Decided before the design

Bringing the other side to the front is the design technique: fold so that a panel showing the reverse of the paper lands where it can be seen, and the model has a beak of a different colour from its body. That essay named the two-colouring as what decides which panels are available, and the conservation law that makes visible area cost something.

This is the measurement of what the constraint actually is, and it is not the one the account implies.

Two facts that are the same fact #

The first is the two-colouring. The panels of a flat-foldable crease pattern can be coloured black and white so that panels sharing a crease differ, and the sheet has two sides showed that this is Maekawa’s parity rather than a coincidence: every interior vertex has even degree, so the panel graph has no odd cycle, so a two-colouring exists. It is a property of the flat pattern; nothing about folding is in it.

The second is what the folding does. A flat folding is a composition of reflections, so a panel arrives at the folded state either the same way up as it started or turned over — and which of the two is exactly the two-colouring.

That is not a restatement. It is an agreement between two computations of very different kinds: one walks the panel graph assigning alternate colours, the other composes reflections and asks the sign of the resulting motion. They agree on all eight printed patterns, and the check requires it — a single panel reversed by hand must break it, or the agreement would be a formality rather than a measurement.

What that says about a folded model #

The consequence is worth stating flatly. A flat folded model shows one side of the sheet from above and the other from below, panel by panel, and which panel does which was fixed by the crease pattern.

So a folder choosing which face of the paper to start with is choosing what the finished model’s colours will be, in the strong sense that no decision made later can change any of it. There is no folding sequence that turns a panel over relative to its neighbours; the parity is in the pattern.

That is the sense in which the palette is decided before the design. It is also, on its own, not much of a constraint — because the two classes are both everywhere.

Availability is not the problem #

Over the folded footprint of the preliminary base, panels of both classes lie over 99.6 per cent of the area. On the Miura it is 100 per cent, on the tapered corrugation 100 per cent, on the waterbomb tessellation 99.9, on the Yoshimura 100.

A designer looking for a place to put a colour change is not looking for a rare panel. At nearly every point of nearly every folded model, a panel of the other class is already there.

What varies is how deep. The mean stack over the mixed area is eight layers on the preliminary base, 9.2 on the Miura, 8.4 on the tapered corrugation, 31 on the waterbomb tessellation and 60 on the Yoshimura. The reverse colour is present at those points and it is under that many sheets of paper.

The numbers, pattern by pattern #

The census is small enough to set down in full, and the spread across it is the part that matters.

The fourth column is the share of the folded footprint with panels of both classes over it, and the fifth is the mean number of layers there:

The two twists are the interesting middle. They cover about two-thirds of their footprint with both classes, which is high enough that availability is not a constraint and low enough that a third of the model has only one colour to offer — and the mean depth there is four layers, which is shallow enough that a colour change is a real option rather than an aspiration.

The Yoshimura is the other extreme, and it is the one where the two framings give opposite answers. Both classes cover every point of it, so on the availability reading a colour change is available anywhere. The mean stack at those points is sixty layers deep, so on the ordering reading it is available nowhere at all.

So the constraint is the layer order #

A colour change is therefore not a colouring problem. It is a layer-ordering problem: the question is not whether a panel of the other class covers the point, but whether it can be brought to the top of the stack there.

Which layer goes on top is the essay about that, and the two forbidden local patterns it names — the taco-taco and taco-tortilla conditions — are what a design has to work around. The distance between the two framings is the distance between a question with a short answer and a question the vertex theorems cannot see at all.

There is an exception in the census and it is the informative one. The fold-and-cut triangle has both classes over only 3.5 per cent of its footprint, and its two classes are unequal in area — a 92-to-8 split, where every other pattern is close to even. That pattern is a straight skeleton with perpendiculars, folded so that a single cut removes a triangle, and its panels are not a tessellation of similar pieces. For it, availability is the constraint, and the reason is that its colouring is lopsided rather than that its stack is deep.

The one thing a folder does control #

There is a decision that changes everything about the palette and it is made before the first crease.

Which face of the paper starts uppermost decides which class shows which colour. Every panel’s parity is fixed by the pattern, but the assignment of colours to the two classes is not — start with the coloured side up and the majority class shows colour; start the other way and it shows white. On a pattern whose split is even that choice is nearly free; on the fold-and-cut triangle it is the difference between a model that is 92 per cent one colour and one that is 92 per cent the other.

That is the whole of a folder’s control over the palette, and it is one bit. Everything else — which panels, how many, how deep — was settled when the crease pattern was drawn.

Which theorem was checked, and how #

Four things, and the second is the one that makes the numbers evidence rather than bookkeeping.

Every pattern is folded, not assumed to fold: the folded state is built by composing reflections and the disagreement between two routes to the same panel is measured and required to be zero.

The colouring and the turning are computed separately and required to agree on every pattern. They share no code and no data structure. The check also requires that turning every panel over at once leaves the agreement intact — because a two-colouring has no preferred colour, so the claim is about the partition and never about which class is which — and that turning a single panel breaks it.

The footprint is sampled on the same grid the site’s layer-map machinery uses, so the shares here and the layer counts elsewhere are comparable, and the sample count is reported with every share.

And the classes are checked for balance rather than assumed even. Six of the eight patterns split their paper within one per cent of a half; two do not; and the two that do not are the two whose panels are of visibly different sizes.

The depth column is the shrink ratio #

The census’s last column looks like an independent measurement and it is not one, and identifying what it actually is turns the table into a design rule.

