A body is all size

Run the scaling probe over the bodies field and every row comes back with an exponent. There is no zero column.

Every quantity has a dimension #

A clearance is a distance between two parts. Exponent one. This site’s own crank rocker has a coupler inside its frame bar by 0.432 at a half-width of 0.16, and both numbers are lengths.

A penetration depth is the same quantity with the other sign. Exponent one.

A swept area is an area. Exponent two — the field measures 4.2349 by lattice count against πR² = 4.2478 for a disc, and doubling the machine quadruples both.

A footprint’s box is a pair of lengths, and the fill — how much of the box the material occupies — is a ratio and is the field’s one shape.

A boss radius, a pin spacing, a bearing pedestal’s admissible width, a stowed height — all lengths.

And the plane count is a count, unchanged by anything, in the third class.

So the field’s output is lengths, one area, one ratio and some counts. That is a different profile from every other field on the site, and it follows from what the field is: a link with a width, and a width is a length.

It is worth being clear that this is not a criticism of the field’s choices. The quantities it computes are the quantities its questions have — do these two parts collide is a question with a distance for an answer, and there is no dimensionless way to ask it. What the classification adds is a warning about carrying the answers anywhere.

A penetration depth is the same quantity with the other sign. Exponent one.

A boss radius, a pin spacing, a bearing pedestal’s admissible width, a stowed height — all lengths.

And the plane count is a count, unchanged by anything, in the third class.

Why the other fields are not like this #

Worth setting the contrast out, because it says what makes this field’s results hard to transfer.

Every other field computes something about a mechanism’s motion: where it goes, how fast, at what angle, in what ratio. Motions are described by angles and ratios, and angles and ratios are dimensionless.

This field computes something about a mechanism’s material: how much room it takes, how close two parts come, what box it fits in. Material is described by distances.

There is a second reason and it is the sharper one. Every other field’s quantities are functions of the mechanism’s own parameters alone, so a scaling acts on everything at once and the ratios survive. This field’s quantities compare the mechanism against something brought to it — a half-width from a bar-stock catalogue, a boss radius from a pin’s diameter, a stud somebody bolted to the bench. A comparison between a machine and an outside object cannot be scale-free, because scaling the machine does not scale the object.

So the division is not arbitrary. A field about motion produces shapes and a field about matter produces sizes, and the bodies field is the site’s only field about matter.

Worth setting the contrast out, because it says what makes this field’s results hard to transfer.

The exponents in full #

Running the probe over the field’s own quantities gives a table with no zero row but three exponents in it.

clearance between two parts          1
penetration depth                    1
boss radius                          1
admissible pedestal width            1
pin spacing                          1
stowed height of a scissor stack     1
guide length for a given stroke      1
face width a flat follower needs     1
swept area                           2
footprint box area                   2
profile curvature at a corner       −1
fill fraction                        0
plane count                        none

Eight rows at exponent one, two at two, one at minus one, one shape and one count.

That distribution is worth comparing against the site-wide survey, whose twelve rows are six shapes, four lengths, one area and one curvature. The site is mostly shapes and this field is mostly lengths, and the two profiles could hardly be further apart.

The Lipschitz bound is a rate #

One row deserves separating because it is the field’s cleverest quantity and its exponent is not obvious.

The sweep certificate rests on a bound V on how fast the gap between two parts can change per radian of drive — measured at 1.256 per radian on the site’s own machine. A radian is dimensionless, so V is a length per unit of nothing, which is a length. Exponent one.

The certificate’s conclusion — that a sweep at n samples missed nothing — compares V times a sample spacing against a measured gap. Length against length, so the comparison is a ratio and is dimensionless.

So the bound is a size and the certificate is a shape. A machine scaled up needs the same number of samples to certify, because both sides of the comparison scale together, which is a genuinely useful and non-obvious result: the sample count is a property of the design rather than of its size.

What a protractor gets, which is none of it #

Apply the field’s own question and the answer is stark.

An angle-only measurement of a mechanism recovers three of its four lengths’ worth of information — its shape — and no size. Every quantity in this field is a size. So an angle-only measurement recovers nothing at all about the bodies field’s results.

Not a clearance, not a footprint, not a swept area, not a stowed height. Every one of them needs the machine’s size, and the size needs one reading with a length in it.

That is the sharpest contrast the survey has produced. The transmission field’s whole output is recoverable from a protractor; this field’s whole output is not recoverable from any number of protractors.

The half-widths are not the machine’s #

There is a second reason the field’s results do not transfer, and it is independent of the first.

Every clearance in this field is computed at a stated link half-width — 0.16 on the site’s own machines — and that half-width is not a property of the mechanism. It is a choice about how fat the bars are, made separately from the lengths, and it is what turns a distance constraint into a body.

