Every flap on one axis

The circle argument is presented as a fact about paper: a flap of length L uses every point within L, therefore two flaps whose circles overlap cannot both exist. Stated that way it sounds unconditional.

It is not. It is true of a particular kind of folded object, and every account of the method assumes that kind without naming it.

What a uniaxial base is #

The definition is short and the consequences are not.

A base is uniaxial when all of its flaps lie along one common line — the axis — and each flap is folded flat so that it projects onto that line as a segment. Look at such a base edge-on, along the axis, and every flap collapses to a point; look at it from the side and the flaps hang in a row like the teeth of a comb.

The bird base is uniaxial. The frog base is uniaxial. Nearly every traditional base is, and so is nearly every complex insect designed in the last thirty years.

A model that is not uniaxial is one whose parts point in genuinely different directions in the folded state — a box, a cube, a curved sculpture, a tessellation. None of those has an axis, and none of them is what the tree method designs.

The projection is the tree #

Now the fact that makes the theory work, and it is worth stating as an identity rather than as an intuition.

Project a uniaxial base onto the plane square to its axis. Each flap, being flat and square to the axis, projects onto a segment whose length is the flap’s length. Each junction projects onto a point. The flaps that meet at a junction project onto segments meeting at that point.

What comes out is a stick figure with the flap lengths as its edge lengths — which is exactly the tree the design started from.

So the tree is not an approximation of the base, or a schematic of it, or a way of thinking about it. It is a projection of it, in the strict sense, and the correspondence is exact.

Which is why paper is conserved #

The conservation argument follows immediately, and this is where the restriction earns its keep.

Take two flap tips. The distance between them measured through the folded model is at least the distance between their projections, which is the distance between them through the tree. And the distance between them through the folded model equals the distance between the corresponding points of the flat sheet, because folding does not stretch.

Therefore two points of the flat sheet that become flap tips must be at least the tree-distance apart. That is the packing condition, and it drops out of the projection in two lines.

Remove the uniaxial restriction and the first step fails. If the flaps do not lie square to one axis, their projections are shorter than they are, the tree-distance is no longer a lower bound for the paper-distance, and the condition is simply not implied. Nothing in the argument survives.

What the restriction costs, in paper #

The projection argument makes the condition true; it also makes it possible to price the restriction, which the usual account does not attempt.

Take two flaps of length $L$ from one junction and tilt each by an angle $\varphi$ out of the plane square to the axis, toward one another. The angle between them becomes $180° - 2\varphi$, so their tips are

$$2L\cos\varphi$$

apart instead of $2L$. The paper only has to supply that, so their circles may overlap by a fraction $1 - \cos\varphi$ — 13% at thirty degrees, 29% at forty-five — and the area a pair of flaps demands falls as $\cos^2\varphi$.

A design free to tilt its flaps by forty-five degrees needs half the paper of the uniaxial one. That is not a marginal saving; it is larger than every efficiency gain the packing literature argues about.

Which is why the restriction is a real loss #

So uniaxiality is not a harmless normalisation adopted for convenience. It is a substantial constraint, and the whole packing theory holds only because the constraint has been accepted.

The trade is visible once stated. A uniaxial base is flat, which is what makes its layers analysable, its creases constructible from the packing, and its design problem a two-dimensional one anybody can draw. Tilting the flaps buys paper and immediately gives up all three: the object is no longer flat, so there is no shadow to be a tree, no projection to make the conservation argument, and no construction from packing to crease pattern.

That is the honest statement of what a non-uniaxial theory would be worth, and of why nobody has one. The prize is up to a factor of two in paper. The price is that every tool in the method was built on a projection that no longer exists.

What the restriction excludes #

It is worth listing what the theory does not reach, because the list is longer than the summaries imply.

Anything with volume. A cube, a box, a vessel. These have parts pointing in three independent directions, and no axis exists.

Tessellations. A Miura fold has no flaps at all in the design sense; it is a surface, and its structure is periodic rather than tree-shaped.

Curved work. A curved crease produces a surface with no flat state, so there is nothing to project.

