What the square saves

A stretch keeps crossings sorts folding constructions into two kinds and gives the sorting a reason. A rectangle is a square stretched along its edges; the stretch carries lines to lines, crossings to crossings and midpoints to midpoints, and a fold along a line parallel to an edge to a fold along its image; it does not carry a fold across a slanted line, because the perpendicular to a slant is exactly what a stretch along the edges bends. So a recipe made only of the first kind lands at the same fraction of every rectangle, and a recipe that uses one slanted fold — Haga’s, a corner halved, a corner brought to its opposite — has a sheet it was written for.

That essay ends on a question its classification makes answerable. Haga’s fold marks two thirds in two creases on a square, and on an A-series sheet held tall the same two creases mark two sevenths. Two sevenths is rational, so a construction using only the portable kind of step certainly reaches it. How many creases does it take? The difference between that number and two is what knowing the sheet was worth, and it has not been measured for any fraction.

It can be measured for all of them at once, and the answer is not the one the classical recipes suggest.

Two toolkits, and what each is allowed to read #

The count needs both toolkits stated exactly, because the difference between them is the result.

The full toolkit is the first four of the folding axioms on a named sheet. A crease may run through two points already marked; it may bring one marked point onto another, which creases their perpendicular bisector; it may lay one crease or edge onto another, which creases the line that bisects them; and it may run through a marked point square to a crease or edge. A mark is made wherever two creases cross, wherever a crease meets an edge, and — the step Haga’s construction depends on — wherever an edge of the sheet lands while a fold is held, which is a pinch made against the folded flap.

The portable toolkit keeps from that list exactly what the stretch keeps. A crease through two marked points stays, since a line is carried to a line. A point may be brought onto a point only when the two are level with each other, along or across the sheet, so the crease runs parallel to an edge. A crease or edge may be laid onto another only when they are parallel, which creases their midline. A crease square to another is allowed only when the other runs along an edge’s direction. And an edge’s landing may be read only under a fold along an edge’s direction. Every one of these is carried by the stretch to the same step on every rectangle, so a portable count is computed once, on the square, and holds for every sheet with four right-angled corners.

That last phrase is the precise extent of the claim. The portable steps use the corners’ right angles — a crease square to an edge is found by laying the edge onto itself — and nothing else about the sheet. A parallelogram is also a stretched square, but folding its edge onto itself does not crease a line parallel to its other edge, so the portable count is a statement about every sheet a guillotine can cut, and about no other.

What is counted is creases. A midline made by bringing one edge onto the other is one crease. A pinch where a folded edge lands is free with the crease that put the edge there, and gone once the fold is opened: the landing is not a line any later crease can refer to. Both conventions favour the full toolkit slightly, since Haga’s construction gets its reading for nothing.

The search, and why its counts are minima #

For each toolkit the search starts from the four corners and four edges and adds one crease at a time, trying every crease the toolkit specifies from what is marked so far, and keeping each distinct set of creases once. After each crease it records every fraction of an edge that has become marked. A fraction first marked by crease number $k$ is marked by no set of $k - 1$ creases, because every such set was tried at the level before; so the first count recorded is the minimum, and the search stops at the first level on which all twenty-three fractions up to twelfths are marked.

A fraction and its complement are one target, since a mark two sevenths from one end is five sevenths from the other. The full toolkit marks every one of the twenty-three within four creases on both sheets. The portable toolkit needs six. Each level of the search is thirty to fifty times the size of the one before — the portable search holds thirty-six thousand distinct sets of five creases and one and three quarter million of six, and the full search on the A-series sheet over one and a half million of four — which is why nothing past twelfths is in the table: a seventh portable crease would multiply the work by about fifty again.

Two checks are made on the table itself rather than on the search. The full toolkit contains the portable one, so no fraction may cost more creases on a named sheet with every axiom than with the portable steps alone; none does. And each portable sequence the search returns is carried onto an A-series sheet held both ways, a three-to-two sheet and a triple square, and required to mark its fraction on every one — which checks the invariance on the actual sequences rather than trusting the argument that says it must hold.

A third, portably: four creases #

The cheapest portable third is worth reading step by step, because every portable construction is built from the same few moves.

