Closing is not building

A strip creased along parallel lines has no interior vertex at all, so every local condition in the subject holds with nothing to evaluate — and rolled far enough it laps itself. That is the sharpest case for the theorems and the weakest case for a sceptic, because a strip creased along parallel lines is barely a pattern. The obvious reply is that a real pattern, solved properly, does not behave like that.

Here is the other end of the same argument, on a mesh with no two vertices alike, whose every closure condition has been solved rather than satisfied by symmetry. The answer is that one of them cannot be built anywhere in its motion, and that nothing which was solved says so.

Two conditions that are usually run together #

A rigid folding of a quadrilateral mesh is a set of fold angles, one per crease, such that walking round any loop of the pattern and turning the paper at each crease it crosses brings it back to where it started. Panels instead of paper is the whole of the model, and it is a good one: hardware does not bend, so a mechanism made of plates and hinges is what a deployable actually is. That is the closure, it is one equation per interior face, and solving all of them at once is what turns a mesh that nearly folds into a mesh that folds exactly.

Every quantity in that sentence is an angle. Nothing in it mentions where a panel is.

Solidity is the other condition, and it is a statement about positions: two pieces of the sheet may not occupy the same region of space. It is not a weakened form of the closure, not implied by it, and — this is the part worth stating plainly — not checked anywhere in this collection, in any published account of the tree method, or in any of the standard conditions the field states for rigid origami.

The two are usually run together because on the patterns everybody draws they agree. A Miura is solid throughout its motion; so is a fan; so is every corrugation with a repeating cell. Agreement over a family is not implication — and a Miura’s every property is the property of one vertex repeated, which is exactly what a solved mesh does not have.

Following a motion rather than sampling one #

The measurement has a trap in it that had to be avoided before anything could be measured, and it is the same trap a residual plot on this site once fell into.

A vertex of a quadrilateral mesh has two configurations at a given fold angle, which is the same doubling that gives a Miura two folded states at one angle. Solving each angle independently takes whichever configuration fits best at that angle, so a sequence of solved states is not a motion — it hops between branches, and any quantity plotted along it jumps for a reason that has nothing to do with the sheet.

So the states here are followed. The driving angle is stepped in small increments, the whole previous state is carried forward, and at each step the configuration nearest the previous one is chosen — nearest over every crease at once, rather than vertex by vertex, because a vertex picking its own nearest branch can assemble into something that is not near the previous state anywhere else. If any crease moves further in one step than a stated bound allows, the sweep refuses rather than reporting a collision between two states that are not on the same motion.

On the meshes here the largest single step is 0.12 radians against a bound of 0.35, so the branch is followed cleanly and the refusal never fires. That it can fire is the point: a jump would otherwise be drawn as a collision.

The mesh that is never solid #

Of six meshes solved from independent starts, five have no overlap at any of the twenty angles sampled between 0.08 and 2.7 radians. The sixth has an overlap at every one of them.

The number is a chord: the length of the segment two panels share where they interpenetrate, in the sheet’s own units, where a panel is about one. On this mesh it starts at 0.47 and grows to 1.18 — so at its worst two panels are sharing more than a panel’s width of the same space, which is not a numerical artefact of anything.

And the pair is not adjacent. Panels that share a crease meet along it by definition and are excluded from the test; the pair here is two steps apart in the grid, on opposite sides of a panel that touches neither of the interesting ones. No condition evaluated at a vertex — and every condition in this subject is evaluated at a vertex — has both of those panels in its neighbourhood.

Why the gearing makes it happen so early #

The overlap is already 0.47 at a driving angle of 0.08 radians, which is a nearly flat sheet, and that reads at first like a mistake.

It is not. The driving angle is one crease’s, and a quadrilateral mesh is geared: setting one fold angle sets every other, by factors the sector angles fix. On a Miura those factors are all alike, so a small driving angle means a nearly flat sheet everywhere. On a mesh with no two vertices alike they are not, and a crease four steps away from the driven one can be folded most of the way while the driven one has barely moved.

So “nearly unfolded” is a statement about one crease and not about the sheet. The mesh that collides is one whose gearing carries a small drive into a large fold somewhere else, and the panels that arrive in the same place arrive there almost immediately.

Which theorem was checked, and how #

Four things are checked and none of them is checked by the code that produced the mesh.

The closure, by a residual: every loop of every solved mesh comes back to itself within 4×10⁻¹² radians, and a mesh that fails to solve is reported as unsolved rather than swept.

The placement, by the gap: the four copies of every vertex, produced by four different walks over the sheet, have to arrive at the same point. Nothing in the walk makes them meet, so a placement whose gap exceeds a ten-millionth is refused and there is no folded object to test.

The overlap, by triangulating both panels and running every edge of each against the other — so what is reported is a segment two panels genuinely share, and panels that merely touch along a hinge give a chord of zero to floating-point accuracy.

The control. Five of the six meshes are solid at every angle, and a Miura is solid throughout its own motion. Both halves are needed: if nothing collided the detector would be untested, and if everything collided the detector would be measuring itself.

Where the model stops #

One in six is a count over six. Six meshes were solved and one collides; nothing here is an estimate of how often it happens, and the honest statement is a count with its denominator attached rather than a rate.

Twenty angles is twenty angles. The sweep samples the motion rather than proving anything about the whole of it, so “solid at every angle sampled” is what the five clean meshes have earned, and a collision that opens and closes between two samples would be missed. What is not at risk is the positive finding: an overlap at a sampled angle is an overlap.

