The last free parameter

The tree method decides almost everything before a single crease is drawn. The tree fixes the flaps, the packing fixes where they sit, and the hinges and ridges follow from the packing by construction. What is left over is a set of polygons with nothing in them, and filling one is the last thing anybody gets to choose.

Even that choice turns out to be smaller than it looks. There is a construction that always works, and once it is adopted the only thing still free is a single number: how far the shrink runs.

What a molecule is for #

An axial polygon is a region of the sheet belonging to no flap, left over once every flap has been placed on the axis. Its corners are the centres of discs in the packing and its edges are the lines between touching discs, and the paper inside it has to be folded so that it disappears into the base’s axis along with everything else.

A molecule is a small crease pattern that does that: a pattern filling the polygon whose creases meet the hinges already fixed around the boundary, and which folds the interior flat onto the axis.

For a triangle the answer is old and unique. Three angle bisectors meet at the incentre; folding along them brings all three edges onto a point, and the resulting flap is the rabbit ear, which folders have used for centuries without calling it a molecule. For a quadrilateral there are several answers — the waterbomb, the arrowhead, the gusset — and choosing between them changes how the layers pile up.

For a general polygon with many corners, the question is whether any molecule exists at all.

The shrink #

The construction that always works is to make the polygon smaller.

Move every edge inward along its own normal at the same rate. The edges stay parallel to where they started, the corners move along fixed straight lines, and after a while the polygon is a smaller polygon of the same shape — or nearly, since edges can vanish on the way. The creases are the corners’ traces plus the successive outlines.

The traces are straight, and that is not an accident of drawing: an edge moving along its own normal at a constant rate sweeps a strip, and the corner where two such strips meet moves along the locus of points equidistant from both edges, which is a straight line. The whole construction has this one property behind it, and it is why the resulting pattern is made of straight creases rather than curves.

The figure asserts it rather than assuming it. At each intermediate outline every corner is checked to be exactly the requested distance from the edge it came from, to a billionth; a corner that had drifted would mean the offset was being computed some other way, and the traces would not be straight.

How far it goes #

Shrinking cannot continue forever. Edges shorten as the outline moves inward, and eventually one of them reaches zero length, at which point the polygon has fewer corners than it started with and the construction has to continue with the smaller one. Continue far enough and there is nothing left.

That distance is the last free parameter of the design. Every crease in the molecule is determined by the polygon and the offset; the polygon came from the packing; and so the only number a designer still holds is where to stop.

The figure computes it by bisection on a predicate the outline can answer about itself — whether the inset is still a polygon wound the same way as the original. A drawing that stopped at an offset chosen by eye would look identical and would be reporting nothing.

What the number buys #

Since the shrink distance is the only thing left, it is worth asking what changes when it moves.

Stop early and the molecule is shallow: few intermediate outlines, few layers where they meet, and a residual polygon in the middle that has not been folded away. That residual has to go somewhere, and in practice it becomes a small flap nobody asked for — often visible on a finished model as a stub in the middle of the back.

Stop at the collapse and the molecule is deep: the polygon has been folded entirely into the axis, no paper is left unassigned, and the layers where the traces meet are as many as the construction can produce. That is the choice most design software makes, because it is the one that leaves nothing to explain — and it is the choice that makes the paper budget come out as advertised.

Between the two there is a genuine trade. A shallow molecule is easier to fold and wastes paper; a deep one uses everything and concentrates thickness. The mathematics has no preference, and this is one of the few places in the method where a designer’s judgement is doing something the optimiser cannot.

The path condition, and what it rules out #

Universal does not mean unconditional. The shrink produces a valid molecule — one whose folded state actually lies on the axis, with every flap the right length — only when the polygon satisfies a condition on its edges.

Informally: for every pair of corners, the distance between them measured around the polygon must be at least the sum of the flap lengths those corners represent, and the polygon’s edges must be at least as long as the differences. This is the same conservation argument that gives a flap its disc, applied to a boundary instead of to a region: paper that has to stretch between two flaps cannot be shorter than the flaps demand.

