One piece, and still not reachable

Six phases ago this site started answering a particular question by counting connected components. Can this four-bar be got from one assembly to the other without taking it apart? Can this platform reach that pose? Can this arm’s elbow go from up to down? Every one of those was settled the same way: build the free space, find its components, and see whether the two configurations are in the same one.

The last essay of the arm’s field put the general form of it plainly — two configurations in different components cannot be joined by any motion — and it is true, and it is half of something.

The mechanism, which everybody owns #

The example is a cable tie. A toothed strap runs through a housing and a pawl inside the housing drops into the teeth; the strap can be pulled tighter and cannot be let out. It is the simplest one-way mechanism there is, it has a finite travel, and it has no wrap-around to confuse the argument.

Its configuration is a single number: how far the strap has been pulled through, in teeth. Between teeth is a perfectly good configuration — the strap is somewhere, the pawl is riding up a ramp — so the space is a continuous interval, not a set of discrete stops. What the teeth decide is not where the mechanism can be but where it can be left.

The free space is one piece #

Ask the question the site is used to asking. Is there an obstacle anywhere in this space? Is there a configuration at which the mechanism cannot pass?

No. The strap slides through the whole of its travel; the pawl rides up each ramp and drops into each root; nothing binds and nothing interferes. Build the free space, run a flood fill over it, and it has one connected component. Every state is connected to every other state, in the strict sense that a continuous path of admissible configurations joins them.

Two configurations in the same component, then — and the mechanism cannot go from the second to the first.

Reachability is an order #

The relation the site has been computing is connectivity, and it is symmetric: if a path joins $a$ to $b$ then the same path traversed backwards joins $b$ to $a$. That symmetry is so automatic that nothing in six phases of this site ever named it.

A one-way contact removes it while leaving the connectivity untouched. The pawl does not remove any configuration from the space; it removes the reverse of every motion past a tooth. So the relation that actually matters — can this state be reached from that one — is no longer an equivalence. It is a partial order.

The difference is measurable. Sample the twenty-four-tooth strap at four hundred points and count the ordered pairs:

The first row is what the site’s existing machinery reports. The second is what the mechanism does. The third is the pairs that are genuinely equivalent — states between the same two teeth, where the strap can be slid back and forth freely — and it is 4.2% because there are twenty-four teeth and a state can only return to within one of them.

Where the numbers come from #

Two of the three fractions in the table are arithmetic and worth deriving, because a measured number whose closed form is available should be quoted with it.

The one-way fraction. From a state at $x$ on a strap of travel $T$, the reachable set is $[\lfloor x \rfloor, T]$ — everything forwards, plus the small amount of backlash to the last tooth. Averaging its length over $x$ gives a fraction just above one half: exactly a half from the forward part, plus the average backlash, which is half a tooth divided by the travel. On twenty-four teeth that is $0.5 + 1/48 = 52.08\%$, against 52.09% measured over 160,000 pairs.

The both-ways fraction. Two states are mutually reachable only if neither has passed a tooth the other has not, which means both lie between the same pair of teeth. The chance of that for two states drawn independently is one over the number of teeth, so $1/24 = 4.17\%$ against 4.17% measured.

Both come out of the sampling to two decimal places, which is the check that the sampling is measuring the thing the argument describes. And both say the same thing about the mechanism: the mutually-reachable fraction is $1/n$, so a finer ratchet is a more one-way ratchet, and the limit of a very fine ratchet is a mechanism in which no two distinct states can be joined in both directions at all.

Which is the resolution essay’s trade seen from a completely different direction. There, more teeth bought a finer index and cost tooth depth. Here, more teeth buy a more strictly one-way mechanism and cost the small freedom of being able to back off — and a ratchet that cannot be backed off at all is a ratchet nobody can adjust.

What this means for the earlier result #

The result from the arm’s field survives entirely and needs a companion. Written out, the two halves are:

Different components ⇒ no motion joins them. True, unconditional, and the useful direction: it is a negative result, and negative results about mechanisms are rare and valuable. If the flood fill says two poses are in different components, no cleverness will find a path.

Same component ⇒ some motion joins them. True only when every joint is reversible. Every mechanism the site had built before this field satisfied that, because every joint was a pin or a slider or a screw, and all of those go both ways. It is not true here, and it is not true of any mechanism with a one-way element in it.

So the machinery is not wrong; its second half has a hypothesis that was invisible because nothing violated it. This is the shape of a great many results in this fleet — a statement that is exactly true within the class of objects it was written for, and silently partial the moment the class widens.

A test that had to be able to fail #

The claim being made — connected everywhere, reachable half the time — is the kind that a check can confirm without ever having been able to deny, so it is worth saying what the check actually does.

It asserts three things and each of them can go wrong independently. That every pair is in one component: a mechanism with a genuine barrier fails this immediately. That the ordered fraction is $0.5 \pm 0.05$: a reachability relation that had quietly been made symmetric would return 1.0 and fail, and one that had been made empty would return 0 and fail. And that the mutually-reachable fraction is under 10%: this is the one that catches the subtle error, which is a reachability test that ignores the teeth and lets the strap slide back anywhere, giving 1.0.

The middle assertion is the interesting one, because the obvious way to write it is the ordered fraction is less than one, and that version passes for any relation at all that is not the whole square. Bounding a quantity from both sides is the difference between a check and a comment, and this field has already produced one assertion that could not fail by getting exactly that wrong.

The other direction, which is not symmetric either #

There is a second asymmetry hiding in the same mechanism and it belongs to the transitions rather than to the states.

An escapement’s wheel advances by half a tooth per beat and never retreats by a whole one — the recoil during a supplementary arc is a fraction of a degree and is given back. So the wheel’s coordinate is monotone over a cycle without being monotone at every instant, which is a weaker property than the ratchet’s and a different one: the ratchet cannot go back at all, and the escape wheel can go back a little and cannot go back a lot.

