Bringing the other side to the front

Origami paper is usually coloured on one side and white on the other, and most finished models are one colour. That is a choice the geometry makes easy: fold a square without thinking about it and the outside is nearly all the same face of the sheet.

A colour change is the deliberate opposite. The reverse of the paper is brought to the outside in a chosen region — a black beak, a white belly, a two-tone pattern on a tessellation — and it is the technique that separates a model that looks designed from one that looks folded.

It is also expensive, in a way with a number attached.

The arithmetic, which is not intuitive #

Fold a flap of width a back over a unit sheet. The flap shows the reverse over an area a. The paper it lands on was showing the front, and is now hidden.

So the front loses 2a — the flap itself, which used to show its front, and the region underneath, which is now covered — while the reverse gains a. Every unit of visible reverse is paid for with two units of visible front.

Three consequences follow immediately, and each is a design fact rather than an observation about arithmetic.

The two are equal when the flap is a third of the sheet. Below that the model is mostly its front face; above it, mostly its reverse. Designers who describe a colour change as “a bit of the other side” are describing flaps well under a third.

The whole visible face can be reverse, at the cost of half the sheet. Fold exactly half back and the front is entirely hidden. That is not a colour change any more; it is a model made of the other side of the paper, with the front spent on nothing.

The total visible face shrinks throughout. A sheet with a flap folded back covers less than a sheet without one. Colour is bought with size, and this is the conservation law in its smallest possible instance: the paper is all still there, and some of it is underneath.

What it costs in the size of the model #

The three consequences are stated in area, and a folder judges a model by its size, so the conversion is worth making.

Visible face after the fold is $1 - a$, of which $a$ is reverse. So the reverse share of what shows is $a/(1-a)$, and inverting it gives the flap a design needs for any desired balance.

A model that is half reverse — the ordinary two-tone animal, dark above and pale below — needs $a = 1/3$, and then the visible face is $2/3$ of the sheet. In the dimension anybody notices that is $\sqrt{2/3} = 0.816$: a balanced two-tone model is eighteen per cent smaller than the one-tone model from the same square.

A quarter reverse needs $a = 1/5$ and costs eleven per cent of linear size. A model that is one-tenth reverse costs five.

And the rate never improves #

The marginal arithmetic is the discouraging part, and it is the reason there is no clever region to work in.

Differentiating, the visible face falls by exactly one unit for every unit of reverse gained — $d(1-a)/da = -1$ against $d(a)/da = +1$ — and the rate is constant across the whole range. There is no cheap first colour change followed by expensive later ones, and no threshold past which the technique starts paying for itself.

That is unusual enough to be worth stating plainly. Most costs in this subject accelerate: a flap’s circle grows as the square of its length, a packing’s difficulty grows with the count, layers pile up faster than creases do. Colour is the one resource that is sold at a flat price, and the flat price is one for one.

Which means a designer can decide how much reverse a model shows last, without re-deriving anything — the only quantity it moves is the finished size, and it moves it by a factor known in advance.

Which panels are even available #

The arithmetic says what a colour change costs. It says nothing about where one can be put, and that question has a complete answer that most accounts of the technique leave out.

The panels of a flat-foldable pattern take two colours, and the colours are exactly the two faces of the sheet: cross a crease and the panel beyond it is showing the other side. So the pattern already contains a map of what is available. A panel of the first colour shows the front, a panel of the second shows the reverse, and no amount of ingenuity moves a panel from one class to the other without changing the creases.

This reframes the design problem in a useful way. A designer wanting a black region does not need to invent a fold; they need a panel of the right colour in the right place, and the creases are the means of getting one there.

It also explains why colour changes cluster where they do in the traditional repertoire — at tips, at edges, at the ends of flaps. Those are the places where a single extra crease turns one panel over without disturbing anything else. A colour change in the middle of a large panel needs the panel divided first, and dividing it changes everything built on it.

The balance the patterns start from #

There is a measurement worth putting next to the design advice, because it says how much work a colour change is doing.

