A state no motion reaches

A folded state is a place. A folding is a journey. Most of this subject’s theory is about places, and almost everything that gets manufactured cares about journeys.

For four of the five vertices in the figure the counts agree. For the fifth they do not, and the fifth is the one with equal sectors — which is to say the symmetric one, which is to say the one anybody would draw.

Two different questions #

The conditions at a vertex ask about the flat state and nothing else. Do the sectors close, do alternating sectors sum to a straight angle, do the letters differ by two, is any strictly smallest sector flanked by two of the same letter. All four are statements about the sheet lying flat with the creases fully folded.

Rigid-foldability asks something else entirely: is there a continuous family of configurations, starting from the unfolded sheet, ending at the folded state, in which no panel ever bends. The intermediate states have partial fold angles and are not flat, so none of the four conditions applies to any of them.

There is no reason in advance for the answers to agree, and they do not.

The odd vertex is the one the lemma cannot refuse #

The five vertices are not five arbitrary shapes, and reading their sectors says exactly which one will disagree before anything is enumerated.

Each is given as a pair, and the four sectors run $a$, $b$, $180° - a$, $180° - b$. Take them in turn. Sixty and ninety gives 60, 90, 120, 90 — the smallest sector is 60 and both its neighbours are 90, so it is strictly smallest. Thirty and 120 gives 30, 120, 150, 60: strictly smallest. Fifty and seventy: strictly smallest. Eighty and fifty-five: strictly smallest.

Forty-five and forty-five gives 45, 45, 135, 135. The two smallest sectors are adjacent and equal, so neither is smaller than both its neighbours, and the smallest-sector lemma has nothing to act on.

The one vertex where the counts disagree is the one where that lemma is silent, and it is the only one of the five in that position.

Which says how large the gap has to be #

That identification makes the gap predictable rather than merely observed.

At a generic degree-four vertex the counting theorem leaves eight assignments and the lemma halves them to four. Where the lemma is silent, all eight survive the local conditions.

The rigid motion is a property of the linkage and does not know about the lemma; it reaches the same four it reaches anywhere. So the gap is exactly the four assignments the lemma would have removed — flat states that exist, satisfy every condition the subject states, and have no continuous rigid path to them.

That is a considerably sharper statement than the counts sometimes differ. It says the discrepancy is not a general looseness between two notions but a specific one: the smallest-sector lemma is the condition that happens to agree with rigid-foldability, and where it falls silent the agreement fails.

It also predicts where else to look. A tie between the two smallest sectors is a coincidence between continuous quantities, so a vertex drawn at random never has one — and a vertex drawn on a grid has them constantly, since a coarse grid offers too few sector sizes to avoid a repeat. So the states that exist and cannot be reached are concentrated in exactly the patterns designers draw, and absent from the ones a sampler produces.

What a vertex actually does #

A degree-four vertex is a closed spherical linkage. Project the four creases onto a small sphere centred on the vertex and they become four points; the sectors become four arcs of fixed length; and folding is the motion of a closed spherical quadrilateral with four fixed side lengths.

Such a linkage has one degree of freedom, and its configuration space is a curve. Driving one fold angle determines the others, which is what the kinematics essay works out in full.

The point that matters here is that the curve is a curve. It has ends, where the linkage jams; it may have more than one component; and a folded state not lying on the component containing the flat sheet is unreachable from the flat sheet, however respectable it looks.

Counting both sides #

The comparison in the top figure is worth describing precisely, because its value comes entirely from the two sides being independent.

The combinatorial count enumerates all $2^4 = 16$ assignments of the vertex’s four creases and tests each against developability, Kawasaki, Maekawa and the big-little-big lemma. It never solves anything in three dimensions; it works with sector angles and letters.

The kinematic count solves the spherical linkage across its whole travel, samples the configurations, and reads the mountain-and-valley assignment off each one by the sign of its fold angles. It never enumerates anything; it works with unit vectors and rotations.

They agree on four of the five vertices at four assignments each. On the vertex with equal sectors the combinatorial count is eight and the kinematic count is four, two of which have a crease left flat.

The gate requires that every assignment the linkage reaches with all four creases folded is one the enumeration accepts. If the linkage ever produced a state the theorems reject, one of the two would be wrong, and it would matter a great deal which — so that is checked rather than assumed, on every vertex, at every build.

