One sheet down

Decided before the design found both sides of the paper present over nearly every point of most folded footprints, and priced a colour change by what stood in the way: the mean depth of the pile. On the preliminary base eight layers lie over every point, and the essay’s summary was that the other side is not scarce, it is under eight layers of paper. On a Yoshimura it put the reverse colour under sixty.

Which side arrives then ordered the piles and read off the top of each. The preliminary base shows the side that started face up over one part in a thousand of its footprint; the square twist over just under half; and nobody chose either number, because on each pattern the rules admit exactly one order.

Ordered piles answer the earlier question too, and more sharply than it was asked. The mean depth of a pile says how much paper is over a point. It does not say where in that paper the other side is, and the other side’s position is what a colour change has to reach. Reading one layer further into the ordered pile — at each point, how many panels lie above the highest one showing the other side — gives the number the first measurement was standing in for.

The census, taken over every state and both faces #

The census is taken on every pattern whose pile can be ordered: the four printed patterns small enough — the preliminary base, the square twist, the hexagon twist and the fold-and-cut triangle — and four patches of the Miura fold, from two by two to four by three, which is as large as a Miura patch gets before the ordering search runs out of room. The Miura patches have one, three, six and eleven folded states; the preliminary base and the twists one each; the triangle two. A ninth case, the letter fold, is added for a reason the census itself supplies.

Each pattern is folded, its panels are placed by composing reflections, and every order of the panels that the non-crossing rules allow is found. For every such order, the folded footprint is sampled on a grid, and at each sample the panels over it are sorted by height. The panel on top shows one side of the paper. The census records how far down the list the first panel showing the other side is, or that there is none. It then does the same from below, since a folded object has two faces and a colour change can be wanted on either.

On every pattern with one folded state the answer is one, at every sample, from both faces. The preliminary base, both twists and the two-by-two Miura never put the other side more than one layer down anywhere it is present at all. On the four patterns with more than one state it is one almost everywhere and two over a sliver: 0.9 per cent of the fold-and-cut triangle’s samples, 3.6 per cent of the three-by-two and three-by-three Miura patches’ and 5.7 per cent of the four-by-three’s, counted over every state and both faces. Nothing measured puts it three down.

Where the bar is grey, the other side is absent rather than deep. That is about a third of the square twist’s footprint, over a quarter of the hexagon twist’s and nearly all of the fold-and-cut triangle’s, and it is the region decided before the design found carrying only one class of panel. There a colour change is not a matter of depth at all; there is no paper of the other colour to bring up.

The pile is deep and the other side is not #

Put the two numbers side by side and the old price comes apart.

The preliminary base has eight layers over every point and the other side one layer down. The three-by-three Miura patch averages five and a half layers and puts the other side at 1.04. The four-by-three averages six and puts it at 1.06. The number of layers over a point and the number above the reverse colour have nothing to do with each other, and the second is the one a colour change pays.

The earlier essay was measuring something true — how much paper is there — and reading it as something it is not. Eight layers over a point of the preliminary base are eight sheets, alternating sides, and the reverse colour is not under all of them; it is under the first. The Yoshimura’s sixty layers could not be ordered and are not in this census, so what the Yoshimura’s reverse colour is under is not known, but the reason given for its being under sixty was the pile’s mean depth, and that reason is now known to be the wrong kind of number.

Why one: every crease turns the paper over #

The preliminary base shows the mechanism most plainly.

The panels alternate, top to bottom; across the whole footprint, 99.6 per cent of the preliminary base’s piles alternate all the way down, and the remainder are samples on the edge of a panel. A fold along a crease reflects the paper beyond it, so two panels joined by a crease always show opposite sides of the sheet — the two-colouring every flat-foldable pattern has is exactly this fact, stated for the whole pattern at once. On the preliminary base every panel is wrapped round its neighbours in the pile by a crease: the base is a square folded into quarters along its creases, and each layer of the stack turns at an edge into the layer below. A pile built entirely of wraps alternates, and in an alternating pile the other side is always the next sheet down.

The Miura patches say something subtler, and it is the more useful half. On the two-by-two patch every pile alternates all the way down, as the preliminary base’s does. On the three-by-two only 92.5 per cent of the piles do, on the three-by-three 77.5 per cent and on the four-by-three 70.4 per cent: in the rest, two panels showing the same side lie against each other somewhere in the pile, where two columns of the zigzag interleave. In the first state the ordering search returns they are never at the top or the bottom. Every patch keeps the other side one layer down from both faces in its first state, so there the same-side pairs are buried and the surfaces of the pile are wrapped even where its interior is not.

