Two corners compete across an edge

A corner turned shows the sheet beneath found the cheapest change that recolours part of the preliminary base. Relettering alone turns the whole face one colour or the other, so the next cheapest change adds a crease, and the angle condition allows exactly one kind: a crease across a diagonal at right angles, cutting a corner off. Placed $s$ along each edge from the corner it divides the face into three regions — where the corner lay, where it lands, and the rest — and the letterings of the result colour those three in all eight possible ways.

That essay ended on the obvious next step and a guess about it. Two creases cut the face into more regions, and the open question was whether their letterings still reach every colouring, or whether two corners competing for the same layers beneath would put some combinations out of reach. It proposed the pairs worth trying first: corners that share paper under them, and corners that do not.

Both have now been counted, every lettering of each folded, ordered and looked at from both faces. The guess was right about the split and wrong about where the competition would come from. Two opposite corners colour their five regions in all thirty-two possible ways, on the upper face and on the lower. Two adjacent corners reach twenty-nine on each face, and the three each face cannot show are the other face’s three with every side exchanged. What separates the two pairs is an edge: adjacent corners are the two ends of one edge of the square, which the base folds in half, and opposite corners share no edge at all.

Where two corners go when the base is folded #

The regions come from the folded geometry before any letter is chosen, and the geometry of the preliminary base makes them unusually tidy. Folded, the base brings all four corners of the square to one point — the base whose pile which side arrives ordered layer by layer — and it lays the four half-diagonals along one line, the base’s long axis. A crease cutting a corner off runs at right angles to that corner’s diagonal, so in the folded base every such crease lands at right angles to the same axis, a distance along it set by how far the crease was from its own corner. Two corner creases, wherever on the square they were drawn, fold to two parallel segments across the same narrow end of the triangle.

So the small corner’s triangle sits inside the large one’s. The large corner, cut $s_A$ from its tip, folds along its crease and lands as the mirror image of its triangle on the other side of the crease; the small one, cut $s_B$ from its tip, does the same at its own distance. Five regions follow, and each has an area the geometry fixes:

• where both corners lay, the tip of the folded triangle, $2s_B^2$ of the footprint;
• where the small corner lands, the mirror of that tip across the small crease, also $2s_B^2$;
• the band only the large corner left, between the two creases and outside the small corner’s landing, $2s_A^2 - 4s_B^2$;
• where the large corner lands, $2s_A^2$;
• the rest of the face, $1 - 4s_A^2$.

The count of five depends on the small corner’s landing staying inside the band. Its mirror triangle reaches $2s_B$ along the axis from the tip, so the arrangement above holds while $2s_B \le s_A$. The census here uses $s_A = 0.3$ and $s_B = 0.15$, the largest small corner that keeps it, which gives the five regions 4.5, 4.5, 9, 18 and 64 per cent of the face.

The measured shares match the formula to within the sampling grid, and that agreement is the check that the regions are what the probes say they are. Each region is identified by carrying a point of the flat sheet through the folding — a point near a corner once through that corner’s own motion, to where it lands, and once through the motion of the panel across its crease, to where it would have stayed had the crease been left flat. The second route is how “where the corner lay” is located on a sheet whose corner has gone.

Every lettering of two creases #

The census is the one-crease census grown by three creases. The added creases cross a diagonal each and are cut there into two pieces, so the pattern has fourteen creases where the base had eight, and $2^{14} = 16{,}384$ letterings. Every one that passes the vertex conditions is folded by composing reflections, every stacking the non-crossing rules allow is found — the rules that make a collision an order rather than an event — and each region is read from above and from below: the top panel over a region is the same panel at every point of it, because the region is defined as the points with the same panels over them, so a face is exactly one side per region.

Two opposite corners give 1,456 letterings that fold, with 3,424 folded states between them. Every one of the thirty-two colourings appears from above, and every one from below. A designer who wanted the small corner shown and the large one hidden, or the band uncovered and the tip left as printed, or any other assignment of the two sides to the five regions, has a lettering for it. The two corners do not compete at all. Each region changes colour for the two reasons the single corner showed: where a corner lands it shows its back, because folding a panel over reflects it and the sheet has two sides; where a corner lay, the sheet beneath shows, and one sheet down found that the sheet beneath shows the other side at every point of this base.

