Which side is showing

The shadow does not name the pattern asked what a folded object records. The observation it gave the observer was the profile — the outline together with the number of layers over every point of it, which is what a photograph of a silhouette carries — and the answer was that 71 of the 233 profiles four creases on a twelfths grid can reach are produced by more than one genuinely different pattern.

The order does not name it either gave the observer more: the complete layer order, meaning which two layers of the stack every fold joins. That is what somebody who may take the model apart layer by layer learns, and it separated 69 of the 71.

Both are the right question and neither is an observation anybody can actually make. A silhouette throws away too much and a layer order is not visible at all. What a person holding a folded model has is a photograph, and this rung is about what one is worth.

What a photograph carries #

Fold from duo paper — coloured on one face, white on the other — and the folded model shows both. The two-colouring of a crease pattern’s panels is the same partition seen in the flat: panels sharing a crease differ, and the reason is Maekawa’s parity rather than a coincidence. Which face a given piece of the sheet presents is decided before anything is folded: walking along the strip, the direction of travel reverses at every crease, and so does which side is uppermost. So the sides alternate along the strip, segment by segment, whatever the letters are and wherever the creases were placed.

That is what makes the colour a real second observation rather than a restatement of the first. Where several segments land on the same place, the colour visible from above is the side shown by whichever of them is on top — so the colour reads one bit of the layer order at every point of the outline, and nothing anywhere else.

It does vary. Of 116 folded states examined across a handful of crease sets, 66 show both sides of the paper from above at once. A reader who has folded duo paper knows this already: a model of any complexity shows patches of the reverse colour without anybody having designed one in, and the design technique that arranges those patches deliberately exists precisely because the alternative is that they turn up wherever the folding happens to put them.

And it is worth nothing #

Over the census at four creases on twelfths there are 233 profiles, of which 71 are reached by more than one genuinely different pattern. Adding the colour — from above and from below, so the observer is allowed to turn the model over — separates none of them. Not few. None.

At three creases the census is smaller and the answer is the same: 69 profiles, sixteen ambiguous, and the colour separates none of the sixteen.

That is a stronger result than it looks, and it is worth resisting the obvious explanation. The colour is not constant — the figures above show it varying across a folded strip and it varies between states of the same strip. It is not redundant in the sense of being computable from the outline: two states of one pattern can have the same profile and different colour words. It carries information. It carries information that never happens to be the information that separates two patterns folding to the same outline.

The one thing the colour does settle #

There is a question the colour answers completely, and it is worth separating from the one it fails at, because a reader who has folded duo paper will already know it.

Which side a folded model shows from above, overall, is decided before anything is folded. The segments alternate, so a strip with an even number of segments shows one colour from one end and the other from the other; a strip with an odd number shows the same colour at both ends. A folder choosing which face of the paper to start with is choosing what the finished model’s dominant colour will be, and nothing about the folding sequence can change it.

That is real information and it is information about the strip, not about the pattern. Every pattern with the same number of creases carries it identically, which is exactly why it separates nothing: it is constant across the groups the census is asking about.

The colours over a point alternate, and that is why the observation is thin #

The result reads as a surprise, and there is a lemma underneath it that makes it much less surprising — and that can be stated and checked independently of the census.

Take any point $x$ of the folded line and walk the strip from one end to the other. The walk crosses $x$ once for each layer over it. Between two consecutive crossings the path stays entirely on one side of $x$: it leaves going right and comes back going left, so it turns around an odd number of times in between. A turn is a crease, and a crease flips which face is uppermost.

So the segments covering any point alternate in side, in strip order. Not approximately, and not usually — always, and for the same reason the sides alternate along the unfolded strip.

That fixes the composition of the stack over $x$ completely. If $L$ layers cover $x$ then exactly $\lceil L/2 \rceil$ of them show one side and $\lfloor L/2 \rfloor$ show the other. A point with four layers has two of each; a point with five has three and two.