That is why the patches’ piles can grow from three layers to six without the other side moving far. What a colour change has to reach is not the pile’s structure in general but its outermost two layers, and on every one-state pattern measured those two layers are a sheet and the sheet it is folded round.

Two same-side sheets are a choice #

The rule has an exception, and the exception is what the ninth case is for. The ordinary letter fold — a sheet creased in thirds and both outer thirds folded onto the middle — puts two panels showing the same side against each other.

The two flaps are not wrapped round each other. Each is wrapped round the middle panel, from opposite edges, so each shows the side opposite to the middle, and therefore both show the same side, and they meet. From the flaps’ face the other side is two sheets down.

And nothing decides which flap is on top. The rules that order a pile relate panels joined by a crease, or panels whose creases land inside one another, and the two flaps of a letter fold are neither; either can go on top, so the letter fold has two folded states. That is not a coincidence of this example. On the census the pairing is exact in both directions: every pattern with more than one folded state puts the other side two down somewhere, and no pattern with one does. The fold-and-cut triangle does it in one of its four views — two states, two faces — the three-by-two and three-by-three Miura patches in a state after the first from one face, and the four-by-three in seven of its twenty-two, over about a fifth of the footprint in each. Same-side sheets meeting at the surface are sheets the crease pattern does not order, and a pattern that has them has a choice in it for the same reason it has them.

That is an observation over eight patterns and a mechanism that explains it, not a theorem. A pattern could in principle order two same-side neighbours through a longer chain of constraints and still have one state; none of the patterns measured does.

The cheapest change: relettering the creases #

If the other side is almost always one sheet down, the price of a colour change is the price of moving one sheet. Which side arrives showed the pile itself cannot be rearranged: there is no height to swap on a pattern with one ordering. The change has to be made to the pattern.

The smallest change a crease pattern admits adds no crease and moves none. It changes letters — turns some mountains into valleys and back — and folds the same creases the other way. Every lettering of the preliminary base, the square twist and the fold-and-cut triangle can be enumerated, sieved for the ones that fold, ordered, and looked down on from above, so this change can be tried exhaustively.

The preliminary base has 112 letterings that fold, with 336 folded states between them. They show exactly two faces. Half the states show the printed face — the reverse side over 99.6 per cent of the footprint — and half show the face-up side over all of it. Two letters are enough to get from one to the other. There is no lettering between: none shows the reverse on one flap and the face-up side on another. A relettering of the preliminary base changes the colour of everything or of nothing.

The fold-and-cut triangle is the same with even less in it. Eighteen of its letterings fold, with forty-four states, and every one of them shows one of the two faces its printed lettering already shows.

The square twist turns its face round #

The square twist is the case where relettering does something, and what it does is instructive.

Eight of its letterings fold, and they give four faces. Every one shows the face-up side over exactly 48.6 per cent of the footprint, and the three that differ from the printed face are the printed face turned about its centre by a quarter, a half and three quarters of a turn. Changing six letters moves 17.6 per cent of the footprint to the other colour and moves the same area back elsewhere; eight letters move a third. Nothing is recoloured. The pinwheel of colours on top of a square twist is fixed by its creases up to a rotation, and relettering chooses which way it points.

That is what the twist’s symmetry predicts once it is said aloud. The square twist’s creases are unchanged by a quarter turn, so turning a folding lettering a quarter turn gives another folding lettering of the same creases, and it folds to the same object turned — which is the same face turned. Each face is reached by two letterings, and one lettering four letters away from the printed one folds to exactly the printed face. What the census adds to the symmetry argument is that there is nothing else: no lettering outside that family folds at all.

So on all three patterns small enough to try it, no relettering recolours part of a face. Every face any lettering shows is the printed face, the printed face turned, or the whole face changed at once. The reverse colour is one sheet down almost everywhere and no change of letters reaches it anywhere in particular.

What a colour change actually costs #

That locates the cost more precisely than any earlier essay could. It is not the depth of the pile, which is irrelevant; it is not the depth of the other side, which is one; and it is not a choice among the pattern’s letterings, which offer nothing local. It is new creases.

