Packing is the hard part
The design problem has been reduced. A subject becomes a tree with lengths, the tree becomes circles and rivers, and what remains is to fit them into a square.
That reduction is a genuine achievement, and it relocated the difficulty rather than removing it.
Circle packing in a square is old, well studied, and hard. Origami’s version is harder, because the circles are unequal and there are connectivity constraints between them.
The reduction, restated
A flap of length consumes a disc of radius . An internal edge of the tree — a body segment — consumes a strip of constant width running between the relevant discs.
So the input is:
- a set of radii , one per appendage;
- a set of required separations, one per body segment;
- a unit square.
And the question is whether everything fits, and if so how tightly. Tighter is better, because the leftover paper is wasted and a design that packs at 70% produces longer flaps from the same sheet than one that packs at 50%.
Why the rivers make it hard
Packing unequal circles into a square is difficult and it is at least a clean problem. Rivers change its character.
A river is a strip of a given width that must connect two specified regions without being crossed. That is a connectivity requirement, not a geometric one, and it interacts with the packing in a way that resists local optimisation: moving one circle a little can block a river entirely, which is a discontinuous change in feasibility.
The practical symptom is that designs with long thin bodies — a snake, a centipede, anything with segments — are much harder to pack than designs with many short appendages, even though the total area required may be lower. A dozen legs is a packing problem; a long body between two clusters of legs is a routing problem with a packing problem inside it.
What TreeMaker does
Robert Lang’s TreeMaker is the reference implementation, and knowing what it actually does is more useful than knowing that it exists.
It takes the tree and the lengths, sets up the packing as a nonlinear constrained optimisation, and runs a numerical solver. The variables are the circle centres and a global scale factor; the constraints are that no two circles overlap, that every circle stays inside the square, and that the rivers have room; and the objective is to maximise the scale.
The output is a packing, and then a separate construction turns the packing into a crease pattern.
Two things follow from it being an optimiser rather than an algorithm.
It finds local optima. Run it twice from different starting points and it can return different packings with different efficiencies. There is no guarantee that either is best.
It can fail without proving anything. If it does not find a packing, that is not a proof that none exists — only that the solver did not find one. The area bound gives genuine impossibility results; the solver does not.
Nobody knows the optimum
Circle packing in a square has known optimal solutions for equal circles up to about 30, established by a mixture of proof and exhaustive computation. Beyond that, and for unequal circles, the optima are conjectural.
Origami packings are unequal, constrained, and typically involve ten to thirty circles. Every one of them is in the region where the optimum is unknown.
That means a designer cannot answer “is my packing the best possible?” and neither can anybody else. Two designs of the same subject on the same square can differ in efficiency by ten per cent, and the better one is better by measurement rather than by proof.
It is an unusual position for a field to be in: the impossibility results are exact, and the possibility results are heuristic.
The bound that is exact
There is one thing the theory says with certainty, and it is worth having because it is so restrictive.
The circles must fit inside the square, so their total area is at most one:
That is a hard bound requiring no packing at all. Six legs of length need of the sheet on their own. Add a head, an abdomen and the body joining them and the square is gone.
It explains something that puzzles people looking at complex origami: why the finished insect is so much smaller than the sheet it came from. A 60 cm square producing a 10 cm beetle looks wasteful and is close to optimal — the sum of squares leaves no alternative.
What the method cannot design
The tree method solves one family of problems completely and does not touch the rest.
Non-uniaxial forms. The construction assumes the flaps all lie along a single axis in the folded base. Boxes, curved shells, tessellations and anything with volume are outside it.
Shaping. The method produces a base — a flat arrangement of points of the right lengths. Turning that into something that looks like a beetle is entirely craft, and it is most of the work and all of the artistry.
Layers. The packing says nothing about how thick the result is anywhere. A tight packing can produce a region twenty layers deep, and twenty layers of anything is a problem.
Foldability. Non-overlapping circles mean the paper is available. Whether the resulting crease pattern actually folds is a separate and much harder question.
What a designer does with it
In practice the method is used less as an oracle and more as a check and a starting point.
A designer with an idea for a subject uses the area bound first: add up the squares and see whether the sheet can possibly carry it. That takes a minute and rules out a good fraction of ideas.
Then a packing is found — by the solver, by hand, or by both — and adjusted. Hand adjustment is common, because a solver optimises efficiency and a folder cares about symmetry, about creases landing on convenient references, and about the sequence being followable.
Then the pattern is generated, and then it is modified. Almost no published complex model is exactly what the algorithm produced.
Efficiency is not the only objective
The optimiser maximises flap length per unit of paper. Designers routinely accept less, and the reasons are worth listing because they are invisible to the mathematics.
Foldability by a person. A pattern with creases at arbitrary angles is far harder to fold accurately than one on a grid, and inaccuracy compounds.
Layer distribution. A packing that concentrates paper produces a model that is thick in one place and thin elsewhere, which folds badly and looks worse.
Symmetry. A symmetric subject folded from an asymmetric pattern is unpleasant to make and often visibly lopsided.
Sequence. Some patterns collapse in a satisfying order and others require holding six things at once. Nothing in the optimisation knows about this.
So the algorithm answers a question that designers care about and is not the only question they care about, which is the normal fate of an optimisation applied to a craft.
The contact graph carries the pattern
There is a step between the packing and the creases that is worth drawing out, because it explains why the packing determines so much.
When two circles touch, the point of contact is a place where two flaps meet on the finished base, and the line between their centres crosses that point. Those lines — one per touching pair — form the contact graph of the packing, and the crease pattern is built on it.
The axial creases run along the perpendiculars to those lines. The regions between circles are triangulated by the contact graph, and each triangle folds by a standard construction determined by its three radii. So a packing where the circles touch in many places determines the pattern almost completely; one where they float with gaps leaves regions the construction has to fill arbitrarily.