Why the gap opens at equal sectors #

The mechanism of the gap is specific and it is the interesting part.

Big-little-big says that a strictly smallest sector may not be flanked by two creases of the same letter. When all four sectors are equal there is no strictly smallest sector, so the condition has no subject and forbids nothing. Maekawa alone then admits all eight of the three-and-one splits, and the combinatorial count is eight.

The linkage does not care whether a sector is strictly smallest. It cares about whether the four arcs can close in three dimensions in a particular cyclic arrangement, and for four equal sectors half of the eight arrangements do not correspond to any closed configuration with all creases folded. They are not near-misses; there is no configuration at all.

So the discrepancy is not a subtlety of tolerance or sampling. It is the local conditions being weak in exactly the case where the pattern is symmetric, and symmetry is what folders design with.

The two branches, and what they are not #

There is a related fact about a degree-four vertex that is easy to confuse with this one, and separating them is worth a section.

A vertex’s linkage generically has two solution branches through the flat state: two different ways the four creases can leave flatness, corresponding to the two signs of a square root in the closure. Both are legitimate rigid foldings and a vertex passing through flat can come out on either.

That is a statement about choice, and it is the opposite of the statement in this essay. Branching means a vertex has more places to go than the naive account suggests. Unreachability means it has fewer. Both are properties of the same configuration space and they arise at different places on it — branching at the flat state, where the curve crosses itself, and unreachability away from it, where the curve simply does not go.

The practical upshot is that a designer has to worry about both, and about them separately. A branch is a risk of the mechanism deploying into the wrong shape; unreachability is a risk of it not deploying at all.

Which theorem was checked, and how #

Neither count is a theorem and both are computations, so what is checked is agreement — and the checking is arranged so that agreement is difficult to fake.

The linkage solver was written to answer a different question (what a vertex does part-way through its motion) and shares no functions with the enumerator. Its own correctness is checked elsewhere against two things it was not built to reproduce: it recovers Maekawa’s three-to-one split throughout the motion rather than only at the flat state, and it reaches a flat state exactly when Kawasaki holds.

The figure refuses to draw a comparison in which no case shows a gap. That is a small guard and it earns its place: a later edit that replaced the equal-sector vertex with another generic one would leave a figure that draws beautifully and argues nothing, and nothing else in the build would notice.

Counting the whole travel, not just the ends #

One more refinement makes the counting sharper, and it is the reason the kinematic side reports partial states as well as complete ones.

Sampling the linkage across its whole travel produces configurations at every fold state, not only at the flat ones, and reading an assignment off a partly-folded configuration gives a string in which some creases are not yet folded at all. Those show up as an F — flat, uncreased — and they are not flat-foldable assignments in the combinatorial sense, because that enumeration works over mountains and valleys only.

For the equal-sector vertex two of the four reachable states are of this kind, which is worth saying plainly: the vertex can reach a configuration in which one crease has never been used. That is a legitimate rigid folding and a slightly disconcerting object — a four-crease vertex behaving as a three-crease one, which is to say as a sheet with a single fold and a couple of scored lines that stayed flat.

The gate distinguishes the two kinds and only requires the fully-folded ones to be accepted by the enumeration. Requiring it of the partial ones would fail immediately and for an uninteresting reason, and a check that fails for an uninteresting reason gets loosened until it fails for no reason at all.

What the picture cannot show #

The figure counts assignments and does not show configurations. A reader cannot see from it which four of the eight are reachable, and the answer is not interesting — the labels are strings of four letters and any of them could be listed.

What is genuinely missing is the whole-sheet version. Everything here is one vertex, and a tessellation is many vertices sharing creases. Reachability for a whole pattern is a much harder question: the configuration space is a variety cut out by every vertex’s closure conditions at once, and it can be disconnected in ways no single vertex predicts. The single-vertex gap is the smallest instance of a phenomenon that gets worse.

The figure also says nothing about self-intersection. A configuration on the linkage curve is one in which the four sectors close in three dimensions; it is not necessarily one in which the panels avoid each other, and checking that is a separate problem the solver does not attempt.

Why the enumeration is the generous one #

It might be expected that a physical criterion is stricter than a combinatorial one, and here the expectation holds — but not for the reason it usually does.

