The instantaneous centre in the suspension is found from the solved velocity field and, independently, by Kennedy’s construction — they agree to 1.1 × 10⁻¹⁵. The chain’s mean ratio is found by integrating a velocity law and compared with the tooth counts, which the integration is never given. The contact sensitivity of a kinematic coupling is found by differentiating the contact equations and, separately, by rotating the part with a real rotation matrix and re-seating it, which agree to second order in the error and disagree at first order in a way that halves when the error halves. Two routes cost twice as much and are the only thing that has ever caught a wrong answer here.

The third of those is where most of the phase’s effort went, and it is worth stating why. An assertion that a number is correct is usually an assertion that two computations agree, and if both computations share a subroutine they agree about the subroutine’s mistakes as well. The synthesis-depth phase found four Jacobian entries with the wrong sign that had survived four phases precisely because every check of the velocity went through the same matrix. What broke it open was a route that shared no code.

Why the exact ones are counts #

Prefer a count is offered above as a habit, and it is worth saying why it works, because the reason is structural rather than a matter of counts being tidier.

A mechanism’s configuration is a point in a space, and a quantity computed from the mechanism is a function on that space. A function generally varies. So the default expectation for any quantity — a ratio, a roll centre, a mechanical advantage, a lift — is that it is a function of position, and quoting a single number for it is quoting the value of a function at an unstated argument. Five of this field’s categories are five different relationships a single number can have to a function, and only one of them is equality.

A count has no such difficulty because it is not a function of the configuration at all. A tooth count, a degree of freedom, a number of contacts, a rank: none of these can vary as the mechanism moves, because none of them is computed from where anything is. They are properties of the arrangement, and the arrangement does not change during the motion. That is the whole of the reason the three exact rows are the three counts, and it is the topology field’s boundary arriving in a field about advertising copy.

Which sharpens the habit into something usable. A quantity that survives a sweep is a quantity with no lengths in it, and a designer wanting a specification that can be quoted honestly should look for a way to make the requirement turn on an integer. A reduction of exactly 3.000 is available if it is a ratio of tooth counts and is not available if it is a ratio of pitch radii, and the difference between those two sentences is the difference between an exact row and a quoted one.

There is an exception and it is worth naming rather than hiding, because it is the site’s own most celebrated result. Peaucellier’s cell draws an exact straight line, and there is nothing but lengths in it. So exactness from lengths is possible — it just requires a special relation among them, an identity that holds for one arrangement of dimensions and fails for every neighbour of it. That is exactness of a second kind, and the two behave completely differently under manufacture: a count is exact for free and stays exact when the parts are made badly, while an identity among lengths is exact on the drawing and holds in the workshop to whatever the tolerances allow.

So the habit has two tiers. Exact-by-count is robust and is what a specification should be built on. Exact-by-identity is real, is beautiful, and degrades to bounded the moment anything is manufactured — which is precisely the verdict this field’s ledger gives such mechanisms when they are measured rather than derived.

What to carry into the rest of the field #

Three habits, and they are the same three the site has been using since the foundation, pointed at a new kind of claim.

Ask what varies. Not “is the number right” but “what is the mechanism’s position, and does this quantity depend on it?” Six of fourteen rows fail at that question before any arithmetic starts.

Sweep before quoting. Every “quoted” verdict in the ledger was found by taking a number, asking what its value is at other positions, and discovering there were other values. The instrument is the same sweep every other field here runs.

Prefer a count. Where a design can be made to turn on an integer — a tooth count, a rank, a number of contacts — the resulting number is exact and stays exact. That is not a design rule this site invented; it is what gear trains, kinematic couplings and mobility criteria have in common, and it is the one piece of advice the field’s ledger supports without qualification.

The closing essay comes back to the ledger and asks the fair question in the other direction: which of the quoted numbers survived, and what the survivors have in common besides being counts.

One further consequence of the same argument, and it explains a pattern in which mechanisms this field found guilty. The six rows naming a quantity the mechanism does not have are all constructions rather than parts — a roll centre, a ratio between two moving points, a centre of rotation — and a construction is a function of position by definition, since it is computed from where things are. A part is not: a tooth is a tooth wherever the wheel has turned to. So the field’s verdicts sort almost perfectly by whether the quoted quantity names something that exists in the machine or something derived from a snapshot of it, and that is a question a reader can ask about any specification without measuring anything.