Most representational sculpture. A model whose interest is in a curve of the back or the set of a wing is not being designed by lengths, and the tree carries none of the information that matters.

Anything with a closed loop. A tree has no cycles. A subject that needs a handle, a ring or a closed body is not a tree, and the method has nothing to say about it.

That leaves a large and important family — anything that can be reduced to a stick figure with measured limbs — which happens to include most of what technical folders design. The method is not narrow in practice. It is narrow in principle, and the difference matters when somebody tries to apply it outside its range.

Which theorem was checked, and how #

The figure at the head of this essay is an identity check rather than a computation, and it is worth saying exactly which identity.

The generator carries the tree as a list of edges with lengths. It draws the tree with those lengths and then draws the base with the same lengths, each flap square to the axis. The check is that the drawn flap lengths equal the tree’s edge lengths exactly — which sounds trivial and is precisely the claim the projection makes. A figure that drew the base with flaps of convenient length rather than of the tree’s length would be illustrating the correspondence instead of exhibiting it.

What the figure cannot show is the hard direction. That every valid packing yields a uniaxial base is Lang’s theorem, and proving it needs the construction of the crease pattern from the packing — the ridge and hinge creases, and an argument that they always assemble. A picture of one base is not evidence about all packings.

It also cannot show the folded state properly. A uniaxial base is a stack of flat flaps sharing an axis, and drawn honestly it is a row of overlapping lines with no depth. The figure draws the flaps side by side for legibility, which is a schematic: in the real base they lie on top of one another.

Where the creases come from #

The packing determines the pattern, and the mechanism is worth a paragraph even though it is the part this site does not draw.

Each disc’s centre becomes a flap tip. The tangencies between discs — where two discs touch — become ridge creases, running between the two centres. The boundaries between a disc and its neighbours become hinge creases, which are the folds that let the flap swing to the axis.

The resulting pattern is a subdivision of the square into regions, one per disc and one per river, with the ridges and hinges as its edges. Every polygon in the subdivision has to be filled with a molecule — a small standard crease pattern that collapses it correctly — and choosing molecules is where the remaining freedom lives.

That last step is the reason two designers working from the same packing produce different crease patterns. The packing is determined; the filling is not.

The surprise: the axis is a choice, and it can be moved #

The restriction sounds like a limitation on what can be designed. It is better understood as a coordinate system, and coordinate systems can be changed.

A base is uniaxial with respect to some axis. Nothing says the axis has to be the one the designer first thought of, and a subject that resists one choice may submit to another. More usefully, a model can be built from several uniaxial pieces joined together, each with its own axis — which is what a designer is doing when a base is folded and then a section is reoriented.

There is also a generalisation in the literature. Lang and Alex Bateman have developed methods for bases with more than one axis and for what are called polygon-packed designs, in which the sheet is subdivided into polygons that each collapse to a piece of the model rather than into discs that each collapse to a flap. Box pleating, taken seriously, is a version of the same idea: designing on a grid is designing with rectangles rather than with circles.

So the uniaxial restriction is the first case of a family rather than a boundary. It happens to be the case with a clean theorem, which is why it is the one everybody learns.

How to tell whether a base is uniaxial #

The definition is easy to state and slightly awkward to apply, so a working test is worth having.

Hold the folded base and look for a line such that every flap, laid flat, is square to it. If one exists, the base is uniaxial with respect to it. In practice the axis is usually obvious — it is the line the model would balance along — and the test is whether the flaps really are all square to it or merely mostly.

The negative test is easier. If two parts of the model point in directions that are not parallel and not opposite, no axis works, and the base is not uniaxial. A model with a head that turns, a wing that sweeps, or a body with any thickness fails immediately.

There is a middle case that causes confusion. Many finished models are shaped after the base is folded, and the shaping moves flaps off the axis. Such a model was designed uniaxially and is no longer uniaxial when finished, which is entirely normal — the restriction applies to the base, not to the result.

That distinction is why the tree method is more general than it looks. What has to be uniaxial is the intermediate object, and everything after it is free.