That gives a designer a reason to prefer tight packings beyond efficiency: a tight packing has a rich contact graph, and a rich contact graph means fewer arbitrary choices later. A loose packing wastes paper and leaves work.
When the paper runs out, change the tree
The most common practical response to an infeasible packing is not a better packing. It is a different subject.
The tree is a model of the subject, and it was made by somebody deciding how long each appendage should be. Those lengths are a judgement, not a measurement, and the area bound is quadratic in them — so shortening a few legs by fifteen per cent frees a surprising amount of sheet.
Designers therefore iterate between the tree and the packing: propose lengths, attempt a packing, find it infeasible, shorten the least important appendage, and repeat. What emerges is a compromise between the subject as imagined and the subject the square can carry.
That is worth saying plainly because it inverts how the method is usually described. The tree is presented as the input and the packing as the output; in practice the packing is a constraint that reaches back and rewrites the input.
What the leftovers become
A packing rarely fills the square, and the gaps are not waste in the way they first appear.
Paper between circles that belongs to no flap and no river becomes structure: the body of the model, the webbing between legs, the thickness of a thorax. A design with a little slack folds more easily than one packed to the limit, because there is paper to absorb the thickness the model accumulates.
Designers sometimes add slack deliberately — packing at ninety per cent of what the solver achieved, and using the difference to make the base foldable. The efficiency number gets worse and the model gets better, which is the same trade box pleating makes in a different currency.
Who found it, and when
The method has two independent origins, and the usual asymmetry in how they are remembered.
Robert Lang developed the tree method through the late 1980s, published Origami Design Secrets in 2003, and released TreeMaker publicly. Toshiyuki Meguro developed circle packing for origami design in Japan at essentially the same time, and Jun Maekawa and Fumiaki Kawahata contributed closely related approaches.
Meguro’s work is less cited in English, for the ordinary reason that it was published in Japanese in origami society publications. The technique is sometimes called the circle-river method to acknowledge both lines.
Whichever name is used, the shift it caused is not in dispute: the complexity of published origami increased sharply in the 1990s, and it increased because designers stopped guessing.
Two designers, one subject
A concrete illustration of the optimum being unknown: give two designers the same tree and the same square, and they produce different patterns.
Both will satisfy the area bound. Both will pack without overlaps. One may reach an eighty per cent efficiency and the other seventy-two, and the difference shows up as flaps that are ten per cent longer from the same sheet — which for an insect’s antennae is the difference between a convincing model and an apologetic one.
Neither can prove theirs is best, and there is no procedure that would settle it. The field handles this the way crafts usually do: by comparison, reputation and the slow accumulation of arrangements that are known to work well for particular shapes of tree.
That is a slightly uncomfortable position for a subject that presents itself as having been mathematised. The mathematics established what is impossible and provided a good search; it did not provide an answer.
Where the model stops
The optimum is unknown. Every packing in use is a good one found by search, not a best one found by proof.
Rivers are simplified. Real subjects have branching bodies, and the river structure is a tree of strips rather than a few isolated ones.
Uniaxial only. As above — a large and popular family, and not the whole subject.
The figure is symmetric and therefore unrepresentative. The packing drawn on this page is solved geometrically: four corner circles and a middle one sized so nothing overlaps. Real packings are irregular, found numerically, and considerably less tidy.
Circles are drawn as full discs. In a real packing many circles are truncated by the square’s edge, and a flap whose circle hangs off the edge is a legitimate and common arrangement that the tidy picture does not show.
The algorithm changed what got folded
It is worth being concrete about the effect, because “turned a craft into an algorithm” is the kind of claim that should come with evidence.
Before 1990, complex origami meant a model with perhaps ten or fifteen identifiable points, folded from a base chosen from a small traditional repertoire. Insects with the correct number of legs were rare and usually cheated — legs made from a single flap split, or a body segment omitted.
After the method spread, published models routinely had twenty or thirty points, correct anatomy, and crease patterns nobody could have arrived at by adjustment. The change is visible in any collection of published work from the two decades, and it was fast.
What the algorithm supplied was not skill — the folding is harder, not easier — but confidence that a base exists. A designer could decide on the subject, verify the paper was sufficient, and then do the work, rather than discovering after a week that the sheet was never going to be enough.
That is the useful summary of what a design method does. It does not do the design; it removes the possibility of wasting a month on something impossible.
What a packing does not decide
A short list of the things left over once the packing is fixed, because the method is often described as though it produced a finished design.
The folding sequence. The pattern says where the creases go. Getting there is a separate problem, frequently harder than the design, and the grid exists largely to make it tractable.
The layer order. Which flap lies above which is not determined by the packing, is a real choice, and decides whether the finished model looks tidy.
The shaping. Everything after the base — thinning, curving, texturing — is craft the algorithm does not touch and is most of what makes a model recognisable.
The proportions. The tree’s lengths were somebody’s judgement about the subject, and the packing takes them as given while quietly constraining them.
So the algorithm produces a base and a guarantee. Everything that makes the result a model rather than a diagram happens afterwards.
The ladder from here
Later rungs: the packing-to-pattern construction. Rivers and the contact graph. TreeMaker’s optimisation, and what it is actually solving. Efficiency bounds. Known optimal circle packings, and where the knowledge stops. Edge and corner flaps. Meguro’s method. Non-uniaxial design, which has no general theory. Layer distribution as a design objective. And the question of whether a good base can be recognised automatically, which nobody can currently do.
The area bound — the sum of squared flap lengths is at most — takes ten seconds to apply and rules out most ideas anybody has for a complex model. It is the single most useful thing the theory produced, and it is a line of arithmetic.