Every repeating pattern this repository folds puts almost exactly half of its area on each side. A Miura fold is fifty–fifty to the last decimal. So is the preliminary base, so is the waterbomb tessellation, so is the corrugated leaf.

This is not a theorem and should not be read as one. It is what repetition does: a pattern built by tiling a unit pairs its panels off, and the balance follows from the tiling rather than from the folding. The square twist misses by seven parts in a thousand because its central polygon is unpaired.

The design consequence is the useful half. A pattern left to itself is two-tone in equal measure and shows none of it, because the reverse panels end up inside the stack. Getting colour onto the outside is not a matter of arranging for reverse panels to exist — they already exist, in quantity — but of arranging for particular ones to finish on top.

Where the reverse panels actually go #

The balance is easy to state and easy to misread, so it is worth looking at where those panels end up.

A folded base is a stack, and a stack has two outer surfaces and a great many interior ones. The two-colouring guarantees that the panels alternate through the stack — each crease crossed turns the paper over — so a stack of eight layers has four of each face, arranged so that no two adjacent layers agree.

That alternation is what makes a colour change possible and also what makes it fiddly. The reverse-showing panels are never scarce; they are simply almost all interior. Getting one to the outside means either arranging for it to be the top of the stack, which the ordering may not permit, or turning it out at an edge, which costs the flap width the arithmetic above measures.

Folders know the second route as reversing a flap or unfolding a layer, and it is worth noticing that both are descriptions of moving a panel to the outside of a stack rather than of creating anything. The paper of the right colour was there the whole time.

What the cost looks like in a whole design #

The single-crease case is the cleanest and the smallest. A real design pays the same rate in a more complicated currency.

Consider a flap that has to be a different colour at its tip — the standard case, and the one every two-tone animal needs. The flap is already claimed by a circle in the packing, whose radius is the flap’s length. Making its last portion show the reverse means folding that portion back on itself somewhere along the flap, which consumes twice the reversed length out of the flap’s own paper.

So the flap has to be longer than the design needs it to be, by twice the coloured tip. Its circle grows by the same amount. And a circle that grows in a packing pushes every neighbouring circle outward, so the sheet either grows or every other flap shrinks.

That is the mechanism by which a small aesthetic decision becomes a re-solve of the hardest step in the design. The cost is not the paper in the tip. It is the paper in the tip, doubled, and then propagated through a packing that has no general algorithm to re-solve it with.

The alternative is to buy the paper up front, by inserting a strip into the pattern where the colour change will be. That keeps the rest of the packing intact and makes the sheet larger, and it is the more common choice in practice for the same reason: it is a local change with a known price, rather than a global change with an unknown one.

Which theorem was checked, and how #

Three claims here are computed rather than asserted, and they are checked against different things.

The linear law is measured on the folded state. For each flap width the pattern is built, folded by composing reflections, and the areas are read off the result; the generator then compares what it measured with what the arithmetic predicts and refuses to draw if they differ by more than a hundredth of the sheet. The arithmetic could have been wrong and the measurement is what would have caught it.

The availability claim is the two-colouring, which is checked three ways: by walking the panels and counting crease crossings, by counting creases at vertices and taking the parity, and by composing the reflections and asking which panels the fold turns over. Three computations sharing no code, agreeing on every pattern here.

The balance figures are areas rather than panel counts, deliberately. Counting panels would be the wrong measure and would give a different answer: two patterns can turn over the same number of panels and differ by a large factor in how much sheet that is.

A tessellation makes the same point at a larger scale.

What decides whether it actually shows #

Everything above is about which panels can show the reverse. Whether a particular one does is a different question, and it is the hard one.

A panel showing the reverse is only visible if nothing lies on top of it. That is layer ordering: the mountain-and-valley assignment says which way each crease turns and says nothing at all about which sheet ends up above which. The folded state computed here places every panel and deliberately does not order them, because ordering is the NP-hard half of flat-foldability.

So a colour change has two independent requirements and the essay has only settled one. There must be a panel of the right colour in the right place — a two-colouring question, answered completely and cheaply. And it must finish on top — an ordering question, answered by nothing here and in general by nothing at all.

