A corner turned shows the sheet beneath

One sheet down asked what the cheapest change to a crease pattern could do to the colour of its face, and found the answer for the preliminary base was nearly nothing. The smallest change a pattern admits adds no crease and moves none; it only changes letters. Every lettering of the preliminary base that folds — 112 of them, with 336 folded states — shows one of exactly two faces from above: the printed face, the reverse side of the paper over all but a sliver, or the same face turned wholly to the other side. Relettering changes the colour of everything or of nothing.

That essay ended with the next question in the order of cost. If relettering cannot recolour part of the face, what is the fewest creases that, added and lettered, can? And it pointed at where to look: try every crease through two of the pattern’s own points, every lettering of the result, and look down.

The answer is one crease, and it comes with more structure than the question asked for. The angle condition leaves that crease a single direction. Placed anywhere along that direction it divides the face into three regions of exactly known size. And the letterings of the result colour those three regions in every way there is.

The angles choose the crease #

The preliminary base has nine points a crease could run between: four corners, four edge midpoints and the centre. Leaving out the creases it already has — both diagonals and both midlines, all through the centre — and the sheet’s edges, there are twelve straight creases through two of them.

Eight are refused before a single letter is chosen. A crease from a corner to the midpoint of a far edge, or from a midpoint to a far corner, crosses the base’s diagonals and midlines at angles other than ninety degrees. Where two straight creases cross, they make a vertex of degree four whose opposite sectors are equal, and Kawasaki’s condition asks the alternate sectors to sum to a straight angle — which, with opposite sectors equal, happens only when every sector is a right angle.

That is the theorem the sheet remembers turned into a rule for folding a second model into a sheet that already carries a first: every crease of the second must meet every crease of the first at ninety degrees or not at all. A colour change is a second pattern, a small one, laid into the first. It inherits the rule.

The four that survive are one crease in four positions. Each joins two neighbouring edge midpoints, cutting a corner off the square, and crosses the diagonal from that corner at right angles. By the square’s symmetry any one of them will do, and the rest of this essay takes the one across the corner at the top right.

And the rule fixes only the direction. A crease cutting the same corner off at any distance — from $s$ along one edge to $s$ along the other — still crosses the diagonal at right angles. It keeps the angle condition until $s$ passes a half, where it starts to cross a midline at forty-five degrees. So the cheapest colour change on this base has one dial, and the census below is taken at several of its settings.

What one crease adds #

With the crease in place the pattern has eleven creases rather than eight: the two pieces of the new one, and the diagonal it crosses now in two pieces as well. Every one of the $2^{11}$ letterings was put past the vertex conditions, and those that pass were folded and every stacking of their panels found.

404 letterings fold, in 1,064 states, against the plain base’s 112 and 336. With the crease joining the midpoints, a half of the way along each edge, the faces fall into exactly three kinds by how much of them shows the side that started face up: none of it, all of it, and half. Half is the new thing. It is a colour change on part of the face, and nothing the plain base’s letterings can show.

Three regions, exactly measured #

The half-and-half faces divide along a line, and the line is the reason the shares are exact.

In the model used throughout these essays every crease is folded completely, so the preliminary base stacks all eight of its triangles onto one: a right triangle whose sides are half the sheet’s, eight layers deep, with the four corners of the sheet together at one of its sharp vertices. The added crease cuts the top right corner of the sheet off, as a triangle of area $s^2/2$, split by the diagonal into two halves that fold onto each other. In the folded stack the corner therefore occupies a triangle of area $s^2/4$, out of a footprint of area $1/8$. Folded over along the new crease, it lands on the mirror image of that triangle on the other side of the crease, an equal area.

So the crease cuts the face into three regions:

$$\underbrace{2s^2}_{\text{where the corner lay}} \;+\; \underbrace{2s^2}_{\text{where it lands}} \;+\; \underbrace{1 - 4s^2}_{\text{the rest}} \;=\; 1.$$

And on every face of every lettering, each of the three regions shows a single side of the paper, so the share of the face showing the face-up side is the total of the regions that show it. At $s = 0.3$ the shares possible are nought, 18, 36, 64, 82 and 100 per cent, and all of them occur; at $s = 0.5$ the rest has vanished, since $4s^2 = 1$, and the two remaining regions are a half each.

