A stretch keeps crossings

A construction assumes its sheet runs Haga’s fold on rectangles and finds two ways for it to fail. On an A-series sheet held tall the corner-to-midpoint fold gives exactly two sevenths where a square gives two thirds — a clean fraction, silently wrong. Held wide, the crease runs off the paper — loud, and harmless. The essay ends with a classification it owes and does not make: which of the standard constructions survive a change of sheet?

It guesses at the shape of the answer. Halving and quartering survive; the division constructions survive along the right edge; Haga’s does not; and a construction is portable when the quantity it finally reads is compared against something that scaled with everything else.

That guess is close, and the precise version is shorter and sharper. There is one operation that turns a square into any rectangle, and the constructions that survive are exactly the ones that operation preserves.

A rectangle is a stretched square #

A w-by-h rectangle is what a unit square becomes if every point’s horizontal coordinate is multiplied by w and every vertical one by h. That stretch along the edges is a particularly well-behaved map, and its good behaviour is the whole of the result.

It carries every straight line to a straight line. It carries two lines that cross to two lines that cross, at the image of their crossing. It carries the midpoint of a segment to the midpoint of the image segment, and more generally a point dividing a segment in any ratio to the point dividing the image in the same ratio. It carries parallel lines to parallel lines.

What it does not carry is anything about angles or lengths in different directions. A square’s diagonal meets its sides at forty-five degrees and a rectangle’s diagonal does not. Two perpendicular lines on the square are not perpendicular after the stretch, unless they run along the edges. A circle becomes an ellipse.

Maps with the first list of properties and not the second are called affine, and the classification this essay owes follows from one observation: a construction whose every step is a line, a crossing or a midpoint is carried by the stretch to the same construction on the rectangle, so it lands at the same fraction of the sheet.

Four constructions that keep their answer #

The first figure runs seven constructions on a square, an A-series sheet held both ways, a three-to-two sheet and a double square held wide. Four of them give the square’s fraction on every one.

Halve it, edge onto edge. Bringing one edge onto the opposite edge creases the midline, at a half. On every rectangle it creases the midline, at a half, because a midpoint is carried to a midpoint.

A third, from two crossing lines. Cross the diagonal of the sheet with the line from a top corner to the midpoint of the bottom edge, and the crossing lies a third of the way across. Dividing without measuring is built on crossings of this kind. On every rectangle the crossing lies a third of the way across and a third of the way up, because a crossing of two lines is carried to the crossing of their images.

A fifth, by repeated crossings. One crossing, and then another walks a mark down the sheet by crossing the diagonal with a line through the last mark: a half, a third, a quarter, a fifth. Every step is a crossing, so every rectangle gets the same fifths at the same fraction of its height.

A third, by Fujimoto’s halvings. Folding a strip into thirds guesses a mark and halves the gap toward alternate edges until it settles on a third. Every step brings a mark onto an edge along the strip, which is taking a midpoint, and a midpoint is carried to a midpoint.

For each of the four, the figure also checks the statement directly rather than by comparing fractions: the square’s point, stretched by the rectangle’s width and height, is the rectangle’s own point, computed separately. The construction on the rectangle is the construction on the square, stretched.

Three that do not #

The other three constructions give a different point on at least one sheet.

Haga’s fold, corner to midpoint. On the square the folded edge crosses the far edge at two thirds. On the A-series sheet held tall, at two sevenths; on the three-to-two sheet held tall, at a quarter; on the A-series sheet held wide and the double square, the fold’s crease leaves the paper.

A corner halved, edge onto edge. Laying the bottom edge along the left edge creases the forty-five-degree line from the corner, which on a square runs to the opposite corner. On the A-series sheet held tall it meets the right edge at 0.707 of the height; held wide, the top edge at 0.707 of the width; on the double square, the top edge halfway across.

A corner onto the opposite corner. On a square the crease is the other diagonal and passes through the bottom-right corner. On the A-series sheet held wide it meets the bottom edge three quarters of the way across; on the double square, five eighths; on the A-series sheet held tall and the three-to-two sheet, it leaves the paper.

All three are folds of a kind the first four are not.

A fold is a reflection, and a slant is not kept #

Every fold is a reflection: the paper on one side of the crease is carried to its mirror image across the crease line. The question of which folds the stretch preserves is therefore which reflections it preserves, and the answer divides by the crease’s direction.

A reflection across a line parallel to an edge — the vertical midline, say — sends a point at horizontal position $x$ to position $c - x$, leaving its height alone. Stretching the sheet horizontally by $w$ turns that into sending $wx$ to $wc - wx$: still a reflection, across the stretched line. So a fold along a crease parallel to an edge is carried to a fold. That covers halving, and every Fujimoto step.

