The motion has no letters to choose

Everything about the cost of a search in this collection has turned on the same fact: a search chooses, and the order it chooses in decides what it costs. One line’s coin toss is worth three orders of magnitude; one branching rule against another is worth four; and a search that never deduces anything has no ordering question at all because it has nothing to choose between.

Rigid folding sits outside that entirely, and it is worth asking why, because the answer says something about what a discrete question costs that is hard to see from inside one.

Nothing to assign #

A flat-folding question is about letters. Each crease is a mountain or a valley, the assignment is one of two to the power of however many creases there are, and finding a good one is a search through that space.

A rigid folding question is about angles. Each crease has a fold angle, a real number between minus and plus a straight angle, and the panels’ rigidity imposes equations relating the angles around each vertex. Panels instead of paper sets out what that changes; what it changes here is that there is no space to search. The equations either have a solution or they do not, and finding it is a solve.

A letter is recoverable afterwards — the sign of the fold angle says which way the crease went — but it is an output rather than an input. Nobody chooses it, so nobody can choose it badly, and every ordering question in the previous four essays evaporates.

There is a way of seeing why this is inevitable rather than convenient. A discrete search exists because the answer is one of finitely many things and there is no formula for which. A continuous solve exists because the answer is a point in a space where the constraints are differentiable, so the constraints themselves say which way to move. The first has to look; the second is told.

Folding sits astride that line in an unusual way. The same crease pattern supports both questions — a flat folding, which is a choice of letters, and a rigid folding, which is a choice of angles — and one of them is combinatorial and the other analytic. That is rare. Most subjects have their problems on one side or the other, and the fact that this one has the same object on both sides is what makes the comparison available at all.

What replaces the ordering question #

Something does replace it, and it is not nothing.

A solve is iterative. It starts from a guess and refines, and three things can go wrong that a discrete search never faces.

It can fail to converge, wandering or oscillating without settling. A discrete search either finds something, exhausts its tree, or runs out of budget, and the three are distinguishable; an iteration that has not settled after a thousand steps might settle at the thousand and first.

It can converge to the wrong thing. The equations often have several solutions — a degree-four vertex has two distinct folding modes and a mesh has more — and which one an iteration reaches depends on where it started. That is the continuous analogue of the value-ordering problem, and it is worse rather than better: a discrete search that takes a wrong branch eventually backs out of it, and an iteration that converges to another solution simply reports it.

And it returns a number rather than a verdict. A residual of 10⁻¹⁴ is a solution; a residual of 10⁻³ is not; a residual of 10⁻⁷ is a judgement about tolerance, and there is nothing in the mathematics to make it for anybody.

The third of those is the one that costs the most in practice, and it is worth dwelling on because it is invisible in a way the others are not.

A discrete search’s answer is checkable in one pass by somebody who does not trust it: write the letters onto the pattern, run the conditions, rebuild the folded sheet, look for a circle. Every claim in the flat-folding half of this collection is verified that way, by machinery that shares no code with what produced the answer.

A solve’s answer is a set of real numbers, and checking it means re-running the same equations. The independent check available — that no edge changes length and no panel bends — is genuine and it is weaker: it confirms the configuration is a folding of something, not that it is a folding of the pattern that was asked about. Getting an independent check as strong as the discrete side’s would mean deriving the configuration a second way, which is exactly what a second solver written to share no code is for.

The trade, stated as a trade #

So the two questions are not one easier and one harder. They fail in different places and the failures have different shapes.

A discrete search’s failures are honest and expensive. It cannot converge to the wrong answer, because it checks; it cannot mistake a near-miss for a hit, because there are no near-misses; and it distinguishes no solution exists from the budget ran out. What it costs for that honesty is the ordering problem — four orders of magnitude of variation from choices nobody thought were choices.

A continuous solve’s failures are cheap and slippery. It is fast, it has no ordering problem, and it will hand back a plausible-looking answer that is a different solution from the one asked for, or a residual that somebody has to interpret. Solving every face at once is the arrangement that makes the solve reliable here, and its reliability comes from checking the answer afterwards against something that shares no code with it.