The first crease halves the sheet; the second is its diagonal; the third lays the left edge onto the halving line, creasing the vertical line a quarter of the way across. The quarter line meets the diagonal at the point a quarter across and a quarter up. The fourth crease runs from the bottom-right corner through that point, and it meets the left edge a third of the way up: a line from $(1, 0)$ through $(\tfrac14, \tfrac14)$ falls a third for every unit it runs left, so it reaches the left edge at height $\tfrac13$.

That is dividing without measuring with the reading step made explicit. The textbook version crosses the diagonal with a line from a corner to a midpoint and finds the crossing a third of the way along; but a crossing in the middle of the sheet is not a mark on an edge, and transferring it to one costs another crease. The search found a route that puts the crossing on the edge directly by choosing which corner the last crease starts from, and it saved the transfer.

On the square, the full toolkit marks a third in two creases, and the two are Haga’s: halve the far edge, then bring a corner to its midpoint and pinch where the folded edge lands on the other side.

On the A-series sheet the same two creases give two sevenths, as the earlier essay on Haga’s fold found, and the cheapest third takes three creases by a route that has nothing of Haga’s in it: a fold along the sheet’s own diagonal, which lands one long edge across the other a quarter of the way along it; a crease straight across the sheet three eighths of the way up; and a slanted crease through two of the points those make, which meets the top edge a third of the way along. The square saves two creases on a third, and the A-series sheet saves one.

Two sevenths: five creases, three, and two #

Two sevenths is the fraction the question was asked about, and its three numbers are the whole result in miniature.

The portable route is five creases, and the arithmetic of the last one is worth doing because it shows where a seven comes from when every move is a halving. The point where the eighth line meets the quarter line is $(\tfrac14, \tfrac18)$. A crease from the top-left corner $(0, 1)$ through it falls $\tfrac78$ while running $\tfrac14$ across, so it reaches the bottom edge after running $\tfrac14 \cdot \tfrac{1}{7/8} = \tfrac27$. Every mark the first four creases made is a dyadic fraction, and the seven appears only in the last crease, as the ratio of two dyadics. That is the general pattern of the portable table: the halvings build a grid of eighths, and one line through two grid points reads off whatever ratio its slope carries.

With every fold allowed the two sheets part company.

On the A-series sheet two sevenths is the cheapest non-trivial mark on the sheet — it is exactly what Haga’s fold produces there, and the fold that a construction assumes its sheet called a silent failure is, read the other way, the cheapest known route to two sevenths on paper of that proportion. On the square it costs three, by a route that starts as Haga’s does and ends in a slanted fold meeting the sides at forty-seven and thirty-five seventy-seconds of their height — a crease nobody would think to make, found only because the search tries all of them. And on any rectangle at all it costs five.

So the answer to the question is three creases, and it has a second half the question did not ask. The square is not where two sevenths is cheapest. Measured against the portable five, the square saves two and the A-series sheet saves three, on the very fraction that sheet was blamed for.

The square is not the cheapest sheet #

That reversal is not a quirk of sevenths. Across all twenty-three fractions the two sheets are cheap in different places, and by the same amount on average.

The full toolkit averages 2.96 creases a fraction on the square and 2.91 on the A-series sheet. The square is strictly cheaper on six fractions — a third, a sixth, two and four ninths, and one and five elevenths. The A-series sheet is strictly cheaper on seven — a quarter, all three sevenths, a ninth, a tenth and three elevenths. They tie on the other ten. The quarter is the starkest: the A-series sheet marks it in one crease, because folding the sheet along its own diagonal lands one long edge across the other exactly a quarter of the way along it. On a square the same fold lays edge onto edge and marks nothing at all.

The sheet decides which points exist found the square at the bottom of five proportions for reach — two rounds of folding mark 565 points on it against more than a hundred thousand on the A-series rectangle — and gave the reason: a square has eight symmetries and a rectangle four, so folds a rectangle keeps apart coincide on a square. The quarter above is that argument in one crease. The square’s diagonal is an axis of its symmetry, so folding along it lays an edge exactly onto an edge and marks nothing; the rectangle’s diagonal is not, and the same fold marks a quarter. In price the gap is much smaller than in reach, because a cheapest route needs only one way to each fraction rather than many, but it runs the same way. The A-series sheet is not an odd shape to be folding on either: it is the one rectangle that keeps its shape when halved, and most of the paper in the world is cut to it.