A polygon satisfying it is called active-path feasible, and a packing that has been optimised properly produces only such polygons, because the packing constraint and the path condition are the same constraint seen from two sides.

When the condition fails — which happens when a packing is arranged by hand and not checked — the shrink still runs and still produces a pattern. The pattern simply does not fold into the base it was supposed to. That is a failure mode worth naming, because it is silent: the construction has no way to notice, and the drawing looks like every other drawing.

Filling a polygon the old way #

It is worth seeing what the universal molecule replaced, because the older constructions are still the ones a folder recognises.

A triangle is filled by its three bisectors, and the fold is the rabbit ear. A quadrilateral is filled by a gusset: two bisectors from one pair of corners, two from the other, and a short segment joining them where they fail to meet at a point. That short segment is the gusset, its length is determined by the quadrilateral, and when it comes out at zero the quadrilateral happens to have an incircle and the four bisectors do meet — which is the waterbomb base, and is why the waterbomb base looks like the special case it is.

For five corners and more the ad-hoc constructions run out. The bisectors from opposite corners miss each other in more ways than one gusset can absorb, and there is no obvious pattern to continue.

The shrink resolves all of them at once. Run it on a triangle and the outlines collapse to a point, the traces are the three bisectors, and the rabbit ear falls out. Run it on a quadrilateral and the outlines collapse to a segment, whose length is the gusset. Run it on any polygon and the outlines collapse to whatever they collapse to, and the traces are the pattern. The older constructions were not wrong; they were the first two cases of this one, discovered separately because nobody had noticed they were a sequence.

Only a triangle collapses in one step #

The observation that the rabbit ear and the gusset are the first two cases of the shrink has a counting argument behind it, and the argument says how many cases there are and why nobody found the third.

An inward offset is described by three numbers: where the shrinking outline’s limiting point is, and how far the edges have moved. Requiring every edge to reach that limit together is one equation per edge.

For a triangle, three equations in three unknowns: generically a unique solution, and all three edges vanish at the same instant. That instant is the incentre, the traces are the three bisectors, and the whole construction is over in one event.

For any polygon with more edges the system is over-determined by $n - 3$, so the edges cannot all vanish together. They vanish one at a time, and each vanishing is an event at which the outline loses a corner and the construction continues on a smaller polygon.

Which is why the sequence stopped at four #

So a quadrilateral has one event before it collapses, a pentagon has two, a hexagon three — and the older constructions map onto the count exactly.

The rabbit ear is the zero-event case: nothing happens on the way and the answer is three bisectors. The gusset is the one-event case: something happens once, and the short segment it leaves is the gusset, whose length is zero precisely when the quadrilateral happens to have an incircle and the system is consistent after all.

The two-event case is a pentagon, and there is no name for it, because an ad-hoc construction has to anticipate every event and by two the case analysis stops being memorable. That is the sense in which the hand-built molecules ran out: not that anybody failed, but that the number of cases grows with the corners and the third one already has more than a folder would carry around.

So the universal molecule’s contribution is the recursion rather than the geometry. The shrink is not a cleverer way of drawing bisectors; it is the observation that an edge event returns a smaller instance of the same problem, so one rule covers $n - 3$ events instead of $n - 3$ rules covering one each.

That also predicts the shape of the figure’s five-sided example. It runs further before collapsing not because it is rounder but because it has two events to get through, and a hexagon would have three.

Which theorem was checked, and how #

Two things are checked and one is not.

The offset is checked, at every corner of every intermediate outline: the distance from the corner to its parent edge must equal the offset to within a billionth. That is the property the straight traces rest on.

The collapse distance is checked in the sense of being computed rather than chosen — sixty rounds of bisection on the orientation of the inset outline, which is well past the precision of anything drawn.

What is not checked is that the resulting molecule folds flat. The figure draws the shrink and the traces; it does not assign mountains and valleys and does not run the pattern past the conditions at a vertex. The full molecule is a complete crease pattern and could be checked, and this rung does not do it — which is stated here rather than left for a reader to assume either way.

What the picture cannot show #

The shrink is a motion and the figure is a set of stills. The intermediate outlines are drawn at equal fractions of the collapse distance, which makes them look like a sequence of decisions; they are samples of something continuous.