The measured shares sit on the curves at every setting tried: at $s = 0.2$, 8 and 16 per cent; at 0.3, 18 and 36; at 0.4, 32 and the complement of 64; at 0.5, half. Nothing else ever appears. A colour change made with one crease is built from the corner it cuts off: each of the two regions the corner touches is its area, and the rest of the face is what they leave.

Why each region changes colour #

The regions do not change colour for one reason. They change for two, and both have already been met.

Where the corner lands, the corner shows its other side. Folding paper over a crease reflects it, and a reflected panel shows the face it had underneath; that is the two-colouring every flat-foldable pattern has, seen at a single crease. If the corner folds over onto the face, the region where it lands turns to the side the corner was hiding. If it folds under, beneath the flap it was cut from, it lands out of sight and that region keeps its colour.

Where the corner lay, the sheet beneath shows. With the corner gone, the paper on top of that region is whatever was under it, and one sheet down found what is under the top layer of a preliminary base: the next sheet down, showing the other side, at every point of the footprint. So uncovering the region usually recolours it. Usually, not always, because the letters that move the corner also reorder the layers beneath it, and some letterings put a sheet of the same side underneath.

The two effects are independent of each other and of what happens to the rest of the face, which the letters can turn over wholesale as they could without the crease. That is why the face is three regions coloured in any of eight ways rather than one picture with a few variants.

The paper it spends, and the paper it does not #

Bringing the other side to the front priced a colour change in paper, not letters: what it costs is twice what it shows, because a flap folded over shows one of its sides and spends the other hiding. The corner here keeps that rule exactly. It is a triangle of $s^2/2$ of the sheet, folded into two layers of $s^2/4$, and where it lands it shows its back over $s^2/4$ of the folded triangle. Twice the area shown, spent to show it.

The region where the corner lay obeys no such rule, because it spends nothing. What appears there is paper that was already on the face’s side of the stack, waiting under the corner, and no part of the sheet is turned over to produce it. When both regions change, the corner shows its back over $s^2/4$ and uncovers another $s^2/4$, so the $s^2/2$ of paper folded over shows the other side over $s^2/2$ of the face: one to one, half the price that rule charges. The uncovering is free.

That is the same fact as the paper is all still there, read for colour. A folded sheet is smaller than it was and none of it has gone anywhere, so the area of a footprint times its depth is the sheet. A colour change does not create a reverse side. It rearranges which of the sheet’s two sides is outermost over each point, and the cheapest rearrangement is the one that lets paper already lying the right way up become the top layer by moving what was on it.

And it is the reason the one-sheet-down finding mattered more than it seemed to. If the other side had lain eight layers down, as the pile’s depth once suggested, uncovering a region would show the same side again and the free half of the colour change would not exist. Because it lies one layer down, every region a corner leaves is a region the corner can recolour by leaving — unless the letters that moved it have also brought a same-side sheet to the top, which no height to swap makes the letters’ business rather than the hand’s, since a stacking on this base cannot be rearranged without relettering.

What each colouring costs #

The cost of a colour change is now countable, and it is small. Every partial colouring at $s = 0.3$ is reached by lettering the new crease and changing two or three of the base’s own letters. Recolouring where the corner lands, or where it lay, or both, takes two; recolouring the rest along with either takes three. The whole face turned costs two, as it did without the crease.

The two letters are not arbitrary, and the reason runs through the folder’s most familiar technique.

The added crease makes a new vertex where it crosses the diagonal: the diagonal’s two pieces, both mountains in the printed base, and the new crease’s two pieces. Maekawa’s count asks three of one letter and one of the other. The cheapest way to satisfy it changes nothing in the base at all: letter the new crease’s two pieces differently, one mountain and one valley, and the vertex has three mountains and a valley. In the folded corner that lettering is a reverse fold — the corner turned inside out between its own two layers.

And it is the one lettering of the new crease that changes nothing anyone can see. A corner reverse-folded between its own layers lands out of sight, and the region it left is covered by the flap it was cut from, folded the way it always was. Every partial colouring needs the two pieces lettered alike — the corner folded over or under as a single packet — and then the new vertex has two creases of each letter, or four of one, unless one of the diagonal’s pieces changes too. The piece that changes is the inner one, between the new vertex and the centre; and the centre, which had five creases of one letter and three of the other, must then change one more to keep that balance. Two letters: one paid at each vertex the change passes through.