A designer who wants the reverse colour on one flap has to give that flap a way to turn over that the rest of the model does not share: a crease across it, along which the one sheet above the reverse colour can fold back — a reverse fold, in the traditional vocabulary, or a pleat that brings a strip of the other side to the surface. Each is a crease the base did not have, and each changes the pattern’s colouring locally, which is the only thing that can change a face locally.

And every such crease is paid for in paper. Paying in paper prices a feature added to a finished design at exactly the strip slid in for it, and a crease that turns a flap over is not free either: the flap loses the length it folds back. That is the price bringing the other side to the front put at twice the area shown, and it can now be said why it is the right currency. The reverse colour is never far away. It is fixed in place by the creases, and only more creases unfix it.

What the census cannot show #

The large patterns are missing. The Miura beyond four by three, the Yoshimura, the waterbomb tessellation and the tapered corrugation all have too many panels for the ordering search, so what lies one layer down in them is unknown. The census covers eight patterns with up to thirteen panels; the claim that the other side is one sheet down is a claim about them and about piles built the way they are, by wrapping.

The samples are a grid. Every share is counted at the centres of a grid over the folded footprint, fine enough that the layer count integrates back to the area of the sheet; a region narrower than a grid cell — a sliver of depth two along an edge — could be missed or miscounted by a cell’s width.

Relettering is tried on three patterns. The hexagon twist has 262,144 letterings and was not enumerated; the Miura patches’ letterings were not tried at all. Whether a larger pattern has a lettering that recolours part of its face is open, and a pattern with more independent flaps is the natural place to look for one.

And a sheet is not a colour. The side that started face up is a label; duo paper makes it visible and plain paper does not, and none of this is about ink or about how a model is lit.

The pile the numbers assume #

Every panel has no thickness and every folded state is flat. A pile is a list of panels in height order over a point, which is a statement about an idealised object; a real model has a crease with a radius and layers that are not quite parallel, and a reader looking at an edge sees the curled edge of the sheet beneath as well as the top one.

A folded state is an order the rules allow. The ordering search finds every order that satisfies the crease rule and the two non-crossing rules, as which side arrives describes; a state that paper could not reach by any sequence of moves is still counted if it satisfies them.

And “the other side” is read panel by panel. A panel shows one side of the sheet over the whole of itself, so the depth at a point is a count of whole panels, never of fractions of one.

How the numbers were checked #

Every depth is read off an ordered pile at a sample, over every folded state the search returns and from both faces, so no pattern is reported by whichever state happened to come first. The census is required to find the other side at most two layers down on every orderable pattern, one down on every pattern with one folded state and two down somewhere on every pattern with more, and the letter fold is required to put it two down — so a census that broke the pairing in either direction would refuse itself rather than claim it.

Each relettering census is required to have searched every lettering to the end, with none left undecided by the ordering search’s budget. Every face is compared sample by sample with the printed face and with the printed face turned by a quarter, a half and three quarters, and the figure is refused if any face changes part of the printed one without being a turn of it.

Still open: the smallest crease that recolours a flap #

The cost is now a number of creases, and the number has not been counted. For each flap of the preliminary base, what is the fewest creases that, added to the pattern and lettered, give a face showing the reverse on that flap and nowhere else? The same census can answer it for a single added crease: try every crease through two of the pattern’s vertices, every lettering of the result, and look down. A traditional reverse fold is one candidate, and whether it is the cheapest is a finite question.

The other direction is the patterns the census could not reach. The Miura’s piles alternate at every size tried, and the account above says why — wrapping — and predicts that a large Miura keeps the other side one sheet down except where a later state lays two same-side sheets together. A cheaper route to a large pile’s top two layers, without enumerating every order, would test that on the Yoshimura and the waterbomb, where the earlier price of sixty layers was quoted.

Sideways from here, the pairing of same-side sheets with free choices belongs beside one marking, many objects, which counts how many folded objects a single lettering makes. If every extra object comes from a pair of same-side sheets the creases leave unordered, then the number of folded states and the number of such pairs are two readings of one quantity, and counting pairs would be much cheaper than counting states.

The habit worth carrying is about proxies. Before pricing something by a quantity, ask whether the quantity is where the thing is, or only how much there is around it. A pile’s depth was a good measure of how much paper lay over a point and was quoted as the depth of the reverse colour, which it is not; the difference between them was eight layers against one, and it took ordering the pile to see it.