The field has no edge

Reachable is not cheap prices a construction and takes the reachable set for granted. The set is a field: the lengths a folder can mark, closed under six operations, with a boundary given by an arithmetic condition on a degree.

A field is an abstract object and the sentence that introduces it says where it lives. The origami numbers are the coordinates of the points a sequence of the seven axioms can specify in the plane — and the plane is unbounded, while a piece of paper is not.

That difference has always been in the definitions and has never been counted. Counting it turns out to matter a great deal.

What a fold line does past the paper #

Every axiom names a fold, and a fold is a line. Axiom one takes two points and gives the line through them; axiom three takes two lines and gives a bisector of the angle between them; axiom six takes two points and two lines and gives a common tangent.

None of those operations knows where the paper is. The line through two points on a small square runs out of the square almost immediately and continues; the bisector of two lines meets them at a point that may be far outside; a common tangent to two parabolas is a line in the plane with no reason to intersect any particular rectangle.

A folder cannot make a fold along a line that misses the paper, and that is the first restriction — but it is not the interesting one, because a fold whose line misses the sheet is a fold nobody would attempt. The interesting restriction is on the crossings.

A reference point is a crossing of two creases. Fold, unfold, fold again, and the two creases cross somewhere; the crossing is a mark, and a mark is what the next axiom aligns to. The crossing is a number in the field wherever it falls, and it is a reference only where there is paper under it. A crossing three sheet-widths away is a perfectly good origami number and it is nothing a folder can touch.

The count, and it is not a correction #

The reachable-points machinery in this collection has enforced that restriction since it was written — only crossings on the sheet are kept — and has thrown the others away without counting them. Counting them is one line and the number is large.

Two rounds of the four linear axioms on a square specify 565 references on the paper and 1,440 crossings off it. Seventy-two per cent of what the plane produces is discarded, and the discarded ones are not marginal cases hovering near the edge: they are scattered across a region many times the sheet.

Admitting the fold that sends a point onto a line at the second round raises both counts enormously and leaves the share almost exactly where it was — 16,890 kept against 42,843 lost, which is again seventy-two per cent.

So the sheet’s edge is not a correction to the reachable set. It is the leading term: the field contains three or four times as many numbers as the paper carries, at every depth and every axiom set tried.

The share is a property of the proportion #

The obvious guess is that a bigger sheet keeps more, and it is wrong in a way worth being precise about.

The construction is scale-free. Doubling every length doubles the crossings’ coordinates and leaves every one of them on the same side of the edge, so the share lost is exactly unchanged. Area does nothing.

What does change it is the shape. Over four proportions the share lost runs from about forty-seven per cent to seventy-two, and the extreme is the square.

The square’s disadvantage has a cause and this collection has already established it in the neighbouring ladder. The sheet decides which points exist finds the square reaching by far the fewest distinct references of any proportion at the same area, because its own symmetry makes folds coincide: two constructions that would give different creases on a rectangle give the same crease on a square, and the marks they would have produced are one mark.

The same symmetry is what sends crossings off the paper. Folds that coincide produce fewer distinct lines; the lines that remain are the highly symmetric ones; and symmetric lines meet at points far from the centre more often than a generic arrangement does. So the square keeps the fewest references and loses the largest share of what it specifies, for one reason.

The consequence for a construction on a real sheet #

The arithmetic above is about a closure. What it does to a particular construction is more concrete and is worth spelling out, because it is where a folder meets it.

A construction is a sequence of alignments, each naming points and lines that must already exist. If the sequence is written in the plane, every intermediate is available; if it is executed on a square, an intermediate may fall off the paper and the sequence stops.

There are two responses and they cost differently.

Rescale. Choose a smaller unit so the whole construction fits inside the sheet. This always works in principle, since the construction is scale-free, and it costs the folds needed to produce the new unit — which is exactly the composition the rung below prices, so the sheet’s edge is paid for in tower height.

Reflect. A point off the edge can sometimes be replaced by its mirror image in the edge, since the edge is itself a line the sheet carries. That costs one fold and works only when the reflected point serves the same purpose, which is a question about the construction rather than about the geometry.