A crease and two letters is therefore the price of the cheapest colour change on this base, at any setting of the dial. It is a long way under what decided before the design priced a colour change at, by the depth of the pile over the point — eight layers. And the reverse fold, the technique a folder reaches for first, is on this corner the one way of folding the crease that buys nothing.

The dog-ear #

The everyday instance of all this is on the corner of a page. A dog-ear is a corner folded over along a line at forty-five degrees to the page’s edges, which is exactly the crease the angle condition chose here, and it does exactly the two things the regions do: the folded corner shows the back of the page where it lands, and the page beneath shows where the corner was. Fold a corner $s$ along each edge and each region has area $s^2/2$.

The preliminary base differs from a page only in what is underneath. A book’s page is printed on both sides and the page beneath it is another page, so the dog-ear shows text either way; the preliminary base is a two-coloured sheet folded onto itself, so what lies beneath is the same sheet, and which colour it shows is decided by the wrapping that which side arrives read off the ordered pile. A folder making a colour change by turning a corner is doing what a reader does to keep a place, with a sheet whose underside has been arranged in advance.

What the census rests on #

The model folds every crease completely. The preliminary base drawn here is all eight triangles stacked on one, not the open square a folder holds with its flaps spread two and two. In the open form the four corners do not lie together and the face from above is a different shape, so the shares measured are shares of the fully folded triangle. The three regions and their sizes are a property of the corner’s crease and would survive in any form that folds the corner over; the exact percentages are this model’s.

The paper has no thickness. A corner folded over a stack of eight layers lands slightly proud of the face, and turning it under means tucking it between layers that are pressed together. Neither changes which side shows, but both are why a real colour change of this kind is easiest on the outermost flap and harder deeper in.

Only straight creases through the pattern’s own points were tried as candidates, and then the surviving one was moved along its diagonal. A crease ending in the middle of the sheet, a curved crease, or two creases added together were not tried, and a colouring this census cannot reach might be reachable by one of them. What the census shows is that one crease already reaches every colouring of the three regions it defines.

The base is traditional, as old as anybody’s record of folding, and nobody designed the corner either; the crease across a corner is the first fold a child makes in a sheet of paper.

How it was counted #

The candidates are judged by the angle condition alone. Each is drawn into the base, the result is planarised so that every crossing becomes a vertex, and Kawasaki’s condition is checked at every interior vertex with no letters assigned. Four candidates pass, and all four are required to join neighbouring edge midpoints at right angles to a diagonal; each of the eight refused is required to fail at a named crossing.

Every lettering is folded and ordered. For each of the $2^{11}$ letterings that pass the vertex conditions, the panels are placed by composing reflections, every stacking the non-crossing rules allow is found, and the face is read from above and from below on a grid over the footprint — coarsely to sort faces into groups, finely to measure each group’s share. No search was cut short. The three regions are located by carrying a point near the corner through the folding with and without the added crease, and each face is required to show one side over each region and the regions’ total share overall.

The shares are required to be the formula’s, at four settings of the dial, to within the sampling grid’s resolution: every partial face at $2s^2$, at $4s^2$, or at the complement of one of them.

Still open: two creases, and the reverse fold #

One crease reaches eight colourings of three regions. Two creases cut the face into more regions, and whether their letterings still colour them in every possible way, or whether some combinations become unreachable when two corners compete for the same layers beneath, is the same census one step further. The pairs worth trying first are two corners of the same flap, which share the paper under them, and two corners of different flaps, which do not.

The reverse fold deserves a second look on a deeper flap. Here it is invisible because the corner has only two layers and folds between them. On a flap of four or more layers a reverse fold can carry its corner between different layers, and whether it then shows anything — and whether that is ever cheaper in letters than a corner folded over as a packet — is the census above on a base with more paper at the corner, such as the bird base’s flaps.

Sideways, one marking, many objects counts how many folded objects one lettering makes. The added crease more than triples the folded states — 1,064 against 336 — while adding three regions to the face; how many of the new states differ only in layers no one can see is the gap between counting objects and counting appearances.

The habit worth carrying is about the cheapest intervention. Before reaching for a technique, add the smallest thing the rules allow and look at everything it does. Here that was one crease, and it reached every colouring its regions permit.