What buys the reach costs the accuracy

How many polygons a fold reaches counts what the seventh axiom buys and finds a widening ratio. The count is an arithmetic fact about two smoothness conditions, and it is answered entirely inside the field: a number is reachable or it is not, and the reachable ones are reachable exactly.

There is no room in that account for a construction being good or bad. A coordinate has no error bars. Two sequences that arrive at the same point arrive at the same point, and the field has no way to prefer one of them.

A folder has a way to prefer one of them, and the preference is large.

What a crossing does to an error #

A reference point is where two creases meet. Neither crease is exactly where it should be — a fold is placed by hand or by machine against landmarks that are themselves marks with a width — so each is displaced by a small amount from its intended position — and a fold needs something to align to before it can be placed at all.

Take two lines crossing at an angle φ and move one of them sideways by ε. The intersection slides along the other line, and the distance it slides is ε ⁄ sin φ. At a right angle it slides by ε. At thirty degrees it slides by twice ε. At six degrees it slides by nine and a half times ε.

So a reference’s conditioning is one over the sine of the angle its own creases cross at, and it is a property of the construction rather than of the point. The same point reached by two different pairs of creases has two different conditioning numbers, and the field cannot tell them apart because both constructions are exact.

That is not an error model of anything in particular. It is the geometry of an intersection, and it applies to a folder’s hands, to a machine’s actuators and to a printed diagram equally.

Measuring it over a whole closure #

An argument about two lines is one thing and a statement about a construction system is another, so the sensible thing is to run the whole closure and look at the distribution.

Take a square, run two rounds of the axioms, and for every reference on the paper find the widest angle any pair of creases through it makes. The widest rather than the narrowest, because a folder who has a choice of pairs takes the squarest one — pricing a reference at its worst pair would be pricing a mistake rather than the construction.

With the four linear axioms — the line through two points, the perpendicular bisector, the angle bisector, the perpendicular through a point — the answer is reassuring. Five hundred and sixty-five references, a median crossing of sixty-seven degrees, and a worst case of thirty-six point nine. That worst case is a multiplier of one point six seven: the least well-pinned reference on the whole sheet is under two thirds worse than the best.

Admit the conic axiom — the one that sends a named point onto a named line, which is where the parabola comes in — and the picture changes at one end and not the other. The median moves from sixty-seven degrees to sixty-three, which is nothing. The worst case falls to six point three degrees, a multiplier of nine point one, and a hundredth of the references sit below twelve degrees.

The typical reference is unchanged and the tail is new. That is a very specific thing to find and it is the reason this rung exists.

Why the conic axiom is the one that does it #

The mechanism is worth naming because it makes the finding predictable rather than empirical.

The linear axioms produce creases whose directions are determined by points and lines already present. A perpendicular bisector of two points is perpendicular to the line joining them; a bisector of two lines splits their angle; a perpendicular is a perpendicular. What each axiom is worth measures how much each of them adds to the closure, and the linear ones add points rather than directions. On a square whose starting lines are at zero and ninety degrees, these operations generate directions at multiples of a small set of angles, and two creases from that set either coincide or meet at one of a few angles — none of them small.

The conic axiom is different in kind. Its crease is a common tangent to two parabolas, and its direction is the root of a cubic whose coefficients are the coordinates of the points and lines it was given. Those directions are not confined to a small set: they take irrational values that vary continuously with the configuration, so two of them can be arbitrarily close together.

Which is exactly the property that makes the axiom worth having. A tool whose creases lie in a small set of directions reaches a small set of numbers; a tool whose creases can point anywhere reaches many more. The reach and the sensitivity are the same property seen twice — richness of the direction set — and there is no arrangement that has one without the other.

That is a stronger statement than “the conic axiom happens to be less accurate”. It says an axiom that extended the reach without introducing shallow crossings would have to produce a rich set of directions that never come close together, which is not something a set can do.

What this does to the heptagon #

The ladder below this one is about the heptagon: a compass cannot reach it, a fold can, and the fold that does it is the conic one.