The pattern is one this collection keeps meeting. An instrument whose only answers are yes, no and not settled is more trustworthy and more expensive than one that always says something.

There is a useful way to name the two failure modes. A discrete search fails loudly: it stops, and it says which of three things happened. A continuous solve fails quietly: it returns, and the return looks like every other return.

A collection that reports what its instruments say will therefore be systematically more confident about its continuous results than about its discrete ones, for no reason connected to how reliable either is. That is a bias with no natural correction, and the only defence is to insist that a solve’s output carries its residual everywhere the output goes — which is a discipline about reporting rather than about mathematics.

The ambiguous residual is a conditioning report #

The middle case — a residual of $10^{-7}$, neither clearly a solution nor clearly not — is presented as a judgement nothing in the mathematics can make. There is a principled answer, and it removes the judgement rather than making it.

A residual is not comparable against a fixed number. It is comparable against what this system could achieve, which is roughly the condition number times machine epsilon. A well-conditioned system reaches $10^{-15}$; one with a condition number of $10^{8}$ cannot do better than $10^{-8}$ however long it iterates.

So a residual of $10^{-7}$ on a system conditioned at $10^{8}$ is a converged solve, and the same residual on a system conditioned at ten is a solve that has stalled. The number alone says nothing; the number divided by the floor says everything.

The tolerance question is therefore not a judgement about how much error is acceptable. It is a measurement of how much error is unavoidable, and it is a measurement the solve can make: perturb the inputs by a rounding unit and see how far the answer moves.

Which restores the honest three-way answer #

That matters because it recovers, for the continuous side, the property the essay credits only to the discrete one.

A discrete search says found, exhausted or budget. A solve with its residual normalised by the achievable floor says converged, stalled or ill-conditioned — three answers, distinguishable, with no interpretation required.

A solve that returns a residual at its own floor has converged and can say so. One that returns a residual far above its floor has stalled and can say so too. The slipperiness the essay describes is a property of reporting a bare residual, not of continuous methods.

And it explains the asymmetry’s root #

There is a reason the discrete side never has this problem, and it is worth naming because it is more fundamental than loudness.

A discrete answer lives in a space with no metric. A lettering is right or wrong; there is no lettering that is nearly right, because the conditions are equalities and equalities have no neighbourhoods. So there is nothing for a tolerance to be about and no floor to normalise against.

A continuous answer lives in a space that has a metric, and having one is what makes nearly a meaningful word — which is a gain rather than a loss. The discipline the essay asks for is therefore narrower than it sounds: not that a residual travel with its output, but that it travel with the floor it should be compared against, which is one extra number and turns an interpretation into a reading.

Where the discrete question comes back #

Rigid folding does not escape the discrete world entirely, and the place it re-enters is worth marking.

Which mode. A degree-four vertex’s motion has two branches, and which one the panels take is a discrete fact about the folding even though nothing discrete was chosen. The vertex is geared describes the relation along one branch; picking the branch is a choice, made by the starting guess rather than by a search.

Self-contact. Whether the panels pass through one another on the way is a question about orderings, and it is the same expensive question the flat-folding side asks. A solve says nothing about it: the equations are satisfied by configurations in which two panels occupy the same space, and detecting that is a separate search over which panel is where.

Rigid-foldability itself. Whether a pattern folds rigidly at all, rather than only flat, is a discrete property, and it is not flat-foldability — the two conditions are different and a pattern can satisfy either without the other.

Why a continuous problem has no ordering #

It is worth being precise about the mechanism, because “continuous problems have no ordering question” is not true in general and is true here for a reason.

An ordering question exists when a procedure makes a sequence of commitments that constrain the ones after it. A backtracking search does exactly that: each letter fixed narrows what remains, and the narrowing depends on which letters were fixed.

An iterative solve makes no commitments. Every variable is adjusted at every step, nothing is held fixed, and the state after a step is a function of the whole state before it rather than of a chosen subset. There is nothing to order because nothing is decided one at a time.

That does not make the solve unaffected by its arrangement — the starting guess matters enormously, and so does the way the equations are scaled. But those are different levers from an ordering, and none of them can be improved by asking which variable to do next.

