The edge was there first

A fold has to be specified by aligning existing things, and at the start of a folding sequence the only existing things are the sheet’s own outline: four lines and four corners. Everything else in the closure — every crease, every crossing, every mark a later fold can be aligned to — descends from those eight objects.

The outline is therefore doing two jobs at once, and they are easy to run together. It is the input to the first fold, and it is also where the first fold’s output lands.

Four of the first five #

Apply the linear alignments to a bare square — fold through two points, fold one point onto another, fold one line onto another — and the closure specifies thirty-eight folds, of which twelve are distinct lines. Their crossings that land on paper and are not already corners come to five new references, and four of the five are on the sheet’s edge.

They are the midpoints of the four sides. The fold that places one corner onto the next is the perpendicular bisector of those two corners, and where it meets the edge between them is the midpoint — a mark on the rim, made by a fold that was itself specified entirely by the rim.

The fifth is the centre of the square, where the two diagonals cross, and it is the only reference of the first round that is not on the boundary.

Forty-eight of the next five hundred #

The second round is a different world. Twelve lines become ninety-two, three hundred alignments are specified, and the crossings that land on paper add 556 new references — of which forty-eight are on the edge.

The share has fallen from four fifths to one in twelve, and the reason is a counting argument rather than anything about folding.

A line meets the rim twice. Whatever fold is made, its crease crosses the sheet’s boundary at exactly two points — so the marks a round adds to the rim grow with the number of folds.

Two lines cross once. A mark inside the paper is where two creases meet, so the marks a round adds inside grow with the pairs of folds, which is the square of the number.

Twelve lines give a couple of dozen boundary crossings and sixty-six potential interior ones; ninety-two lines give a couple of hundred boundary crossings and four thousand potential interior ones. The interior wins, quickly and permanently, and the rim’s share falls like one over the number of creases.

The one-over-L law, checked #

The counting argument predicts a rate as well as a direction, and the two rounds are enough to test it.

With $L$ creases, the rim gains at most $2L$ marks and the interior at most $\binom{L}{2}$, so the edge’s share is at most

$$\frac{2L}{2L + \binom{L}{2}} = \frac{4}{L+3}$$

which falls like one over the crease count.

Now the measurement. The first round has twelve lines and the edge takes four of five new references — eighty per cent. The second has ninety-two lines and takes forty-eight of 556 — 8.6 per cent.

The share fell by a factor of 9.3 while the crease count rose by a factor of 7.7. A law that says the share is proportional to $1/L$ predicts those two factors to be equal, and they agree to within twenty per cent.

Which is as close as the model deserves #

The agreement in the ratio is much better than the agreement in the level, and the discrepancy is informative rather than a failure.

The bound $4/(L+3)$ gives 27 per cent at twelve lines and 4.2 per cent at ninety-two, against measured figures of 80 and 8.6. Both measurements are about double the bound at the second round and three times it at the first, because the bound counts every crossing as a new reference and most are not: creases through the same corner meet there rather than somewhere new, and the interior is where that coincidence is worst.

So the model over-counts interior marks and under-counts the edge’s share, by a factor that is itself falling — which is why the levels disagree and the rates do not.

The useful form is the rate. Whatever fraction of crossings turn out to be new, that fraction applies to both columns, and the ratio of the two columns is $\binom{L}{2}/2L = (L-1)/4$ regardless. A third round with a few thousand creases would put the edge’s share somewhere near a tenth of a per cent, and no correction for coincidences moves that by an order of magnitude.

That is the sense in which the rim’s advantage is a one-round phenomenon. It is not that the edge stops gaining marks — it gains far more at every round — but that the interior gains them quadratically, and a linear quantity loses to a quadratic one at the first opportunity.

Where the interior references come from #

The 556 are worth a moment, because “the interior wins” is a statement about counting and the counting hides what the marks actually are.

Ninety-two lines have four thousand and something pairs, and only 556 of their crossings are new references on the paper. Most of the pairs cross outside the sheet, or cross at a point some other pair also crosses, or cross where a reference already is. So the interior’s growth is not the raw square of the line count; it is that number heavily thinned by the geometry of a square, in which a great many folds are parallel or coincident.

That thinning is what makes the closure computable at all. A round that produced every pair’s crossing would give tens of thousands of points and the third round could not be attempted; as it is, the third round is already past what can be built.

And it is why the rim’s share falls to one in twelve rather than to one in a hundred. The interior is winning against a thinned square rather than against the full one.

What that means for a folder #

The practical reading is the one the craft already knows, and it is worth stating in the counts’ own terms.

Early folds are edge folds. With nothing on the paper, everything a folder can specify involves the outline, and everything it produces lands on the outline. That is why every folding sequence in every book begins with corner to corner, edge to edge, corner to the centre.

The edge is where the accurate marks are. A reference on the rim can be checked against the paper’s own edge — a fingernail against a straight boundary — where a reference in the middle has to be found by eye at a crossing. The distance at which two marks stop being distinguishable is about half a millimetre and it applies to both, but only one of them has a straightedge running through it.

And the edge runs out. After two folds, eleven of every twelve new references are inside the paper, and by the third round the boundary is a rounding error. A sequence that keeps working from the edges is a sequence that has stopped adding references — which is exactly what a folder doing the same four folds over and over finds.

The edge is not a fold and it counts as one #

There is a small piece of bookkeeping under all of this and it is the reason the first round produces anything at all.

The sheet’s four edges are lines in the closure, on the same footing as the creases: an alignment can place a point on an edge, place an edge on an edge, or fold through two corners of the edge. Without that, a bare square has no lines at all and the closure is empty — the first fold could not be specified, because there would be nothing to specify it with.