The price, in letters of the base changed, is small and grows slowly. The printed face is reached with none changed, and every colouring on the upper face is reached with at most five of the base’s eight original creases relettered. The dearest all turn the rest of the face over while leaving the band as printed, and every one of them also changes a region the small corner decides. With one crease the dearest colouring cost three; the second crease adds two to the worst case, which is what the single-crease account would lead one to expect if each added crease, lettered as a packet, has to buy its own vertex’s balance.

Adjacent corners lose three #

The adjacent pair has the same five regions, at the same shares, and slightly more letterings that fold — 1,488, with 3,536 states. It reaches fewer faces.

Three colourings never appear from above. All three show the side that started underneath where both corners lay and in the large corner’s band, the side that started face up where the small corner lands, and the face-up side again in at least one of the two remaining regions — the large corner’s landing, the rest of the face, or both. Put as a designer would put it: with the two vacated places left as printed, the small corner can be folded forward to show its back, but only if nothing else on the face changes.

All three appear from below. And the three the lower face cannot show are exactly these with every side exchanged — the face-up side where both corners lay and in the band, the reverse where the small corner lands, and the reverse again somewhere else. The constraint is one rule, read from two sides, which is what it must be: a lettering with every letter reversed folds to the same stack turned over, so whatever one face cannot show, the other cannot show in the exchanged colours.

What the small corner forces when it is on top #

The rule is easier to read from the states that obey it than from the colourings that break it. Take every folded state whose upper face shows the reverse side where both corners lay and in the band, and the face-up side where the small corner lands: there are 54 of them. In all 54 the top layer where the small corner lands is the small corner’s own tip, folded over onto the face and showing its back. And in all 54 the large corner, where it lands, is buried — at least one layer down and as many as seven, never on top — while the rest of the face is the printed reverse side. Not one of them shows the face-up side anywhere else.

So the three missing colourings are three ways of asking for one more thing while the small corner lies on top over undisturbed ground: the large corner on top as well, the rest of the face turned over, or both. Opposite corners grant all three. Adjacent corners grant none, and since every lettering was tried, no lettering of these fourteen creases does.

What the census does not show is the reason in a sentence. It shows the arrangements that occur and the ones that do not, and the account below is an account of why the pair matters, offered on that evidence rather than proved.

Why an edge, and not the paper beneath #

The essay this one continues guessed that corners would compete when they share the paper under them. In the folded preliminary base that test cannot separate the two pairs, because every corner shares the paper under it with every other: all four corners meet at one point, so the layers beneath the large corner’s band are the same eight-layer pile whichever corner the small crease cuts. If shared paper were the mechanism, opposite corners would compete as much as adjacent ones.

What the adjacent pair has and the opposite pair lacks is a connection along the outside of the sheet. The edge from one adjacent corner to the other is folded in half at its midpoint by one of the base’s half-midlines, so the two half-edges, one belonging to each corner, lie face to face in the folded stack, joined by that fold. A fold is a hinge between two panels, and the rule that refuses crossing creases says that anything lying between the two panels of a fold must stay between them all the way to the hinge. The small corner folded over on top, over a band left as printed, fixes where its own half-edge lies in the pile; the hinge then fixes where the large corner’s half-edge may go, and the evidence above is that it leaves the large corner nowhere but buried. Opposite corners are joined only through the centre of the sheet, by a path through four creases, and nothing ties their two flaps together near the tip. That is the account, and it is consistent with every state counted; it has not been proved, and a proof would have to follow the hinge’s constraint through the eight layers between the two half-edges.

When the two distances match #

Moving the small corner’s crease out to the large one’s distance does something the count of creases cannot predict.

At equal distances the two corners land on each other. Both creases fold to the same segment across the axis, both corner triangles lie over the same tip, and both fold to the same mirror triangle; the five regions merge into three, the same three a single crease makes — where the corners lay, where they land, and the rest. The letterings colour those three in all eight ways, for adjacent corners and opposite ones alike, and no more.

Yet the letters have more to do. Opposite corners at equal distances have 4,960 folded states against 3,424 at unequal ones, and adjacent corners 6,356 against 3,536: moving the small crease out raises the number of folded objects by a half and more, while the face loses two regions. None of the extra states shows anything the eight colourings do not already include. What they differ in lies below the top layer, and at equal distances that is where two whole flaps lie over exactly the same ground, so that any freedom in how the two are ordered there is a freedom no one looking at the face can see.

That is the gap one marking, many objects opened between counting folded objects and counting what they look like, at its widest here. A single crease more than tripled the states of the plain base, 336 to 1,064, while giving its face three regions; a second crease at the same distance multiplies them by between four and a half and six and gives it none. The number of folded states is a count of arrangements of paper, and an appearance is a function of the top layer alone; whenever two flaps can be ordered either way over the same ground, the first count grows and the second does not.

