What joins the flaps

The circle rule is one of the great simplifications in this subject. A flap of length L consumes every point of the sheet within L of where it comes from, so a design is a packing and two flaps whose circles overlap cannot both exist. It turns an art into an algorithm in one sentence.

It is also a special case, and the general rule is barely harder.

The skeleton has internal edges #

The circle argument begins by reducing the subject to a stick figure. That much is familiar: an insect becomes six legs, two antennae and a body, each with a measured length.

What gets glossed over is that a stick figure has two kinds of edge. Some run out to a tip — a leg, an antenna, a horn — and those become flaps. Others run between one junction and another, joining a group of limbs to a different group. A thorax connecting the legs to the abdomen is an edge of that kind, and it does not become a flap.

The circle rule sees only the first kind. It says how much paper each tip costs and stops there, which is correct exactly when there is only one junction for everything to hang from.

The moment there are two, the edge between them is a length that has to come from somewhere, and it is not inside any circle.

The general condition #

Lang’s statement of the rule replaces circles with distances, and it is worth writing out because it contains the circle rule as a corollary.

Give every leaf of the skeleton a place on the square. Then a base with those flaps exists only if, for every pair of leaves, the distance between their places on the sheet is at least the distance between them through the tree.

Distance through the tree means the length of the path joining them along the skeleton: out from one tip to its junction, across whatever internal edges lie between, and back out to the other tip.

Now specialise. If both leaves hang from the same junction, the path is just their two lengths added, and “distance on the sheet at least the sum of the two lengths” is exactly the statement that two circles of those radii do not overlap. The circle rule is the two-leaves-one-junction case, written out.

If they hang from different junctions, the path picks up the internal edges as well, and the required separation is larger by exactly that amount. That extra separation is the river.

A river is a strip, not a circle #

The geometry of the extra separation is worth being concrete about, because “river” is a suggestive word and suggestion is not enough.

An internal edge of length w separating two groups requires every leaf of one group to be at least w further from every leaf of the other than the circles alone would demand. The set of points excluded is not a disc; it is the region within w/2 of a line running between the groups, which is a strip of width w.

That is why the picture is a band across the sheet rather than another circle. And it is why a designer speaks of routing a river: the strip has to get from one side of the packing to the other without crossing any circle, and where it goes is a decision with consequences for everything else.

A river also has a shape that a circle does not. It can bend, it can branch where the skeleton branches, and its two banks are the boundaries of two different groups of flaps. What it cannot do is get narrower, because its width is a length in the subject being modelled.

Reading a packing backwards #

A packing is usually presented as the output of a design process, which makes it easy to forget that it can be read as an input.

Given a packing — discs and strips on a square, satisfying the condition — the subject it describes can be recovered. Each disc is a flap whose length is its radius. Each strip is an internal edge whose length is its width. The adjacency of discs and strips gives the skeleton’s shape. Nothing is lost.

That reversibility is what makes the method a genuine correspondence rather than a heuristic. The tree determines the constraints; the packing determines the tree; and a designer who alters a packing to make it fit is altering the subject, whether or not they intended to.

It also explains a common frustration. A packing that has been nudged to close a gap has quietly lengthened a flap or narrowed a river, and the model that results has a leg slightly too long or a thorax slightly too short. The paper does not negotiate: every centimetre on the sheet is a centimetre in the subject.

The second bound, which the circles cannot see #

Writing the condition as distances rather than discs makes a whole class of impossibility visible that an area argument never reaches.

The condition demands that every pair of leaves be at least as far apart on the sheet as they are through the tree. Take the two leaves furthest apart in the skeleton — the tree’s diameter — and the sheet has to supply that separation from its own diameter. On a unit square that is $\sqrt{2}$, so

$$\text{tree diameter} \leq \sqrt{2}$$

and no packing, however clever, evades it. The paper’s diagonal is a hard ceiling on the longest path through the subject.