Read with the conditioning measurement, the heptagon’s construction is not merely longer than a compass construction of some other polygon; it is drawn from the operation that supplies the badly-conditioned end of the distribution. So a heptagon folded to a stated tolerance needs its creases placed more accurately than a square or an octagon folded to the same tolerance, and the factor is not small.

That is a testable claim about published constructions and this collection cannot test it, because testing it needs the constructions rather than the closure. What can be said is where to look: the step to examine is the axiom-six fold and the quantity to measure is the angle at which its crease meets whatever it is being crossed with. A published sequence that arranges for that crossing to be near square is a good construction; one that does not is exact and hard to execute, and nothing in the sequence’s own terms distinguishes them.

The neighbouring ladder has met the same distinction from the other end. Exact is not accurate compares a division method that lands on the fraction with one that never arrives, and finds the second one’s crease landing nearer the mark past four parts. That is the same lesson about a different pair, and putting the two together gives the general form: exactness is a property of the answer and accuracy is a property of the route, and the subject’s own vocabulary only has words for the first.

The other half of the same problem #

Conditioning is one of two ways a reference can be unusable and the other one is already measured on this site, so it is worth setting them side by side.

Conditioning is about how an error at the crease becomes an error at the point. It is a multiplier, it depends on the crossing angle, and it is what this rung measures.

Crowding is about whether the point can be distinguished from its neighbours at all. Closer than a crease is wide measures that: at two rounds on a sheet of ordinary size, a great many of the crossings the closure produces are within a crease’s width of one another and are one mark on paper rather than several.

They are independent failures and they compound. A reference that is badly conditioned and crowded is worse than either; one that is well conditioned and crowded is still unusable; one that is badly conditioned and isolated can at least be found, and then placed badly.

Neither is in the field. The reachable set has no width and no angle in it, so both failures are invisible to the arithmetic that decides reachability, and both are properties of the same construction the arithmetic calls exact.

What a multiplier of nine actually costs #

A factor is easy to quote and hard to feel, so it is worth converting into the units a folder works in.

A crease placed by hand against a landmark lands within a few tenths of a millimetre of where it was meant to. Call it a fifth of a millimetre, which is generous for a careful folder and pessimistic for a machine.

At a right-angled crossing that fifth of a millimetre is a fifth of a millimetre of error in the reference. At the linear axioms’ worst crossing, thirty-seven degrees, it is a third of a millimetre — noticeable and not serious. At the conic axiom’s worst, six degrees, it is nearly two millimetres, which on a fifteen-centimetre sheet is more than one part in a hundred and is visible in the finished model as a limb that does not meet its neighbour.

And that is one step. A construction that uses such a reference and then folds against it carries the error forward, and the way errors compound along a sequence is measured elsewhere on this site: a systematic error grows in proportion to the number of creases and a random one as the square root. Two millimetres at the first step is not two millimetres at the end.

So the conditioning tail is not a subtlety about numerical analysis. It is the difference between a construction that works on paper and one that does not, arriving from a quantity the algebra has no symbol for.

Two constructions of one point #

The abstract statement — that conditioning is a property of the construction rather than of the point — deserves to be made concrete, because it is the part that is easy to nod at and hard to believe.

Take any reference the closure reaches by more than one route. The site’s own count says a great many are reached that way: of the five hundred and sixty-five references two rounds of the linear axioms produce, four hundred and thirty-two lie on exactly two creases and the rest lie on three or more, with a handful lying on ten.

A point on ten creases has forty-five pairs to be pinned by, and those pairs meet at every angle the arrangement allows. A folder who takes the squarest pair gets the best conditioning available; one who takes the first pair that comes to hand gets whatever that pair is; and the point is the same point either way, with the same exact coordinates and the same place in the field.

The redundancy is therefore worth something and it is worth something no algebraic account can express. A reference on many creases is more robust than one on two, not because its coordinates are better known — they are exact — but because there is a choice of ways to find it and the best of them is better than the only way.