A reflection across a slanted line is different. It carries a point to its mirror image along the perpendicular to the crease, and the perpendicular to a slanted line is exactly the thing a stretch along the edges does not preserve. After stretching, the image of the square’s fold is a skew reflection — a map that swaps the two sides of the stretched crease along a direction that is no longer perpendicular to it — and a skew reflection is not a fold any sheet of paper can make.

So on the rectangle the fold a folder actually makes is not the stretched fold. It is a different reflection, across a different crease, and it lands somewhere else. Every construction that brings a point to a point off an edge-parallel line folds across a slant, and every one of those assumes its sheet.

The proportion, as a formula #

The three constructions that fail do not fail arbitrarily. Each lands at a point given by a short formula in the sheet’s proportion, and the formulas say more than the table does.

Write $r$ for the sheet’s height divided by its width. Haga’s fold brings a bottom corner to the midpoint of the top edge, and the folded bottom edge meets the far side at $2/(4r^2 - 1)$ of the height, measured down from the top. At $r = 1$ that is two thirds. On the A-series sheet held tall $r^2$ is 2 and the fraction is 2/7; on the three-to-two sheet $r^2$ is $9/4$ and it is 1/4; on the six-to-five sheet it is 0.420. Held wide, the A-series sheet has $r^2 = 1/2$ and the formula reaches 2, a point off the sheet.

The corner halved creases the forty-five-degree line from a corner, which meets the far side at $1/r$ of the height on a tall sheet and the top edge at $r$ of the width on a wide one. The corner brought to its opposite corner creases the perpendicular bisector of the diagonal, which meets the bottom edge at $(1 + r^2)/2$ of the width — three quarters on the A-series sheet held wide, five eighths on the double square — and leaves the paper whenever the sheet is taller than it is wide.

Each formula equals its square’s value only at $r = 1$, which is the affine argument written a second way. And the formulas explain something the table only shows. Haga’s point and the corner-to-corner point depend on $r^2$, so an A-series sheet, whose $r^2$ is two, gives them the clean fractions 2/7 and 3/4; the corner halved depends on $r$ itself and gives 0.707. A clean fraction on a rectangle is a property of the formula, not a sign that the construction survived — which is how the silent failure came to look like a result.

The formulas also price a sheet that is only nearly square. Near $r = 1$ Haga’s point moves by sixteen ninths of any change in $r$, so a fifteen-centimetre sheet cut a millimetre out of square puts the mark about 1.8 millimetres from two thirds of its height, whichever way round the error lies. Exact is not accurate measured a folder’s own placement error at about 0.3 millimetres. A millimetre of bad cutting costs Haga’s fold six times that, and costs the crossing construction nothing: its third is a third of whatever height the sheet has.

Why the essay’s guess was close #

A construction assumes its sheet proposed that a construction is portable when the quantity it finally reads is compared against something that scaled with everything else, and that Haga’s fails because its crossing is read against an edge whose distance from the crease depends on the proportion.

That is right as a description of what goes wrong, and the affine statement says why it goes wrong and where else it will. Haga’s crease is the perpendicular bisector of a corner and a midpoint; a perpendicular bisector is built from a perpendicular; a perpendicular is not affine. The reading against the edge is not the fault. The fault is earlier, at the first fold, and every later step inherits it.

The affine version also corrects one half of the guess. The essay suggested the division constructions survive “along the right edge and not the other”. The crossing constructions survive along both, because crossings are carried to crossings in both directions at once: the third from two crossing lines is a third of the way across and a third of the way up on every rectangle. It is only a construction that uses a slanted fold that has a direction it survives in and a direction it does not.

What a straightedge would have kept #

This division is older than folding, and its origin says something about which constructions are fundamental.

Jean-Victor Poncelet’s projective geometry of the 1820s studied exactly the properties of figures that survive when the figure is projected — lines, crossings, which points lie on which lines — and set them apart from the metric properties — angles, lengths, circles — that do not. Constructions made with a straightedge alone, drawing lines through points and marking where lines cross, live entirely on the first side. They survive projection, and so they survive the gentler stretch along a sheet’s edges.

The folding constructions that survive a change of sheet are the ones a straightedge could draw once the midpoints were given. Halving supplies the midpoints; crossing lines is what a straightedge does. Everything a fold adds beyond that — the ability to bring a point onto a point anywhere, to bisect an angle, to solve a cubic — is a metric operation, and every metric operation assumes the shape of the sheet it is performed on.

That is a sharper way of saying what folding beats the compass by. The extra reach of folding is in its metric operations, and the metric operations are precisely the ones that do not carry from one sheet to another. The power a fold has over a straightedge is the power that ties it to its paper.