The standard recipes are written for the square, and they are cheap there because they were found there; nobody searched the A-series sheet for recipes, because nobody folds constructions on it. The square’s reputation as the natural sheet for constructions is a fact about where the constructions were looked for, and the table gives no sign that it is a fact about the square. A folder handed an A-series sheet and the knowledge of its proportion has, fraction for fraction, as cheap a toolkit as a folder handed a square — with different cheap fractions.

The portable column is the other half of that statement. It averages 4.65 creases a fraction, one and three quarters more than either sheet, and it is the same number on both of them and on every other rectangle. Whatever a named sheet saves, it saves against that.

What knowing the sheet is worth #

The difference between the portable count and a sheet’s own is the value of knowing which sheet it is, in creases, and it has a small and definite range.

On the square the saving is nothing on two fractions, a half and a quarter; one crease on five; two on fourteen; and three on two, an eleventh and five elevenths, each reached by a third fold landed against Haga’s crease. On the A-series sheet it is nothing on a half alone, one on seven, two on twelve, and three on three — a seventh, two sevenths and three elevenths.

No fraction up to twelfths saves more than three creases on either sheet. So the whole of what a folder gives up by refusing every slanted fold — every perpendicular bisector off the grid, every angle bisector, every pinch against a flap folded across a slant — is at most three creases a fraction and about two on average, in exchange for a recipe that works on any sheet that comes out of a guillotine.

That is a small price, and it can be put into millimetres. Exact is not accurate followed a hand that places every crease half a millimetre out on a fifteen-centimetre sheet and found the final mark of a chain of crossings 0.31 mm out at three creases, growing only as the square root of the crease count — so two extra creases cost the final mark about a tenth of a millimetre. Against that, Haga’s fold on a sheet cut a millimetre out of square was found in a stretch keeps crossings to move its mark by 1.8 millimetres. The portable recipe loses a fraction of a millimetre to its extra creases and cannot lose anything to the sheet; the sheet-specific recipe saves two creases and stakes the result on a proportion nobody measured.

Where the cost comes from #

The portable column has a shape, and it is the shape of what halving can do.

The dyadic fractions are cheapest: a half in one crease, a quarter in two, three eighths and an eighth in three, each by halving what the last crease left. Every portable construction of anything else begins by laying down a small grid of such lines and a diagonal, and then spends one crease, occasionally two, reading a ratio off it — a line through two points of the grid, whose slope carries the ratio, and sometimes a second crease square to an edge to carry a crossing out to where it can be used. The thirds and sixths come at four creases, from a grid of quarters. The fifths, sevenths, ninths, tenths and twelfths come at five, from grids of quarters and eighths. The elevenths are the only fractions that need six.

The pattern explains why the cost is flat rather than growing. Folding a strip into thirds by Fujimoto’s halvings never arrives at all, and is portable for the same reason the grid is — every step is a midpoint. One crossing, and then another walks down a chain of crossings — a half, a third, a quarter, a fifth — and pays a crease a step, so its seventh is six creases deep. The search finds a seventh at five by building a grid first and reading the seven off a slope, which is a different construction and a cheaper one. A chain pays per denominator and a grid pays per bit: the grid of eighths costs three or four creases however many different ratios are then read from it, and the reading costs one.

The elevenths show the grid failing to be enough. No line through two points of a grid of quarters and eighths reads off an eleventh at an edge, so the portable route to five elevenths first makes a point the grid does not have: two slanted creases, one from a corner to the middle of the far edge and one from the far corner to the top quarter mark, cross two fifths of the way across and four fifths of the way up. A sixth crease through that point and the one where the quarter line meets the diagonal then reaches the top edge at five elevenths. On the square, the full toolkit gets five elevenths in three, because a pinch against a slanted flap can land anywhere and the grid is not needed at all. That is the specific thing a slanted fold buys: a mark wherever a folded edge happens to fall, without the grid that a line through two points has to be read against.