The local conditions are necessary and not sufficient even for the flat state, which is the ordinary gap this site keeps returning to. What the comparison in this essay adds is a second, independent way for them to be generous: they are also blind to the route. An assignment can be a perfectly good flat-folded state of the vertex — the paper really does lie flat that way — and still have no rigid path, because getting there means bending something.

So a locally-valid assignment can fail in two different directions, and a designer needs both checks. The layer-ordering failure is a fact about the folded state; the reachability failure is a fact about everything between the folded state and flatness. Neither implies the other, and a pattern can pass one and fail the other.

That is why this site’s own machinery runs the two computations separately and requires them to agree only where they overlap. Merging them into one checker would be tidier and would lose the information that the disagreement carries.

The idealisation underneath #

Rigid panels — perfectly stiff, hinged along mathematical lines. That is what separates panels from paper and here it is doing all the work, because the whole distinction between reachable and unreachable dissolves if panels can bend.

Paper reaches unreachable states routinely. A folder confronted with a state that has no rigid path simply bends the paper on the way and lets it flatten out at the end, and the finished model is indistinguishable. That is not cheating; it is what paper is for. The distinction matters when the panels are aluminium or silicon and there is no bending available, and this is exactly the boundary at which origami stops being a craft and becomes a mechanism.

There is a second, quieter idealisation: the vertex is treated as a point. A manufactured hinge has a width, a real panel is attached at a finite distance from the ideal line, and both change the linkage’s arc lengths slightly. That is the thickness problem, and it moves the boundary of reachability without changing its character.

What a designer does about it #

The engineering response is not to check reachability after the fact. It is to build patterns whose reachability is guaranteed by construction, and the construction is the same one that has been running through this whole subject: start from something known to fold and modify it in ways that preserve folding.

A Miura fold is the canonical example and its safety is not luck. Its vertices have four distinct sectors, so big-little-big has a subject, the combinatorial and kinematic counts agree, and the sheet’s configuration space is a single arc with the flat state at one end and the packed state at the other. There is nowhere else to go.

Patterns that are symmetric get treated with more care. A waterbomb tessellation has vertices with equal sectors and is famously awkward to deploy — it has states it settles into that are not on the way to anywhere, which is the whole-sheet version of the gap counted here, and which is why waterbomb-based deployables tend to be actuated at many points rather than driven from one.

The surprising connection #

The distinction between “a state exists” and “a state is reachable” is the same distinction that separates two other pairs of ideas in this subject, and noticing the pattern is worth more than the individual instances.

Flat-foldability of a whole sheet is a statement about the existence of a layer ordering; finding one is a search. The existence is what is NP-hard, and no motion is involved at all.

Deployment reliability is the engineering version. A satellite’s array has a folded state and a deployed state, and the question that keeps engineers awake is not whether the deployed state exists — it obviously does — but whether the mechanism goes there, every time, from wherever it starts. A one-degree-of-freedom pattern is favoured precisely because its configuration space is a single curve with no branch to take wrongly.

So the same shape appears three times: a set of valid configurations, a smaller set that can be got to, and a preference for designs where the two coincide. The counting on one vertex is the smallest case where the difference can be exhibited exactly.

Who found it, and when #

That rigid-foldability and flat-foldability differ has been understood since rigid origami became a subject in its own right, through the 1990s and 2000s — Tomohiro Tachi’s work on rigid-foldable surfaces makes the distinction operational, and simulators built on it decide reachability by tracing the configuration space rather than by testing conditions.

The spherical-linkage treatment of a single vertex is older and belongs to kinematics rather than to origami: a degree-four vertex is a spherical four-bar, and spherical four-bars were analysed thoroughly in the nineteenth century for reasons having nothing to do with paper. The branch structure of a spherical four-bar’s configuration space was understood long before anybody folded one.

The ladder from here #

This rung is about one vertex. The rung above it is the whole sheet, where the configuration space is cut out by many vertices at once and where a pattern with a perfectly good folded state can be rigid — the case that motivates every generic-rigidity argument in the subject.

Below it, what the vertex does on the way solves the linkage this rung counts on, and panels instead of paper sets out why any of it matters once the material is not paper.