The cost of portability #

A portable construction of a given number is usually available, and the table gives the price.

Haga’s fold reaches a third in a single fold, on a square. The crossing construction reaches a third in two creases — a diagonal and a line from a corner to a midpoint — plus the halving that supplied the midpoint, and it reaches a third on every rectangle. So on this example portability costs two extra creases, and what they buy is that the answer does not depend on having been handed a square.

Fujimoto’s method is portable too and never arrives exactly; exact is not accurate found it more accurate in a folder’s hands past four parts anyway. So a folder working on a sheet of unknown proportion has a portable exact construction and a portable approximate one for thirds, and Haga’s fold is neither.

Every proportion, one verdict #

The sorting does not depend on which rectangles were chosen, and it is worth running on proportions far from the square to see that nothing new happens.

Close to square does not mean close to the square’s answer. On a six-to-five sheet held tall Haga’s fold lands at 0.420 of the height instead of two thirds, and held wide it leaves the paper; the corner halved meets the far edge at five sixths; the corner brought to its opposite meets the bottom edge at 0.847 held wide and leaves the paper held tall. On a triple square held tall Haga’s fold lands at 0.057. The affine four return the square’s fraction on all of them, exactly, because the argument that carries them from the square to a rectangle does not care how far the stretch goes.

That is the difference between a classification and a table. A table records what seven constructions did on five sheets. The classification says what any construction will do on any rectangle, from the list of operations it uses, without running it.

What the table cannot show #

The table runs each construction alone, and constructions are used in sequences.

A sequence that begins with a portable step and continues with a slanted fold inherits the slanted fold’s dependence; one that uses a slanted fold only to produce a mark that is then used along an edge may or may not, depending on what the mark is. The classification is exact for each operation and needs to be applied step by step to a sequence, which the table does not do.

Nor does it show accuracy. A portable construction lands at the right fraction of every sheet in the model, and on a real sheet it still has the conditioning of its crossings: a crossing of two nearly parallel lines is badly placed by a folder’s hands however portable it is. A long thin rectangle makes some crossings shallower, so a construction can be exactly portable and practically worse on a stretched sheet.

And seven constructions are seven. The reference-point closures, the heptagon, the cube root and the box-pleating angles are not in the table; the classification predicts that every one of them that brings a point to a point off an edge-parallel line assumes a square, and that is a prediction rather than a measurement.

The sheets the table assumes #

Every sheet is an exact rectangle and every construction is exact: a crease is a line, a crossing is a point, a fold is a reflection. The proportions include one irrational, the A series, to show the portable constructions returning the same rational fractions on a sheet whose sides are incommensurable.

Fractions are read as fractions of the sheet’s own width and height. A construction portable in the affine sense returns the same pair of fractions on every rectangle; returning the same length would be a different and much rarer property.

And a construction is its sequence of alignments, as a fold needs something to align describes. A recipe that says take a square is a different construction from one that does not, and the table runs the second kind.

How the sorting was checked #

Each construction is computed from its alignments on each sheet, not scaled from the square: crossings are solved, reflections are carried out, and crease lines are clipped to the sheet to see whether they stay on it.

The four affine constructions are required to return the square’s fractions on every sheet, to a part in a trillion, and the three that fold across a slant to return a different point on at least one. The sorting is stated as a condition on the table existing, so a construction on the wrong side of the line would stop it being drawn.

For the affine four, the stretch is applied directly: the square’s point, multiplied by each rectangle’s width and height, is required to be the rectangle’s own point, computed separately. That checks the reason as well as the result.

Still open: the cheapest portable version of each number #

The classification turns the owed table into a design question. For every fraction a slanted-fold construction reaches in a few folds on a square, what is the fewest folds that reach it using only crossings, midpoints and edge-parallel folds — and so on every rectangle? For a third the answer is a small number and is drawn above. For Haga’s two sevenths on A4 — which is a rational number, and so certainly reachable by crossings — the portable construction may be long, and the difference between its length and Haga’s one fold is exactly what the square was saving.

The second direction is the reverse of this essay. The sheet decides which points exist measured how a sheet’s proportion changes the reachable set. The affine argument says that the part of the reachable set built from crossings and midpoints is the same on every rectangle, fraction for fraction, so every difference that study found lives entirely in the slanted folds.

The habit worth carrying is a way to read a recipe. List the operations and ask which of them an edge-wise stretch would keep. Lines, crossings and midpoints travel between sheets; angles, perpendiculars and point-to-point folds do not; and a recipe’s dependence on its paper is exactly the list of the second kind it contains.