So a clearance is a comparison between something that scales with the machine (the joint positions) and something that does not automatically (the half-width). Scale the mechanism and hold the bars’ thickness and every clearance improves; scale both and clearances scale with everything else.

Which of those a designer means is the practically important question and it is exactly the question the field does not currently ask. A machine at twice the size built from the same stock bar is a different design from one at twice the size built from bar twice as thick.

Those two ask how fat one bar may be. The same length asks a harder question as soon as there are two of them: what a construction is prepared to put between two pins is a distance, what the two pins themselves need is a distance, and the comparison between them is where a body’s size stops being a detail of the drawing and starts deciding whether the machine can be built at all.

Where a shape hides inside a size #

Several of the field’s results have a dimensionless statement hiding in them and the field states the dimensional one, and it is worth pointing at three.

The gap’s corners. The gap between two parts is piecewise smooth with a corner wherever the closest pair of features changes, and the site reports four corners on a four-bar passing a post, with the minimum at 2.11 radians rather than at an end. The gap values are lengths; the crank angles at which the corners fall are dimensionless. Half the result transfers and half does not.

The colouring. Of a crank rocker’s six proper three-plane colourings, two are buildable, and of Peaucellier’s 192, exactly 96. Those are counts, so they transfer entirely — and the overlap measurements that produce the conflict graph are lengths, so which graph a machine has can change with the half-width.

And free space. Two studs cut a four-bar’s turn into two arcs covering 74% of it. The 74% is a fraction of a turn, dimensionless, a shape; the studs’ positions and radii are lengths.

The pattern is that the field’s conclusions are often shapes computed from ingredients that are sizes, so a conclusion transfers only if the ingredients are scaled together — which for a machine and its studs means scaling both.

Nothing about this makes the field less useful #

A note before the recommendation, because an essay saying a field’s results do not transfer reads as a complaint.

The bodies field found things nothing else could. That this site’s own four-bar cannot be built in the plane it has been drawn in for twenty-three fields, by 0.432 at a half-width of 0.16. That full rotation and a bearing at each pivot are incompatible. That a twelve-sample sweep can report a machine clear by 0.007 when it is 0.010 inside a stud.

Every one of those is a statement about one machine at one size and every one of them is a discovery about mechanisms in general, because the shape of the finding transfers even when the number does not. A four-bar’s coupler crossing its frame bar is a fact about four-bars; 0.432 is a fact about this one.

A field of sizes still produces general knowledge, and what does not transfer is the arithmetic rather than the lesson. The recommendation below is about making the arithmetic portable too.

The one dimensionless number #

The field has one and it is the useful one.

The fill — the fraction of a machine’s bounding box that its material occupies — runs 0.50 to 0.78 across the site’s nine machines. It is a ratio of an area to an area, exponent zero, and it is a genuine shape: unchanged by scaling and responsive to proportions.

So the fill transfers between machines of any size and every other quantity in the field does not. A machine that fills half its box does so at any scale; one whose coupler is 0.432 inside its frame bar does so at one scale and by twice as much at twice the scale.

Every quantity in this field would transfer if it were divided by a length, and the natural length is the machine’s own frame bar or its box’s diagonal. A clearance of 0.432 units on a machine with a 4-unit frame is a clearance of 0.108 frame bars, and that is a number about a design.

The field quotes the first form throughout. That is correct for the machines it draws and it is the reason none of its results can be used on a machine of another size without redoing the computation.

Two ways to make a machine bigger #

The half-width point deserves a section because it is the field’s own version of a distinction the whole survey keeps drawing.

Scale everything. Multiply the lengths and the half-widths together. Every clearance scales, every area scales as the square, the fill is unchanged, the plane count is unchanged, and the machine is the same design at a different size. That is a similarity and it changes no design decision.

Scale the lengths and hold the bars. Multiply the joint positions and keep the stock bar. Every clearance improves — the parts are the same thickness and further apart — the fill falls, and a machine that could not be built in one plane may now be buildable in fewer.

The second is what actually happens when a design is scaled up, because bar stock comes in sizes and a designer reaches for what is available. It is not a similarity and it produces a genuinely different design.

The field’s results are stated for the first operation and the second is what a shop does, which means a clearance figure carried from one size to another is wrong in the safe direction and by an unstated amount. Naming the two operations is the useful part; computing the second is a rerun of the whole field’s machinery at the new proportions.

What that means for the ratios that matter #

Three of the field’s results are already comparisons and they are the ones that come closest to transferring.

The rockers admit widths of 0.099 and 0.198 of their shortest link — quoted as a fraction, deliberately, and therefore a shape. That number transfers.

A machine’s material box is 13% to 36% larger in area than its joints’ box — a ratio of areas, a shape, transferable.

And the fill is 0.50 to 0.78 — a shape.