So the outline is four free lines and four free points, handed to the folder before any work is done. On a square that is a considerable gift, and it is one the sheet’s proportion decides the value of: a square’s four edges are parallel in pairs and meet at right angles, so the alignments among them are highly degenerate and produce few distinct folds. A sheet with no symmetry would give more.

That is the trade the counts here sit inside. A square’s edges are the cheapest references available and the least productive; the interior references are expensive and multiply.

What “on the rim” means here #

The split is made by position rather than by provenance, and the difference matters for reading the numbers.

A reference counts as being on the rim when its coordinates put it on one of the sheet’s four edges — not when the fold that made it involved an edge. Nearly every fold in the first two rounds involves an edge, because there is little else, so a provenance split would put almost everything on one side and say nothing.

By position, the split is clean and the counts mean what they appear to. The four midpoints of the first round are on the rim because they lie on it; the centre of the square is not, although both diagonals that make it start and end on corners.

One consequence: a corner is on the rim twice over and is counted once. The four corners are the sheet’s own, present before any fold, and are excluded from the counts of what each round adds — otherwise the first round would score four free marks it did not make.

Where a hole changes it #

A square with a hole in it offers eight lines and eight points rather than four and four, and the hole’s edges are not redundant: each is parallel to one of the outer edges at a distance the hole’s position decides, so the extra alignments produce new marks rather than repeating old ones.

The same argument applies to those edges as to the outer ones. They are lines that existed before any fold, they yield references immediately, and they saturate — a hole’s rim is a curve like the sheet’s, so marks on it grow with the folds while marks in the paper grow with the pairs.

What a hole adds is therefore a second early harvest rather than a change in the long-run behaviour. Two rounds in, a holed sheet has more references than a plain one and the same shape of growth, which is the shape any closure of lines in a plane has.

The marks the edge gives are the useful ones #

There is a second reason edge references matter more than their share suggests, and it is about what a reference is for.

A reference is used to specify the next fold, and a fold is specified by aligning two things. An alignment involving a point on the edge can use the edge itself as the second thing — bring this corner to that mark on that edge — which is one object rather than two. An alignment between two interior marks needs both of them found and held simultaneously, on paper that is by then covered in creases.

The exact divisions bear this out. Folding a strip into thirds and the ladder of exact divisions both work along an edge, marking it, and the marks they produce are on the edge because that is where the division is being made. The classic constructions of the subject are edge constructions almost without exception.

So the count says the edge saturates and the practice says the edge is where the work is. Both are true, and together they say what a folding sequence is: a few early folds that harvest the edge, and then a long tail of folds specified from interior crossings, which is where the accuracy goes.

Why the share is worth measuring at all #

It could be dismissed as a fact about small numbers — five references in the first round is not much of a sample, and four fifths of five is four. Two things make it more than that.

The mechanism is exact and the counts confirm it. Boundary marks grow with the creases and interior marks with their pairs; that is a statement about lines in a plane and it predicts the fall from 80% to 8.6% before any of it is computed. A measurement that agrees with an argument this simple is worth having precisely because it could have disagreed — a closure that clustered its marks near the rim for some geometric reason would have shown up here.

And it explains a practice. Every folding diagram ever drawn starts at the edges and works inward, and the usual explanation is that the edges are easy to see. The counts say something stronger: for the first fold there is nothing else, and for the second the edge is still where a fifth of the new marks are. The practice is not a convenience, it is what the closure makes available.

The same shape as the boundary’s other measurements #

This is the fourth time in this collection that a quantity has been split by on the rim against inside, and the four disagree about which side is favoured, which is what makes the split worth making rather than assuming.

The rim lies over less: a panel carrying a raw edge stacks under fewer other panels, on every printed pattern. The rim loses.

The rim carries less folding: the outermost band of every printed pattern holds less crease length than its share of the paper. The rim loses again.

Most of a patch is rim: at any size a page or a sheet can carry, a third or more of a tessellation is cut units. The rim dominates.

And the rim is where the first references are: four of the first five. The rim wins, early, and then stops winning.

Four measurements, three quantities, two directions. The boundary is not simply “less” or “more” — it is a different kind of place, and which way a measurement goes depends on whether the quantity is about neighbours (where the rim has fewer) or about lines that already exist (where the rim is all there is at the start).

What the first fold is worth, in one picture #

The whole argument fits into the first round, and the first round is small enough to draw.

Four corners, four edges. The alignments among them specify thirty-eight folds; the distinct lines are twelve — the two diagonals, the two midlines, four corner-to-corner-across bisectors, and four that coincide with the edges themselves. The crossings on paper are the centre and the four side midpoints.

Everything a folder does for the rest of a sequence is descended from those nine points, and eight of them are on the boundary — the four corners the sheet came with and the four midpoints the first fold makes. The gift is small, it is almost entirely at the edges, and it is enough.

What is left out #

Only the linear alignments. The closure counted here uses folds through two points, one point onto another, and one line onto another — deliberately, because the question is what the edges are worth and the bisector-of-a-circle operation would answer a different one. The conditional operations behave differently again, and adding them changes the counts and not the shape.

Two rounds, not three. A third round would ask the alignments for a fold from every pair of five hundred and sixty-five references, which is past what the closure can be computed at. The trend is established by the first two and is not confirmed beyond them.

The square is the only sheet measured. The counts are for a unit square, and the whole argument turns on the outline being four lines. A sheet with more edges — a holed one, or one with a bite out of it — starts with more, and one with a curved edge starts with none that any of these alignments can use, since every operation here takes straight lines.

And a reference is not a mark. These are exact crossings of exact lines. What a folder has is a crease with a width and a crossing they can see, and ninety-four per cent of the marks three folds reach have another within a fifth of a millimetre. The rim’s marks are no better in that respect — they are just as crowded — but they are the ones with an edge to check against.