What an engineer takes from it #

The practical reading is about which question to ask of a mechanism, and it comes out the opposite way round from the mathematical one.

For a design question — will this pattern deploy, how far, through what path — the continuous solve is the right instrument and its weaknesses are manageable. A residual can be watched, a starting guess can be chosen near the configuration wanted, and a mechanism that has been driven through its whole range without the residual growing has been examined more thoroughly than any discrete search could examine it.

For a feasibility question — can this pattern fold at all, is there a configuration nobody has found — the discrete search is the right instrument, because it is the one that can say no. A solve that fails to converge has said nothing.

And for self-contact, neither is sufficient alone. The solve says where the panels are; the ordering search says whether that arrangement is physically possible; and the meshes that fold at no lettering are the standing example of a pattern whose equations are perfectly satisfiable by a configuration matter cannot occupy.

Which theorem was checked, and how #

The rigid folding of a mesh here is never asserted from a formula. The panels are placed by composing the motions round each face, and the check is that going round a face returns to where it started — a residual measured in sheet widths rather than a verdict, with the tolerance stated.

The independent check is the isometry: no edge changes length and no panel bends, verified at every point of the motion by a computation that shares no code with the solve. A solve that converged to something satisfying its own equations and failing that check would be reporting a fold that stretches the material.

And the flat-folding side is checked as it is everywhere: every lettering written back onto its pattern and put past the four vertex conditions and a folded sheet rebuilt from scratch.

What the picture cannot show #

A residual. Every figure of a folded mesh here shows panels in positions, and whether those positions satisfy the closure equations to a hundredth or to 10⁻¹⁴ is invisible — two pictures a reader cannot tell apart can be a rigid folding and a mechanism that nearly closes. The number is in the assertion rather than in the drawing, which is the honest arrangement and not a satisfying one.

Nor is the comparison here between two instruments on the same object. A flat-folding search and a rigid solve answer different questions, and a pattern that fails one may pass the other. What is being compared is the shape of the difficulty, and that comparison is legitimate precisely because the objects are the same crease patterns even where the questions are not.

The counting that survives the change #

One thing does carry across the divide, and it is worth ending on because it is the only quantity that means the same on both sides.

Degrees of freedom. A rigid folding of a mesh has some number of independent parameters — one for a single degree-four vertex, one for a whole Miura however large, more for a mesh with the right irregularity — and that number is a fact about the pattern rather than about any instrument pointed at it. It is computed from the equations on the continuous side; on the discrete side the corresponding quantity is how many creases the vertex conditions leave genuinely free after propagation, which is what decides the search’s cost.

They are not the same number and they are the same kind of number: a count of what the constraints have failed to determine. On both sides it is the quantity that decides how much work remains after the cheap reasoning has been done, and on both sides it is a property of the pattern.

That makes it the right thing to quote about a pattern when a reader wants to know how constrained it is, and it is available cheaply on either side. A pattern whose conditions leave one free parameter is nearly determined; one that leaves fifty is not; and everything about cost, on both the discrete and the continuous side, follows the same number.

What is genuinely lost #

It would be tidy to say the two approaches are complementary and leave it there, and one thing is genuinely given up by moving to the continuous side.

A discrete search can be exhaustive. It can visit every lettering a pattern admits and report that none works, and that report is a proof — nothing is left for a longer run to find. Nine crease patterns in this collection are known to have no consistent lettering at all, and they are known in exactly that sense.

A solve cannot do this. There is no finite set of angles to enumerate, so a solve that fails from a thousand starting guesses has established that a thousand guesses failed. Proving that a mesh has no rigid folding requires an argument about the equations rather than a search — a rank condition, a degeneracy, something algebraic — and such arguments exist but are made by hand rather than computed.

So the continuous side is missing the strongest kind of negative result the discrete side has, and there is no cheap substitute. That is the deeper reason both instruments are kept: not that they answer different questions, but that only one of them can conclusively answer no.

Where the ladder goes next #

The discrete side has one more reading left in it, and it is on the family where a search is most obviously the wrong instrument: the plant’s pattern is not a hard case, where a leaf’s corrugation costs one step per panel and the difficulty is entirely in the geometry the plant has to grow rather than in any question about letters.