That gives a small piece of practical advice which is the closest this ladder comes to one. Where a construction has a choice of which pair of creases to take a reference from, the squarest pair is the right one, and the improvement available is the ratio of the sines — which on the numbers above runs to a factor of nine.

Which theorem was checked and how #

The conditioning is read off the closure rather than modelled. For every reference the creases through it are found by testing each fold line for passing through the point, their directions are taken, and the widest angle any pair makes is the reference’s number. Nothing is assumed about how a folder places a crease.

The two axiom sets are compared as shares rather than as counts, because admitting the conic axiom multiplies the reference count thirtyfold and a comparison of raw counts would say nothing.

The medians are asserted to agree. The figure requires the two distributions to agree about the typical reference to within twelve degrees, and refuses if they do not — because the finding is that the conic axiom adds a tail, and a conic axiom that shifted the whole distribution would be a different and weaker result.

And the worst cases are asserted to differ by more than a factor of three. A conic axiom whose shallowest crossing was close to the linear axioms’ would make this rung’s claim false, and the figure would say so rather than drawing it.

Where the model stops #

Two rounds is two rounds. Deeper closures have more references and presumably longer tails, and nothing here says how the distribution behaves with depth.

The error model is a displacement of a whole crease. A real fold is misplaced in position and in angle, and an angular error grows along the crease so that a crossing far from the landmarks is worse than one near them. That effect is not in this measurement and it acts in the same direction.

The widest pair is the fair figure and it is also the optimistic one. A folder who does not notice that a better pair exists uses the pair they thought of, and the achieved conditioning is somewhere between the best and the worst pair through the point.

And a distribution over a closure is not a statement about any construction. No published sequence is analysed here. What is measured is the population of references a folder has to choose from, and the essay’s claim is about what that population contains.

What the picture cannot show #

The histogram counts references by their crossing angle and cannot show which references are the shallow ones. If the badly-conditioned tail were entirely made of points nobody would want, the finding would be much weaker — and separating that would need each reference’s coordinates checked against what constructions actually use.

Nor can it show the compounding. A construction of several steps has an error at each, and the errors propagate through the sequence in a way that depends on the sequence; a distribution over single crossings is the first term of that and not the whole of it.

The most conspicuous absence is the sequence itself. This essay is about what makes a construction accurate and draws no construction, because the closure is a population and a construction is a path through it — and drawing one path would present a choice as though it were the finding.

The idealisation, named #

The closure is computed with exact arithmetic on an unbounded plane and then the results are filtered to the sheet. The conditioning is then computed on the exact positions.

That combination is worth naming because it is slightly odd. The conditioning of an exactly computed reference is a statement about what would happen if it were not exact, which is a counterfactual rather than a measurement — the multiplier one over sine is the derivative of the crossing with respect to a displacement, evaluated at zero displacement.

For small errors that is the right thing and it is the standard thing. For errors large enough that the crossing moves appreciably, the linearisation is no longer exact and the true sensitivity is worse, because sine is concave near zero.

And the whole calculation is on a square, which is the worst sheet this collection has measured for several other purposes. Whether a rectangle’s crossings are better conditioned is a question this rung does not answer and the machinery would.

Where the ladder goes next #

This closes the ladder’s account of the two tools and leaves it pointing at a construction rather than at another population.

The measurement that would settle it is a conditioning number for each published heptagon construction: take the sequence, find the crossings it uses, and compute the multiplier at each. That is an afternoon’s work with the sequences in hand, it has a definite answer, and it does not appear to have been done — which is the same shape of gap the ladder below this one found, where the reach was illustrated by an example and never counted.

Sideways, the finding belongs beside what a machine that folds one crease at a time can reach. A machine has a different error model from a pair of hands — repeatable, biased, and small — so the conditioning multiplier matters to it more rather than less, since a systematic error at a shallow crossing is amplified without any averaging to soften it.

The habit worth carrying is the question this rung is an instance of. When a theory says two things are equivalent, ask what quantity the theory has no words for. Reachability has no words for conditioning; exactness has no words for accuracy; and in both cases the missing quantity is the one that decides which of two equivalent things anybody should use.