Every point of the sheet lands somewhere in the folded footprint, so the depth summed over the footprint equals the sheet’s area. Dividing by the footprint’s area:

$$\text{mean depth} = \frac{\text{sheet area}}{\text{footprint area}} = \frac{1}{\text{shrink}}$$

The identity is exact, and two rows of the table check it. The preliminary base has eight panels and folds to exactly one of them, so its shrink is one-eighth and its mean depth should be 8.0 — which is the number in the table. The waterbomb tessellation is drawn into a thirty-first of its area elsewhere on this site, and its mean depth here is 31.0.

So the fifth column is the packing ratio wearing different units. It is not telling a reader anything the shrink census did not already say; what it is doing is saying it in the units a colour change is paid in.

Which gives the rule the table was for #

Put the identity next to the finding and the design consequence writes itself.

A colour change costs a number of layers to disturb, and that number is the reciprocal of how well the pattern packs. The better a pattern packs, the more expensive its colour change, in exact proportion. The Yoshimura draws itself into a sixtieth of its area and buries its reverse colour under sixty sheets; the twists draw themselves into about a quarter and bury it under four.

That is a trade with no slack in it, because the two quantities are the same quantity. A designer cannot look for a pattern that packs tightly and offers a shallow colour change — the second is defined as the reciprocal of the first — and the search is instead for a pattern whose depth is unevenly distributed, so that some region of the footprint is shallow while the pattern as a whole is not.

The twists are the two rows where that already happens. They cover about two-thirds of the footprint with both classes at a mean of four layers, which means the remaining third carries the rest of the paper — and it is that unevenness, rather than the average, that makes them the patterns where a colour change is a real option.

So the useful column is not the mean but the spread, and the census as printed does not contain it. That is the measurement this rung should have made and did not: the distribution of depth over the mixed area, per pattern, rather than its average.

Why the split is nearly even #

Six of the eight patterns divide their paper between the two classes to within one per cent of a half, and that is not a coincidence about these patterns.

A two-colouring alternates across every crease, so on a pattern whose panels are all of similar size the two classes hold similar totals — the colouring is an alternation and an alternation over many similar pieces is close to even. The exceptions are patterns whose panels differ a lot in area, and there is exactly one of those here.

That gives a designer a rule with no arithmetic in it. A pattern of many similar panels offers the two colours in equal amounts; a pattern of a few large panels and many small ones does not. Which colour is the scarce one is then decided by which class the large panels fall into, and that is decided by parity — so it can be flipped by adding a single crease somewhere the model does not care about.

Where the model stops #

Everything here is about a flat folded state. A real model is not flat: it has a beak that stands away from the body, and a panel that is buried in the flat folding may be perfectly visible in the finished thing. The census therefore measures the hardest version of the question and not the one a folder faces.

The patterns are also the site’s own eight, which are mathematical objects rather than designs — tessellations, bases and constructions. A designed model has a deliberately uneven distribution of panel sizes and its answer would differ; what the census establishes is the behaviour of the patterns this site can print, which is a stated population and not a sample of origami.

Nothing here computes what a colour change costs. The earlier rung measured that: visible area of the reverse colour is paid for in paper, by a conservation law. This rung is about whether the panel is there and how deep, which is a different question with a different answer.

What the picture cannot show #

The census figure draws a share and a depth and cannot draw where on the footprint the mixed area is. On most of these patterns it is nearly everywhere, so a map would be a solid block; on the fold-and-cut triangle it is a thin region, and that is the one case where a map would say something a bar cannot.

The depth column has the same problem in reverse. A mean of sixty layers is a number a reader can read and not one anybody can picture, and the figure that would make it vivid — a cross-section of a folded Yoshimura — is a picture of sixty lines. Nothing shows a colour change. Every figure here is of a pattern or of a census, and the object a reader wants — a folded model with a differently coloured beak — is a photograph of somebody’s design, which this site does not print.

The generalisation #

The useful form of this is a distinction between two kinds of scarcity, and it applies well outside folding.

A resource can be scarce because there is not much of it, or because what there is sits behind something. The first is a counting problem and is solved by finding more; the second is an ordering problem and is solved by rearranging. They feel similar from the outside and they have nothing in common as problems: one is about the distribution of a quantity and the other about a permutation.

The colour change looks like the first kind and is the second. That is worth knowing because the two have different difficulties attached: counting problems here are decided by the two-colouring, which costs one pass over the panel graph, and ordering problems are decided by the layer order, which is where the hardness of this whole subject lives.

There is a design reading too. If both colours are available nearly everywhere, then a design’s palette is limited by how much of the stack a folder is willing to disturb — which makes the colour change a thickness technique, competing for the same budget as everything else that adds layers, rather than a geometric one.

Who found it, and when #

The two-colouring is old and its equivalence with which way up a panel lands is folklore among anybody who has folded duo paper: the alternation is visible as soon as a sheet is creased. Neither half is new here.

What is new is the census, and the reason it has not been made is that the question does not arise for a designer. A designer works on a particular model and knows, by looking, whether the panel they want is available. The general question — over a population of patterns, what fraction of the folded footprint has both classes present — is one a person cannot answer by looking and a computation can.

Where the ladder goes next #

The next rung is the ordering half made concrete: given a folded state and a point, what is the shallowest panel of the opposite class over it, and what does bringing it to the top cost in extra folds. That is a well-posed question about a stacking and it turns this rung’s depths into a price.

A third question is the one the inverse-problem ladder leaves next door. If the colour visible from outside a folded model carries no information about which pattern made it, and the colour is decided entirely by the pattern’s own parity, then the two facts fit together: the colour is a function of the pattern that every pattern of the same size computes identically. Making that precise would join two ladders that have been circling the same object.

The other direction is the lopsided case. The fold-and-cut triangle’s 92-to-8 split is the only unbalanced pattern here, and what makes a two-colouring unbalanced is a question about the pattern’s panel areas rather than about its graph — a pattern of many small panels and few large ones has an uneven colouring, and that is a property a designer could aim at deliberately.