Those three are the field’s most quotable results and they are quotable precisely because somebody wrote them as ratios. The clearances, the footprints and the swept areas are quoted absolutely and are about one machine each.

That is not a criticism of a decision; it is an observation that the field’s own reporting already distinguishes the two and does so implicitly. The results a reader remembers are the dimensionless ones, which is a fair test of which form is more useful.

Where a clearance comes from #

Every clearance in this field is a number the field is handed, and it is worth asking where it comes from, because the answer changes how large it is by a factor of two or more and nothing here currently accounts for it.

A clearance is a hole’s radius less a pin’s radius. Neither of those is the clearance, and the clearance’s own error is not either of theirs: two independently made parts give a clearance whose spread is √2 times each dimension’s, larger than either, because independent errors add. That is the number every clearance figure on this site is drawn at, and it is the worst case rather than the usual one.

A shop has two ways out, and only one of them is the argument a length between two holes makes.

Matched machining turns the pin to fit the hole that was actually bored. The two errors then share a component, the shared part cancels out of the difference exactly as it does for a length derived from two holes in one setup, and the band falls as √(1−f) in the shared fraction f. At a sixty per cent share that is 0.00894 against 0.01414, tighter by 1.58.

Selective assembly makes both to ordinary tolerance, measures them, sorts each into bins and pairs within a bin. Nothing is made more accurately at all — the batch has exactly the hole sizes it had before anybody sorted it — and the correlation is imposed after the parts exist. Four bins give 0.00612, tighter by 2.31; sixteen bins give 0.00153, tighter by 9.24, and the band falls as roughly one over the bin count.

The second has no analogue in the feature-based work, and that is the point of raising it here. Two holes bored in one part cannot be sorted and paired after the fact; a hole and a pin can, because they are separate objects until somebody puts them together. A clearance is the one derived quantity on this site whose error can be reduced without making anything more accurately, and it is reduced by a factor a drawing has no way of expressing.

So the honest reading of every clearance figure here is that it is an upper bound. A machine assembled by a shop that matches or sorts has clearances several times tighter than these figures show, its gaps are correspondingly larger, and its interference margins are better than anything computed from the tolerances on the drawing. The field cannot say by how much, because how much is a fact about a factory and not about a mechanism — which is the same shape of external input the shared setup fraction already is, arriving one component further down.

What it would take to make the field transferable #

The recommendation is a change of units and it is worth setting out because it is cheap and nobody has done it.

Divide every length in the field by the machine’s frame bar. A clearance of −0.432 on a machine with a 4-unit frame becomes −0.108 frame bars. A half-width of 0.16 becomes 0.04. A swept area of 4.2349 becomes 0.2647 frame bars squared.

Every one of those transfers. Two machines of different sizes with the same proportions have identical numbers in the normalised form and different numbers in the current one, and the normalised form is the one that says something about a design.

The choice of reference is a convention and the frame bar is the natural one for a four-bar. For a mechanism with no frame — an open chain, a floating assembly — there is no natural choice, which is presumably why nobody has standardised one.

Two numbers per result rather than one, then: the value and the reference it was divided by. That is the honest form, it is what the three ratios the field already quotes do implicitly, and it is the difference between a result about a machine and a result about a design.

A field about matter, in a collection about motion #

This is the one place in the survey where the apparatus finds nothing to recover and everything to warn about, and the two facts turn out to be the same fact.

Every other field on this site is about motion, and a motion is a relation between configurations that a scaling leaves alone. This field is about matter — how fat the bars are, whether two of them occupy the same place, how much room a pin needs — and matter has a size in it necessarily. So every quantity here carries an exponent, uniquely on the site, and the reason is not an accident of how the field was written.

The consequence for measurement is severe and clean. A protractor determines a machine’s proportions completely and says nothing whatever about whether its parts collide. Not approximately nothing: the entire content of this field is invisible to the instrument that recovers most of the rest of the site. A measurement campaign planned around angles, however many poses it takes and however well conditioned it is, produces a machine model that cannot answer the one question this field asks.

That division is worth knowing before a measurement is planned rather than after. If what matters is whether the thing fits, the instrument is a rule and not a protractor, and no amount of the second substitutes for the first.

And it is why this field was the last one the site opened. A collection about kinematics can go a long way on shapes — mobility, classes, ratios, curves, the whole of the transmission and synthesis fields — before it has to admit that the machine is made of something. The moment it does, every result becomes a result about one machine at one size, and the transferability the rest of the site takes for granted has to be earned back by dividing each number by a length and saying which length.

Three of this field’s results are quoted as ratios and they are the three that travel. Everything else — every clearance, every half-width, every gap in every caption — is about the machine it was computed on. That is correct, it is unstated, and it is the reason a clearance of 0.432 cannot be carried anywhere at all.