Which leaves the colour one bit to carry #

And $L$ is part of the profile. The layer count over every point is exactly what the profile observer already had.

So the colour visible from above is not a new quantity. The set of colours present over a point is a function of the profile; the only thing the colour adds is which of them happens to be on top — one bit per point, and the bit is a single parity of a single layer of the order.

Two things follow immediately, and both are visible in the figures above. Where $L = 1$ the colour is forced, because there is only one layer and its side is decided by the crease count before it. And the colour can vary between states of one pattern only where $L \geq 2$, which is why a folded model shows reverse-colour patches at its thick places and never at its thin ones.

Put that beside what the layer order carries and the 0-against-69 stops looking like an accident of the census. The full order over an $L$-layer point is one of $L!$ arrangements. The colour reads $\lceil \log_2 \rceil$ of one of them — the parity of the first — and it reads it after the profile has already announced how many of each side are down there. It is the thinnest non-trivial observation the interior admits, and the measurement says thin enough to be worth nothing.

Why the invisible half is the half that matters #

The comparison is what makes the result mean something. The complete layer order separates 69 of the 71, and the layer order is the one part of a folded model no photograph can ever contain: it is a fact about the interior of a stack of paper.

So the evidence a folded object carries divides into two very unequal parts. The part on the surface — outline, thickness, colour, from both sides — is worth 0 of 71. The part inside is worth 69. An observer who wants to know which pattern made a model must take it apart, and taking it apart destroys the object being observed.

There is a practical reading of that, and it is about the historical record rather than about geometry. The documentary evidence for a folded model is nearly always a photograph or a drawing of the finished thing, and this measurement says what such a record establishes about the crease pattern behind it: the outline and the layer count, and nothing more. A published photograph of a model is not evidence of a pattern.

What the two counts have in common #

It is tempting to read 0 and 69 as a large gap between two similar things, and they are not similar things. They are answers to questions of different kinds.

The outline and the colour are both functions of the folded object as an object: two models that are congruent, with the same face uppermost, have the same outline and the same colour word. The layer order is not a function of the folded object in that sense at all — it is a description of how the object is assembled, and two models can be congruent as shapes and differ in it.

So the honest statement of the result is not “looking harder does not help”. It is that the outline and the colour are the same kind of evidence, and the whole of that kind is worth 0 of 71. What separates the patterns is evidence of a different kind, and the only way to obtain it is to unmake the thing.

Which theorem was checked, and how #

The census is exhaustive rather than sampled: every choice of four crease positions from a grid of twelfths, every lettering of each, and every legal stacking of each folded strip. A crease set read from the other end is the same crease set, so sets are counted up to reversal — without that, every pattern is reported as ambiguous with its own mirror image, which is true and is not the question.

The photograph’s description is canonicalised under the two symmetries the object itself has. It can be turned end for end, and it can be turned over. Turning it over swaps what is seen from above with what is seen from below and does not swap the two colours, because the coloured face of the paper stays the coloured face. Reversing the strip does swap them when the segment count is even, because the parity of the segment indices reverses with it. Getting either of those wrong would make two identical photographs look different and inflate the colour’s apparent worth.

The check that keeps the result honest is a refusal rather than a confirmation. If the colour word had accidentally encoded which segment was on top rather than which side it showed, it would separate exactly as much as the complete layer order and the comparison would be vacuous. So the check requires two segments of the same parity to be indistinguishable to it, and requires some folded strip to show both colours from above — a constant would be a different kind of vacuous.

The last requirement is the ordering one: a weaker observation cannot learn more than a stronger one. The photograph’s count must never exceed the object’s, at any size. It does not, at either size tried, and if it ever did one of the two would be wrong.

What it would take for a surface observation to work #

The negative result invites the question of what an observation would have to be to separate these patterns, and the answer sharpens what the layer order is.