The picture also shows the construction on a convex polygon. Axial polygons from real packings are usually convex, because they are bounded by lines between mutually touching discs, but not always — and a reflex corner changes the story. The inward-moving edges can collide with a different edge rather than with their neighbours, which splits the polygon into two, and the construction has to continue on both halves. The version drawn here does not handle that, and refuses rather than guessing.

The idealisation underneath #

The molecule assumes paper of no thickness, and here the assumption bites in a specific and well-known way: the shrink stacks layers.

Every intermediate outline is a crease, and the paper between two successive outlines becomes a layer in the finished base. A polygon shrunk through six outlines contributes six layers at the point where they all meet, on top of whatever the flaps are already contributing. A base whose molecules are deep is a base with a hard centre, and folders describe exactly this: designs that go together easily at the extremities and become impossible at the middle.

The sheet has a thickness, and the molecule is where a design’s thickness is decided. Nothing in the construction sees it.

Why “always works” is the property that matters #

There are better molecules than the universal one for particular polygons — molecules with fewer layers, or with the layers distributed more kindly, or with a symmetry the universal construction destroys. A designer working by hand will often use one, and a designer working on a grid will use a different set again, because a grid supplies its own molecules.

The universal molecule’s claim is not that it is best. It is that it exists for every polygon that could have come out of a valid packing, and that claim is what turns the tree method from a technique into an algorithm.

The distinction is the same one that separates a heuristic from a procedure everywhere else. A method with a case that sometimes has no answer needs a person at that case; a method with an answer in every case can be handed to a machine and left. TreeMaker takes an arbitrary tree and returns a crease pattern precisely because the last step never fails, and every other step was already determinate.

That is worth saying plainly because “universal” sounds like a boast and is actually a modest claim, and the modest claim is the useful one. It is the difference between a subject where designs are found and a subject where they are computed.

The surprising connection #

The shrink is the straight skeleton, and the straight skeleton is the object at the centre of the fold-and-cut theorem.

The two uses look nothing alike. Fold-and-cut takes an outline and folds it so that the whole outline lands on one line, so that a single straight cut releases the shape. The universal molecule takes a polygon of leftover paper and folds it so that the whole boundary lands on the base’s axis. They are the same operation with different names for the line.

That equivalence has a practical consequence: the fold-and-cut construction is a special case of the design method, or the design method is a general case of fold-and-cut, depending on which one somebody learned first. Both are answering fold this polygon’s boundary onto a line, and the straight skeleton is what answers it.

There is a second connection, less famous. The offset construction — move every edge inward at a constant rate and see what happens — is how a machinist computes a tool path, how a cartographer buffers a coastline, and how a chip designer checks a layout for clearance. It is one of the standard operations of computational geometry, and origami design uses it for the same reason everybody else does: it turns a shape into a family of shapes with a single parameter.

Who found it, and when #

The universal molecule is Robert Lang’s, published as part of the tree method through the 1990s and set out fully in Origami Design Secrets (2003). The proof that it always produces a valid molecule for a polygon satisfying the path condition is the result that makes the tree method complete rather than a collection of cases, and it is why TreeMaker can be handed an arbitrary tree and produce a pattern.

The rabbit-ear molecule for a triangle is traditional and has no author. The quadrilateral molecules were catalogued by Meguro and others in Japan in the 1980s, as part of the same movement toward systematic design that produced circle packing.

The straight skeleton itself is older than any of this in spirit and younger in name: it was formalised by Aichholzer, Aurenhammer, Alberts and Gärtner in 1995, and the fold-and-cut application came from Demaine, Demaine and Lubiw a few years later.

The ladder from here #

This rung fills the polygons and finishes the pattern. The ladder above it is about what the finished pattern costs: how many layers accumulate where, which is the question a folder asks and the design software does not answer, and how a molecule can be chosen to move the layers somewhere less crowded.

The rung below it, and the one to read first if this was read first, is the construction that produced the polygons — because the polygon is not a shape somebody drew, and the fact that it came out of a packing is what makes the path condition hold.