Neither is free and neither is guaranteed, and both are invisible in the field. The reachable set says a number is reachable; the sheet says whether that construction is executable; and the gap between them is paid in folds.

The unbounded plane is doing more than holding the losses #

There is a second consequence of defining the field on the plane, and it is easier to miss than the first because nothing is thrown away by it.

A construction on the plane may use a reference arbitrarily far from the sheet as an intermediate and come back. Two nearly-parallel folds meet a hundred widths away; the point where they meet is a legitimate origami number; a later axiom aligning something to it produces a fold that crosses the sheet perfectly ordinarily.

So the plane is not merely a place where surplus crossings go. It is working memory. A construction that would be impossible with only on-sheet intermediates may be routine with distant ones, and the field’s closure is proved in a setting where every intermediate is available.

That means the loss counted above understates the difficulty twice over. The sheet discards seventy-two per cent of the crossings, and it also removes the option of routing a construction through a distant point — which is not a crossing anybody wanted to keep, but a step somebody wanted to take.

The practical shape of this is familiar to anybody who has followed a published sequence on a sheet that is too small. The construction is right, every step is legal, and one of the alignments asks for a point that is not there. Nothing is wrong with the construction and nothing is wrong with the paper; the sequence was written in a setting the paper is not.

A finite sheet has a boundary and the boundary is useful #

The account so far treats the edge purely as a loss, and that is half of it. The edge is also four lines the sheet carries for nothing.

A bare square starts with four corners and four edges, and every one of those edges is a line that axioms three and four can take as an argument. The edge was there first is the ladder that measures what that is worth, and the answer is that it is worth a great deal at the first round: almost everything reachable in one fold from a bare sheet is reachable because the sheet has edges.

So the finite sheet gives with one hand and takes with the other, and the two effects act at different depths. At shallow depth the edge is an asset, because there is nothing else to align to. At depth the edge is a liability, because the construction has outgrown the paper and most of what it specifies is outside.

The crossover is not measured here and it would be worth measuring. What can be said from these counts is that it is early: at one round the square loses twelve crossings and keeps five new ones, which is already a majority lost, and by two rounds the ratio is nearly three to one.

Which of the axioms throws most away #

The four linear axioms are not alike in this and the difference is instructive.

Restrict to axiom one and axiom two — the line through two points, and the perpendicular bisector of two points — and the loss falls sharply. Both produce lines that pass through or between points already on the paper, so their crossings tend to land on it too, and at two rounds the square keeps 133 references against 64 lost: about a third rather than seventy-two per cent.

Add axiom four, the perpendicular to a line through a point, and the loss rises. Add axiom three, the angle bisector, and it rises further, because a bisector of two nearly parallel lines runs almost parallel to both and meets things a long way off.

The axioms that make lines out of lines are the ones that throw work away. Axioms that make lines out of points keep their crossings near the points — which is why the third fold cannot be listed and the second can. That is not a deep fact but it is a usable one, and it says which part of the axiom list a folder working on a small sheet is really paying for.

The refusal is worth stating too. The figure will not draw a comparison in which the loss is smaller than the keep, because a figure whose whole content is that the edge is the leading term should not be able to draw a case where it is a correction. Ask it for axioms one and two alone and it refuses, naming the numbers — which is the check working rather than a limitation.

What the losses would have to be worth to matter #

A count of discarded crossings is only interesting if the discarded ones are different numbers, and it is worth asking what would have to be true for the loss to be as bad as the count makes it look.

Three cases, and they have very different consequences.

A discarded crossing duplicates a mark already on the paper. Losing it costs nothing whatever: the number is in the sheet’s reach by another route, and the crossing was redundant.

A discarded crossing is a scaled copy of one on the paper. Losing it costs a rescaling — some folds, and nothing that the field would notice.

A discarded crossing is a number the sheet reaches no other way at this depth. Losing it is a genuine reduction of the reachable set on that sheet, and the number becomes available only at greater depth or not at all.

The count made above does not separate them, and the honest reading of seventy-two per cent is therefore an upper bound on the damage rather than a measurement of it. What the count does establish is that the arithmetic of the closure — how many crossings a round produces — is dominated by points that are not on the paper, so any statement about how fast the reachable set grows is a statement about the plane.