What the census rests on #

The model folds every crease completely. The base here is all eight triangles stacked on one — the folded footprint whose depth times area is all the paper there is — not the open square a folder handles with its flaps spread, and the regions and their shares are shares of that fully folded triangle. In the open form the corners do not lie together, the two creases would not fold onto one axis, and the regions would be different shapes; the five-region arrangement and its formula are this model’s.

The paper has no thickness. Two corners folded over the same eight-layer pile land at different heights, and tucking one inside the other means sliding it between layers that are pressed together. Neither changes which side shows, but both change what is easy: a colouring that needs the small corner tucked under the large one is a colouring a folder reaches by feel.

The corners were cut in one arrangement. Every census here keeps $2s_B \le s_A$, so the small corner lands inside the large one’s band. With the small crease further out its landing reaches into where the large corner lands, and the face has six regions rather than five; those pairs were not counted, and the rule for which colourings they lose need not be the rule above.

The printed lettering is one of many, and it is lopsided. One sheet down counted 112 letterings of the base that fold, and the printed one is only one of them. The base’s printed letters put three half-diagonals one way and the fourth the other, so its four corners are not alike when the price of a colouring is counted against the printed letters. A crease at the odd corner cannot be lettered to fold at all without changing one of the base’s own letters. The pairs here were chosen to avoid that corner, so that each census includes the printed face; the counts of colourings do not depend on the choice, because every lettering is tried, and the prices do.

The base is traditional, its corners folded over as long as anyone has folded it, and nobody designed the pair of creases either. What has been measured is a property of the folding, not of a design, which is the stance decided before the design took toward colour change from the start: measure what the folding allows before asking what a technique adds.

How it was counted #

The regions are read from the folded geometry, once per pattern. A grid of 120 by 120 sample points is laid over the folded footprint and the points are grouped by the set of panels over them; groups smaller than one per cent of the footprint are slivers along a shared edge and are dropped. Each region is then named by where a probe point lands, and its share is required to match the formula above.

Every lettering is folded and every stacking found. For each of the $2^{14}$ letterings that pass the vertex conditions, the panels are placed by composing reflections, a lettering whose letters force a loop is set aside, and every stacking the non-crossing rules allow is listed; no search was cut short. For each stacking the top panel over each region is read from above and from below, and its side — the one that started face up or the other — is that region’s letter in the colouring.

The counts are required, not merely observed. The opposite pair is required to reach all thirty-two colourings on both faces; the adjacent pair to reach exactly twenty-nine on the upper face, with the missing three obeying the rule stated, appearing on the lower face, and matching the lower face’s missing three exchanged; and the equal pairs to give three regions and eight colourings with more states than the unequal ones.

Still open: the reverse fold on a deeper flap, and the six-region pairs #

Two leads are left by these counts, and one older one is still waiting.

The six-region pairs are the obvious continuation. With $s_A < 2s_B$ the small corner’s landing overlaps the large corner’s landing, and the overlap is a region where two flaps folded forward lie over the same ground. Whether the adjacent pair’s rule survives there, or whether a new constraint appears where two flaps land together, is the same census at a different setting, and the region count can be read from the geometry before any letter is tried.

The reverse fold on a deeper flap is the lead the previous essay left and this one did not take. On the preliminary base a corner has two layers and a reverse fold turns it inside out between them, which changes nothing anyone can see. On a flap of four or more layers, such as the bird base’s, a reverse fold carries its corner between different layers, and whether that ever shows something a corner folded over cannot — or reaches one of the colourings an adjacent pair loses — is a census on a base with more paper at the corner.

The paper-price of all this is still the one bringing the other side to the front set: every flap folded over spends twice what it shows, except where uncovering shows the sheet beneath for nothing.

Sideways, the extra states at equal distances belongs beside no height to swap, which found that a stacking of this base cannot be rearranged without relettering. Two flaps over the same ground that can lie in either order are exactly the place where a stacking might be rearranged without changing a letter, and whether the coincident corners’ extra states are reachable from one another by moving paper rather than by changing letters would say whether they are different models or one model held two ways.

The habit worth carrying is about guesses that name a mechanism. When a prediction says which cases will differ, test the reason as well as the outcome. The guess that shared paper would make corners compete predicted the right split, and the reason was wrong: every corner of this base shares its paper, and the difference was an edge.