Two patterns folding to one profile differ somewhere in where their creases are. Their folded images cover the same intervals with the same multiplicities, so every difference between them is a difference in which segment covers which part, and that is exactly the correspondence a photograph does not record. The colour records a shadow of it — the parity of the top segment — and parity is a two-valued function of an index that runs to eight, so it discards nearly all of the correspondence before the comparison starts.

An observation that recorded the index rather than its parity would separate the groups as well as the layer order does. Nothing about a real model records an index. That is the whole difficulty in one line: an observer can see which paper is on top and cannot see which paper it is.

Where the model stops #

Everything here is one-dimensional. A strip of paper with creases across it is what this repository can enumerate exhaustively — the profiles, the orders, the colours, all of it — and a two-dimensional pattern is a different and much larger problem. The two-dimensional census is a much larger piece of work, and until it exists the claim is about strips.

The second limit is about what “a photograph” means. Real duo paper has a thickness, so a real model shows the edge of the stack as well as its top; a real photograph is taken in perspective and shows some of the sides; a real observer can prod. Every one of those adds information, and none of them adds the layer order. What the model here calls a photograph is the strongest observation that is purely of the surface, which is the right idealisation for the question and is not a description of a camera.

What the picture cannot show #

The pair figure shows two patterns photographing identically, and the only honest way to draw it is to draw the same picture twice. That reads as a mistake — a reader checks the two rows for a difference and finds none — and the absence is the content. A figure whose subject is that two things are indistinguishable has no way to be interesting to look at, and making it interesting would be making it wrong.

The pair figures have the same difficulty in a different form: they draw two patterns that are genuinely different, and what a reader can see of the difference is in the crease positions rather than in anything the photograph would carry. Nothing shows the layer order either, in any of these figures. It is the quantity the whole essay turns on and it is the one thing a picture of a folded object cannot contain, which is the essay’s point arriving as a constraint on its own illustrations.

The generalisation #

The useful form of this is a statement about observations rather than about paper. Three observers are nested: the outline is contained in the photograph, the photograph is contained in the complete description. Nested observers give a monotone sequence of how much can be distinguished, and the interesting question about such a sequence is where the jumps are.

Here there is one jump and it is at the last step. Everything visible is worth what the silhouette is worth; the entire discriminating power of a folded object sits in the part that cannot be seen. That is unusual. In most inverse problems the surface carries a graded amount of the answer and more careful looking buys more of it; here more careful looking buys nothing, and the only route to the answer is destructive.

A record is not a proof made the same distinction about the historical record — what a document can establish against what it appears to establish — and this is the geometric version, with a census instead of an archive.

Who found it, and when #

The inverse problem for folded objects is not a classical subject. The forward problem — pattern to folded state — is what the literature is about, and the backward one is asked mostly in the practical form of reconstructing a crease pattern from a photograph of a model, where the answer is understood to be hard and is treated as a matter of skill.

What this rung adds is a count and a comparison, and the comparison is the part that is new. Nobody appears to have asked what the colour is worth as evidence, probably because duo paper is thought of as a design material rather than as an instrument. It is a natural thing to try, it costs nothing to try, and the answer is that it buys nothing at all.

Where the ladder goes next #

The two patterns that survive the complete layer order are the interesting residue. They fold to the same object, so no observation of any strength separates them, and what distinguishes them is not recoverable even in principle. Characterising which pairs of patterns do that — rather than counting them — is the next rung, and it is a question about when two different sets of creases produce identical folded geometry.

The other direction is the two-dimensional census, which is owed and is a body of work rather than a rung. Until then the honest summary is the one this essay has: on strips, everything a camera can see is worth exactly what a shadow is worth.

A third direction is worth naming because it inverts the question. If the colour separates nothing, then a design can be built to show any colour pattern the folding admits without that choice constraining the crease pattern at all — the colour is free where the geometry is not. What the reverse side costs a design is the rung that takes it, and it finds the cost is not in the colouring but in the depth at which the other side is buried.