Separating the three would need the discarded coordinates checked against the kept ones for rational relations, which is a computation this collection could make and has not. It is the measurement this rung owes, and it would turn a bound into a number.

Which theorem was checked and how #

The loss is counted rather than estimated. Every crossing of every pair of fold lines produced at each round is formed, tested against the sheet, and tallied on one side or the other. Nothing is sampled and nothing is extrapolated.

Distinct crossings are counted once. Two pairs of lines meeting at the same point off the paper are one discarded reference, not two, which is the same deduplication applied to the ones that land on it.

The figure refuses a comparison it cannot make its claim about. If the square keeps more than it loses, or if every proportion loses the same share, it throws rather than drawing — so the two statements the essay rests on are conditions for the picture existing.

And the square must be in the comparison, because the square is the sheet every other figure on this site is drawn on and a claim about proportions that omitted it would be answering a different question.

Where the model stops #

Two rounds is two rounds. The closure at three rounds is far larger and is not computable at this cap, so nothing here says whether the share settles, rises or falls with depth. The two depths measured agree, which is weak evidence and is not a theorem.

A crossing is kept if it is on the sheet and nothing else is asked of it. A crossing a hair inside the edge is counted as a reference and is not one a folder could use, which is what the crowding measurements are about and which this count deliberately does not apply, so that the edge effect is separated from the resolution effect.

The sheet is a rectangle. A sheet with a hole, a notch or a curved edge discards a different set, and a hole is an edge for exactly this purpose.

And the fold lines are not required to meet the paper either. A construction specifying a fold whose line misses the sheet cannot be performed at all, which is a stronger failure than losing its crossing; that count is not made here.

What the picture cannot show #

The bars show how many crossings fall each side of the edge and cannot show where the lost ones are. A picture of the whole plane at the scale the crossings occupy would be mostly empty, with the sheet a small square in the middle of it, and the figure would say less than the count does.

Nor can it show which of the lost numbers are lost usefully. Some of the discarded crossings are duplicates of marks the sheet already carries, reached by another route and landing outside; others are numbers the sheet has no other way to reach. Separating the two is the measurement that would say what the edge really costs the reachable set, and it is not made here.

The figure also cannot show the rescaling. A construction that does not fit can be shrunk until it does, and the shrunk version has the same crossings in the same arrangement — so nothing in a count of what fits says whether the number was reachable on that sheet by some other construction.

The idealisation, named #

The sheet is a rectangle with no thickness, the folds are exact lines, and a crossing is a point. Two of those matter here.

Exactness means a crossing either is or is not on the paper, with no boundary case. A real crossing near the edge is a mark partly on the sheet, and whether it is usable is a judgement rather than a predicate.

Zero width means the plane’s crossings are dense wherever the construction sends them, so the count of what falls outside is a count of distinct exact points. Give the creases a width and nearby crossings merge, and the merging happens most where the crossings are densest — which is on the paper rather than off it, so the correction reduces the kept count and leaves the lost one alone.

That direction is worth noticing: every physical correction to this model makes the sheet keep less, so seventy-two per cent is a floor on what the edge costs rather than an estimate of it.

Where the ladder goes next #

This anchor now has two rungs and both have been about the difference between what the field says and what the paper does. The next one is the third member of that family and it is the sharpest.

The field says nothing about accuracy. A coordinate is exact or it is not; there is no room in an algebraic condition for a construction being well or badly conditioned, and two constructions of the same point can be equally exact and differ by a large factor in what a folder actually gets. What buys the reach costs the accuracy measures that on this same closure and finds the tail arriving with the conic axiom — which is exactly the axiom that reaches the heptagon.

Sideways from here, the proportion result belongs with the sheet-shape ladder, which measures what a square costs a designer packing flaps. The two are the same finding about the same object arriving from two fields: the square’s symmetry is expensive, and it is expensive because coincidences destroy distinctions.

The thing worth carrying is a question about any reachability result. What is the set defined on, and is that where the work happens? A closure computed on the plane and used on a sheet is not slightly optimistic; it is describing a different object, and the difference here is most of the object.