The distinction is worth having because it is exactly the difference between the two mechanisms’ jobs. A ratchet holds a position and must never lose it; an escapement counts events and must never lose one, which permits any amount of local wandering so long as the count is right. The first needs a state space with an order on it. The second needs one with a winding number on it, and the winding number is the count of beats.

Which of the field’s mechanisms are one-way #

Not all of them, and the split is informative.

A ratchet is one-way and that is its purpose. So is a cable tie, a hose clamp, a socket wrench and a winch pawl.

An escapement is one-way in the wheel and reversible in the pallet: the escape wheel never runs backwards over a whole cycle, and the pendulum swings both ways as freely as anything. Its state space is a cylinder — pallet angle around, wheel angle along — and the trajectory is a helix that climbs. Two states with different wheel angles are joinable in exactly one direction, and two states with the same wheel angle and different pallet angles are joinable both ways.

A Geneva drive is fully reversible. Drive its wheel backwards and the pin runs back out of the slot; nothing in it prefers a direction. Its transmission function is piecewise but its motion is not one-way, and the reachability relation on its configuration space is the ordinary symmetric one.

A mutilated gear is reversible too, and for the same reason: teeth push both ways.

The property has a name in the count, and the count cannot see it. Grübler’s criterion knows that a higher pair removes one freedom of three; it has no way to know that the pair in question can only push. This is the second time in this field that the mobility count has been asked a question it cannot answer, and it is the same question both times: which motions can be taken, rather than how many exist.

A one-way mechanism is a mechanism with memory #

The way to think about a partial order on a state space, in mechanism terms, is that the mechanism remembers something.

A four-bar at a given crank angle is in the same condition whether it arrived there clockwise or anticlockwise; nothing about it records the history. A cable tie at a given extension has a pawl in a particular tooth, and which tooth it is in records the furthest it has ever been pulled. That is one integer of memory, and it is exactly what makes the reachable set from a state a half-line rather than the whole space.

Which puts this essay’s mechanism and the previous essay’s in the same family after all. An escapement’s discrete label is memory too — which pallet is engaged, which face the tooth is on — and it is memory that cycles rather than accumulating. A ratchet’s accumulates and never resets, which is why a ratchet’s reachability is an order and an escapement’s is only an order in one of its two coordinates.

Every mechanism in this field has some. It is what “intermittent” turns out to mean when the mechanisms are looked at as objects rather than as devices: not that the motion stops, but that the state has a part which is not a position — a contact, a tooth, a label — and that part is what the motion is intermittent with respect to.

What a one-way mechanism does to this site’s own machinery #

The asymmetry is a fact about the mechanism, and it has consequences for the instruments pointed at it — three of them, and each one is a habit this site applies everywhere without stating a hypothesis.

Continuation assumes it can go back. Every sweep here is built by solving at one parameter value and seeding the next from it, and the standing repair for a step that fails is to shorten it and try again. That repair is a step backwards, and on a one-way mechanism it is not available: a solve that has advanced past a tooth cannot be undone by re-solving at a smaller parameter, because the state includes which tooth the pawl is behind. So a sweep of a one-way mechanism has to be built forwards only, never overshooting, which is a different discipline from the one the rest of the site uses.

A drag figure assumes a slider. A slider is a control the reader moves in both directions, and it is the site’s standard way of showing a mechanism’s whole motion. A ratchet cannot be dragged backwards, so a figure with an ordinary slider on one would show a motion the mechanism does not have — a picture of a configuration it cannot reach from where it was, which is precisely the defect the solved-not-drawn habit exists to prevent. The honest control for a one-way mechanism is a step forward, not a slider.

And a component count answers half the question. That is the essay’s own result, and it is worth restating as an instruction rather than as an observation: the machinery reports connectivity, connectivity is the right answer when every joint is reversible, and the hypothesis has to be checked rather than assumed. It was invisible for six fields because nothing on the site violated it.

None of the three is a deep difficulty and all three are the same difficulty. A one-way mechanism’s state space is ordered rather than merely connected, and every tool built on the assumption of a symmetric relation has to be told so. The tools are not wrong; they are answering the question they were written for, on a mechanism that poses a different one.

That is the practical reason this rung sits where it does in the field. The field’s earlier essays establish that a one-way joint is a geometric object with a sign test; this one establishes that the sign propagates all the way up into how the mechanism may be computed and drawn, and that is the part a reader building something would meet.

What is not claimed here #

Two boundaries, since this essay touches ground that belongs to other people.

Nothing here is a planner. Deciding whether a path exists between two configurations, and finding one, is search — roadmaps, trees, sampling — and it belongs to algorithms-data-structures.com. What is taken here is the same half the arm’s field took: what the space is, and a final negative result about it. The one-sidedness above is a property of the space, and how anybody searches a space with a partial order on it is not.

Nothing here is friction. A ratchet is one-way because of a geometry, not because of a coefficient, and the whole of the second essay in this field is the demonstration. A mechanism that is one-way because of friction — a wedge, a capstan, a self-locking screw — has a reachability relation that depends on the pair of materials, and that is a different subject with a different criterion.

The distinction matters exactly here, at the end of the field, because the two look identical from the outside. Both hold one way and slip the other. Only one of them can be drawn.

That is the last thing this field has to say, and it is the same thing its first essay said with different parts. A mechanism whose behaviour follows from its shape can be reasoned about on paper, checked by a computation that knows no materials, and copied by anybody with the drawing. A mechanism whose behaviour follows from a coefficient cannot, and the fact that the two are indistinguishable to a user is why so much of what gets written about ratchets, escapements and indexers is about the wrong one of the two.