That is a genuinely different constraint from the sum of squares, and the two bind in opposite regimes. A bushy subject — many short limbs from one junction — has a small diameter and a large total disc area, so the area bound decides. A chain-like subject — a snake, a centipede, anything whose skeleton is mostly internal edges — has almost no disc area and a diameter approaching the sheet’s, so the diagonal decides and the circles are silent.

Which is why a long subject wants a long sheet #

The diameter bound also settles a question the circle rule cannot even pose: what proportion of paper a subject wants.

A sheet of proportions $a$ to 1 and area one measures $\sqrt{a}$ by $1/\sqrt{a}$, so its diagonal is $\sqrt{a + 1/a}$ — 1.41 at a square, 2.06 at four to one, 2.67 at seven to one, 3.18 at ten to one. The available tree diameter more than doubles across that range at no cost in paper whatever.

So a centipede on a square is not badly packed; it is short of diagonal, and the repair is a proportion rather than an arrangement. That agrees with what the packing efficiency of equal discs says about chain-like subjects, reached from the opposite end — one argument about how discs tile and one about a single pair of leaves, giving the same advice.

What the condition does not constrain #

It is as important to say which choices the condition leaves free, because those are where design happens.

Where a disc goes. The condition constrains separations, not positions. A packing may be rotated, reflected, or rearranged entirely, and any arrangement meeting the separations is legal. Most of the difficulty of packing is that there are many arrangements and no way to enumerate them.

Which corner a flap uses. A disc against a corner of the square uses a quarter of itself and gets the same flap length for a quarter of the paper, which is why corner flaps are cheap and central flaps are expensive. That is a consequence of the boundary rather than of the condition, and exploiting it is most of what a skilled packing does.

How long the flaps are in absolute terms. The condition is scale-free: multiply every length by a constant and it still holds. What a designer optimises is that constant — the scale — and the packing that maximises it is the one that uses the sheet best.

Anything about the leftover. The condition says nothing about the shape of the paper between the discs, and the shape of that paper is what the finished model mostly consists of.

Which theorem was checked, and how #

The packings in these figures are not arranged to look plausible. They are tested against the condition, pair by pair.

The generator carries the skeleton explicitly — which leaf hangs from which junction, how long each leaf edge is, how long each internal edge is — and computes, for every pair of leaves, the distance through the tree. It then measures the distance between their places on the sheet. If any pair is closer than the tree allows, by more than a part in a billion, the generator throws and the build stops.

The number printed beside each figure is the tightest margin: how much room is left at the worst pair. A packing with a margin of zero is exactly critical and is what a design algorithm aims at; a packing with a negative margin is a drawing of something that cannot be folded.

What the figures cannot show is that the condition is sufficient as well as necessary. That a valid packing always yields a crease pattern is Lang’s theorem and it needs a construction — the packing determines a set of ridge and hinge creases, and proving those always assemble into a base is the substantial part. The figures verify the easy direction.

Where the paper actually goes #

The condition says how much paper each part of the subject claims. It does not by itself say what happens to the paper claimed.

A circle becomes a flap by collapsing, which is what the circle was measuring in the first place: the paper inside it gathers to a point at the circle’s centre and the flap grows out of it. The circle’s area is spent, and the flap that results is as long as the radius.

A river becomes the structure joining two groups: a pleated band whose width in the finished model is however thin the folding makes it, but whose length along the model is the internal edge’s length. So the strip on the sheet becomes a strip in the base, and it is what a body or a neck or a thorax is made of.

The leftover paper between circles and rivers becomes the surfaces that connect everything — the parts of an insect that are neither leg nor body but the membrane between them. There is always some, because circles do not tile the plane, and it is the source of both the layer thickness and most of the shaping.

The idealisation this rests on #

The whole argument treats a flap as a length and nothing else, and that abstraction is doing more work than it looks.

A flap of length L is modelled as a segment of the skeleton. Its thickness is not modelled, its cross-section is not modelled, and whether it is a leg or an antenna is not modelled. The base produced by the packing has flaps of the right lengths and of whatever thickness the leftover paper gives them.

That is why a base is not a model. Turning a base into a finished subject is shaping, narrowing and thinning — steps the packing has nothing to say about, and which occupy most of the actual folding time. The design algorithm solves the part that is combinatorially hard and leaves the part that is craft.