This is the honest limit of the account and it is worth being blunt about it. The arithmetic above tells a designer what a colour change costs if it works. It does not tell them whether their arrangement works, and the reason no essay can is that deciding it is intractable in general.

The strip a colour change inserts is not a new kind of object in a packing: it is a river, the same strip two groups of flaps need between them, and it is paid for the same way.

Where it sits in design #

The technique’s natural home is the tree method, where the paper’s regions are allocated before any crease is placed.

A flap costs a circle, and a flap that has to show the reverse of the paper at its tip costs a larger one, because the reversal has to come from somewhere and the somewhere is the sheet. The usual construction adds a strip: extra paper inserted at the appropriate place in the packing, which pushes everything else apart and reduces the efficiency of the whole design. The bill arrives at the packing stage, long before a crease pattern exists.

That is why a colour change is a decision rather than a decoration. Adding one after a design is finished means re-solving the packing, and the packing is the part with no general algorithm.

It is also worth seeing what the packing hands the designer once it is solved, which is two families of crease — the ridges that bound each flap’s paper and the hinges that let it turn. A colour change has to be arranged among those rather than drawn over them.

Who developed it, and when #

Colour change as a named technique belongs to the modern design tradition rather than to the traditional repertoire, and to the second half of the twentieth century. The traditional models use paper’s two faces incidentally — a crane’s white and coloured regions are wherever the base happens to put them — rather than as a variable to be controlled.

The change came with the shift from folding sequences to designed patterns. Once a designer is choosing where the paper goes rather than following a sequence, the two faces become another thing to allocate, and the technique acquires a name and a literature.

This site does not reproduce any designer’s crease pattern, so the models that made the technique famous are named and not drawn. What can be drawn is the geometry underneath, which belongs to nobody: the two-colouring, the arithmetic, and the packing cost.

The vocabulary is worth a note of its own, because it is one of the places where the practical tradition was ahead of the mathematical one and said so more clearly. Folders have talked about which side is showing since long before anybody wrote down that the panels of a flat-foldable pattern two-colour. The instruction “with the coloured side down” at the head of a diagram is a two-colouring statement: it fixes which class of panels will show which face, for the whole model, in five words.

That is a recurring pattern here rather than a curiosity. The theorems have repeatedly arrived after the practice and explained why it worked. The colour change is a mild case — nobody was surprised — but it is a clean one, because the mathematical statement and the folding instruction are so plainly the same sentence.

There is one more reason the technique stayed a craft matter for so long. Its two halves sit on opposite sides of the subject’s hardest boundary: which panels are available is decided by a condition that costs one pass over the pattern, and whether a chosen one can be brought to the surface is decided by layer ordering, which is intractable. A technique whose easy half is trivial and whose hard half is NP-hard does not develop a theory. It develops a repertoire.

One design style makes the arrangement markedly easier and it is worth saying which.

The cheapest colour change of all #

There is one way of getting the reverse of the paper onto the outside that costs nothing at all, and it is worth naming because it is the exception that shows what the price is for.

Turn the whole sheet over before starting. The model is then made of the other face throughout, no flap has been folded back, and no paper has been spent.

That sounds like a joke and it is the actual technique for a great many models: the choice of which side faces up at the beginning is the single most consequential colour decision in traditional folding, and it is free because it changes nothing about the geometry. What costs is mixing — having two colours in one model — and the arithmetic above is the price of the mixture rather than the price of the colour.

The same reading explains why colour changes cluster at extremities. A tip, an edge, a small flap is where a local reversal disturbs the least paper, and where the two units of front spent buy something small enough to be affordable. A colour change across a large central region costs a large central region twice over, which is why almost nobody attempts one.

Where the ladder goes next #

The cost argument generalises past colour. Anything a design wants on the outside of a folded object competes for the same resource, and the accounting is the same accounting — which is the conservation law again, with the visible face in place of the footprint.

The availability argument generalises the other way, into what a symmetric packing costs and whether the sheet should have been square at all. Both are questions about spending the sheet well, and a colour change is one of the more expensive ways to spend it.