What the counts cannot show #

Accuracy is not in the table. Each count is a count of exact creases, and a crease’s error depends on how it was made. A crease through two marked points is only as good as the angle between them, and a crossing of two nearly parallel creases pins its point badly however few creases it took. The portable route to two sevenths ends in a steep line through a point an eighth of the way up, which is a shallow crossing with the bottom edge; the cheapest route is not necessarily the most accurate one, and ranking them by conditioning is a different search.

The axioms past the fourth are excluded. Folding a point onto a line through another point, or two points onto two lines, reaches numbers the first four cannot — cube roots among them, which is where the cubic comes from — and for rationals they might occasionally shorten a route. The full column is therefore the cheapest route with the first four axioms, and a count including the fifth and sixth could only be the same or lower. The portable column is unaffected: neither of those axioms survives a stretch.

A crease is not a fold-and-unfold, and a pinch is not a crease. The count treats a full-length crease, a half-length pinch and a landing read against a held flap as costing one, one and nothing. A folder’s time and error are not distributed that way, and a count weighting them would move the columns — probably toward the portable toolkit, whose moves are mostly edge onto edge.

And every count assumes the recipe is known. The search finds the cheapest route; a folder has to be told it. A five-crease recipe for two sevenths that nobody has written down is worth less than a longer one somebody teaches, and the table says nothing about how discoverable any route is.

The sheet and the search it rests on #

Every sheet is an exact rectangle, every crease an exact line and every fold an exact reflection. The square has side one; the A-series sheet has sides one and $\sqrt{2}$, and since every edge of it counts, holding it tall or wide makes no difference to its column.

A mark is a point on an edge. Interior crossings are used as stepping stones but are not targets, because a division is read off an edge. A fraction is marked when a crease meets an edge there or an edge lands there under a held fold, measured from whichever end is nearer.

A landing is an edge’s image, cut to the part that actually moves. Folding along a crease carries the paper on one side across; each edge of that side lands along its mirror image, and only the image of the part of the edge that was on the moving side is a flap a pinch can be made against.

How the counts were checked #

Every count is the first level at which a breadth-first search over sets of creases marks the fraction, which makes it a minimum rather than an upper bound; the search stops only once every fraction up to twelfths is marked, and a fraction unmarked at the depth cap would be reported as unreached rather than as the cap.

The full toolkit is required never to cost more than the portable one on any fraction, because it contains it. Each portable sequence is stretched onto four other rectangles — an A-series sheet held both ways, a three-to-two sheet and a triple square — and required to mark its fraction on each. The averages are required to differ by more than a crease, and each sheet is required to win somewhere, so a table in which one sheet dominated would stop being drawn rather than be read as this one.

Still open: the recipe ranked by its error #

The counts rank recipes by length, and a folder cares about the error at the end. A search that carried each crease’s conditioning — how sharply each crossing it relies on is defined — could return, for every fraction, the most accurate recipe of each length, and the question worth answering is whether the portable recipes, built from grids of edge-parallel lines crossing squarely, are more accurate than the cheaper sheet-specific ones despite being longer. The two-sevenths comparison suggests they may not be, because its last crease is steep.

The other direction is the thirteenths and beyond. The portable column is flat at five or six through twelfths because a grid of eighths reads off most ratios in one more crease; whether the portable cost keeps growing like the number of bits in the denominator, as the grid account predicts, or whether some denominators need a second reading step, is the next question the search can answer, and it needs a search that reuses the grid rather than rediscovering it for every fraction.

Sideways from here, the axiom that reaches furthest wastes most counts reach by rounds of every fold at once, and reachable is not cheap prices a number by the height of its tower. The crease count here is a third price, and it is the one a folder pays; how the three orderings of the same numbers disagree would say which of them any of the others is a good proxy for.

The habit worth carrying is about what a convention costs. When a practice is standard, price it against the alternative before explaining it. The square’s recipes are cheap on the square because they were searched for there, and the same search on another sheet finds recipes just as cheap for different fractions; a convention that looks like an optimum is often only the place where anybody looked.