It also means the algorithm optimises the wrong thing if taken literally. A packing that maximises flap length maximises the skeleton, not the model, and a base with maximal legs can be worse to work with than one with slightly shorter legs and better-shaped leftovers.

Where the model stops #

Uniaxial bases only. The condition is a theorem about bases whose flaps all lie along one axis, and the grid alternative gives that up in exchange for creases that land where they should. That restriction is invisible in the statement and is doing most of the work.

Necessary and sufficient, but not constructive in practice. A valid packing yields a crease pattern in principle. Producing it involves computing the ridge creases from the packing’s contact graph, and doing so for a packing with dozens of circles is what TreeMaker is for.

Nothing about foldability of the result. The crease pattern a packing yields folds into the base by construction. Whether it can be folded by a person, in what order, and whether the layers can be got into position, is a different problem entirely.

Distances, not areas. The condition is about separations. The area a circle occupies is a consequence, and reasoning about area directly — “the circles use 62% of the sheet, so there is 38% spare” — is not the same statement and is not equivalent to it.

Nothing about the folding sequence. The pattern a packing yields collapses into the base in principle. Getting a sheet of paper into it involves an ordering problem the packing has no view of.

Efficiency is a separate question. How much of the sheet a valid packing manages to claim is an open optimisation problem, and is not settled by the condition being satisfied.

Straight rivers only, here. The figures draw a river as a straight band because a straight band is what the tested condition needs. A real packing routes rivers around circles, and the routing is part of what makes the packing problem hard.

The surprise: it is a conservation law #

The reason the condition has the shape it does is worth stating in the general form, because in that form it stops being about origami.

A flat sheet of paper carries a metric: the distance between two of its points, measured through the paper. Folding is an isometry — it moves the paper without stretching it — so the distance between two points through the paper is the same before and after.

Now consider two flap tips in the finished model. The distance between them through the paper is at least the distance between them through the skeleton, because the skeleton is a path in the model and the paper contains it. And the distance between them through the flat sheet is the same number, because folding preserved it.

That is the condition. It is not a fact about circles or about design; it is the statement that folding does not stretch, applied to a pair of points and read as a constraint on where they may be placed.

Which explains why the rule has no exceptions and needs no adjustment for clever patterns. Any base whose flaps have the stated lengths, made by any method, obeys it, because the sheet cannot stretch.

Who found it, and when #

The tree method has a clear author and a slightly tangled prehistory.

The circle-packing idea appears independently in the work of several designers in the 1980s and 1990s — Toshiyuki Meguro and Jun Maekawa in Japan, Robert Lang and others elsewhere — as a rule of thumb about how much paper a flap needs.

Lang turned the rule of thumb into a theorem, with rivers, with the distance condition in its general form, and with a proof that a valid packing always yields a base. TreeMaker, first released in 1993 and developed through the following decade, implements it: a subject is described as a tree with lengths, the program solves the packing, and it emits the crease pattern.

There is a lesson in the sequence worth carrying. The circle rule was a heuristic that worked; making it a theorem required finding the hypothesis it had been quietly assuming, which is that the base is uniaxial; and generalising it required noticing what the heuristic could not see, which is the internal edges. Both steps are the same step: saying out loud what had been taken for granted.

Meguro’s contribution deserves separate mention because it is often folded into Lang’s. He developed the circle-and-river method independently and taught it as a design technique rather than publishing it as mathematics, which is why it entered the folding community’s practice before it entered the literature.

The ladder from here #

Later rungs against this anchor: the ridge creases the packing determines, and how the contact graph turns into a pattern. The proof that the distance condition is sufficient. Rivers that branch, and packings with several internal edges. Non-uniaxial design, where the condition does not apply and there is no comparable theory. Point-splitting and other ways of trading one flap for two. And the practical question of how much a design loses by insisting on the tree method at all, given that many of the best-known models were designed without it.

The circle rule is the memorable half. The condition it is a case of is the half that is true in general, and it is one sentence longer.