Point splitting, and where the tree comes from #

A question the method does not answer is where the tree comes from, and there is one operation worth knowing because it is how designers get the tree they want.

Point splitting replaces a single leaf with two shorter ones hanging from a new junction. In the subject that turns one flap into two — a leg into two toes, an antenna into a fork — and in the packing it replaces one disc with two smaller discs and a river between them.

The reason it matters is that it is almost free. Two short flaps and their river take up much less paper than one long flap of the combined length, so a designer who needs more appendages can often get them by splitting rather than by finding more paper.

Run the operation repeatedly and the tree grows into whatever shape the subject needs, with the packing adjusting each time. That is how the elaborate insect designs are built: not by drawing a tree and solving once, but by splitting and re-solving until the scale stops improving.

Which is worth noticing because it changes what the method is. Stated once, it is an algorithm that turns a tree into a pattern. Used in practice, it is a loop in which the designer edits the tree in response to what the packing does, and the interesting decisions are all in the loop.

Where the model stops #

Flat flaps. A flap is assumed folded flat. A flap with any thickness of its own — several layers spread apart — projects onto more than a segment.

No shaping. The base is where the method stops. Everything after it — narrowing, thinning, curving, the whole of what makes a model look like its subject — is outside the theory and is most of the work.

Lengths only. The tree records how long each limb is and nothing about how thick it is, where it points, or what it looks like. Two subjects with the same measurements have the same tree.

Layers unaccounted. The number of layers at a flap’s base grows with the design’s complexity, and the theory says nothing about it. A base that is correct on paper can be unfoldably thick in practice, which is the same idealisation that limits everything else here.

Rivers are part of it. An internal edge of the skeleton projects onto a segment of the tree just as a leaf does, and the uniaxial condition covers both — which is why the packing condition has the form it does.

One tree, many packings. Nothing guarantees the packing a program finds is the one a folder would want, and the objective a program optimises is scale rather than foldability.

Who found it, and when #

The idea has an unusual shape historically: the object was in use for centuries before anybody noticed it was a category.

Uniaxial bases are as old as the traditional bases — the bird base is at least three hundred years old — and every folder who has made one has made a uniaxial object. Nobody described the property because there was nothing to contrast it with.

The naming came with the theory. Robert Lang introduced the term in the course of developing the tree method through the 1990s, precisely because the theorem needed a hypothesis and the hypothesis needed a name. Jun Maekawa and Toshiyuki Meguro had by then developed related techniques in Japan, and the same class of object underlies theirs.

It is the same shape as the restriction to one fold at a time in the axiom list, and it has the same effect: an assumption nobody states is an assumption nobody can lift, and naming it is what makes the generalisation available.

The historical lesson is a familiar one. The restriction was invisible while it was universal; it became visible at the moment somebody tried to prove something, because a proof has to say what it assumes.

The hypothesis is where the theory lives #

There is a general point here about how a practical rule becomes a theorem, and this is a clean instance of it.

The circle rule worked for years as a rule of thumb. Everybody who used it was designing uniaxial bases, so the rule was always true for them, and the restriction was invisible because it was universal in practice.

Making it a theorem meant finding out why it was true, and finding out why it was true meant discovering what it had been assuming. The hypothesis was not added to make the proof work; it was already there, unstated, in every application.

That is worth carrying because it says where to look when a heuristic resists proof. The missing piece is usually not a cleverer argument. It is a condition everybody has been satisfying without noticing, and the payoff for naming it is that the theory then says what happens when it fails — which here is that nothing at all survives, and that is useful to know.

The ladder from here #

Later rungs against this anchor: the ridge and hinge construction, drawn properly. Molecules, and why the choice of molecule is the last free parameter. Multi-axis bases and how the condition generalises. Polygon packing as the successor method. The relationship between uniaxial bases and the tree metric, stated as an embedding problem. And the question of what a design theory for non-tree subjects would even look like, which is open in the sense that nobody has proposed one.

The circle rule is remembered and the hypothesis is not, which is the usual fate of a hypothesis that is always satisfied. It is worth remembering anyway, because the day it fails is the day the rule stops being true.