Nothing here counts a freedom. How many pieces a mesh’s space of configurations has, and whether two positions can be joined by any motion at all, belong to the study of linkages and are not asked. What is measured is a length, along one continuously followed branch, and the measurement stops there.

A collision is not a proof that the pattern is useless. A sheet of paper resolves an overlap by bending slightly, which is what paper does and panels do not; the claim is about a mechanism built from rigid plates, which is what the closure conditions are a model of in the first place. A curved crease makes the same point from the other side: some folded objects have no list of flat pieces at all.

What the picture cannot show #

A drawing of a folded mesh is a projection, and two panels that cross look exactly like two panels that pass near one another. The figure showing the offending pair marks them because the measurement found them, not because the picture reveals them — from most viewpoints the collision is invisible, and from a few it looks like an artefact of the drawing.

That is worth being uncomfortable about. Every other figure on this site can be checked by a reader with a protractor or a sheet of paper; this one cannot, because the folded object it is about cannot be made. What can be checked is the crease pattern, which folds perfectly well in paper — paper bends, and the panels that would have collided simply curve past one another.

The idealisation, named #

A panel is rigid and has no thickness. Both matter here in opposite directions. Rigidity is what makes the collision a collision: a bending panel resolves an overlap by bending, and the whole subject of rigid origami exists because hardware does not bend. Zero thickness makes the measurement optimistic — real panels have depth, so two panels that share a chord of zero in this model can still foul one another, and every one of the five clean meshes is clean only in a model where the plates are surfaces.

So the finding is a lower bound on how often solidity fails, arrived at with the most generous possible assumption about the panels.

The generalisation #

The statement worth carrying is about what a solved system has established.

Solving every equation in a model establishes exactly the properties the equations are about. The closure conditions are about angles; they were solved to fourteen decimal places; and the mesh that satisfies them is not thereby an object, because being an object is a statement about positions that no equation in the system mentions. A solver reports a residual, and a residual near zero is easily read as a verdict on the design rather than on the system that was written down.

That is the same shape as several other findings here — a caption refuted by its own figure, a checker blind to the vertices at the edge of the paper — and the shape is always the same: the machinery is right, the model is right, and the question the reader thought was answered was not in the model.

The practical version, for anybody building one: a rigid-origami solver needs a collision test, and it is not expensive. Sixteen panels, twenty angles, every pair triangulated: the whole sweep above costs less than drawing the figure it produces.

What one in six is worth as a rate #

The essay declines to turn one in six into a frequency and is right to, and the size of the interval it would give is worth stating because it is larger than a reader will assume.

One collision in six independent meshes, treated as the plainest possible sampling question, puts the underlying rate somewhere between about a quarter of a per cent and two thirds, at the usual level of confidence. That is not a narrow range; it spans the difference between a curiosity nobody need worry about and a failure mode that afflicts most designs.

So the honest reading of the count is not about a sixth of solved meshes cannot be built. It is that the rate is not known to be small, which is a much weaker statement and the only one six meshes support — and it is enough to justify running the test, because a test whose cost is a fraction of a second does not need a rate to be worth running.

Sixty meshes would narrow the interval by roughly a factor of three, and the machinery to produce them already exists: the solver runs from independent starts, the sweep is automatic, and the whole of what is missing is patience. That is the cheapest available improvement to this rung and it needs no new idea at all.

Report a clearance rather than a verdict #

The procedure the essay recommends ends with a yes-or-no — do two panels share a chord — and the idealisation section says why that is optimistic. The repair is to change what the test returns.

Panels have thickness. Two panels whose chord is zero in a model of surfaces will still foul one another if they pass within the depth of the plate, and every one of the five clean meshes is clean only in a model where the plates have none. So the useful output is not whether the chord exceeds nought but how far apart the nearest non-adjacent pair actually gets — a minimum clearance, in the sheet’s own units, reported at every angle of the sweep.

That costs nothing extra. The test already computes distances between triangulated panels to decide whether they cross; keeping the smallest of them instead of thresholding it is one line, and it turns a boolean into a margin.

And a margin is the quantity the rest of this anchor is denominated in. A panel with depth needs its hinge somewhere and the room it needs is a length; a manufacturing tolerance is a length; a clearance is a length. A design whose worst clearance is a hundredth of a panel width has told a builder something precise about what thickness of plate it will take, and a design that merely does not collide has told them nothing at all.

What a builder should do with this #

The finding is small enough to state as a procedure, and stating it that way is more useful than another paragraph about epistemics.

Solve the closure, then place the panels, then test the pairs. The placement is the step that is usually skipped, because a solver that reports a residual near zero looks finished; but placing the panels is what turns a list of angles into an object, and it comes with its own check — the four copies of every vertex, produced by four different walks, have to arrive at the same point. A placement that fails that check is a solved system whose solution is not a surface, and it is worth catching before anything is cut.

Test the pairs that share nothing. Panels that share a crease meet along it and are not interesting. Everything else is fair game, including panels far apart in the pattern, and those are where the collisions are: on the mesh here the pair is two steps apart, and a test restricted to neighbours would have reported the mesh clean.

Sweep, do not sample. A single folded state says nothing about the motion, and a motion that is sampled by re-solving at each angle is not a motion at all. Follow the branch, refuse when it jumps, and report what the followed states did.

The cost of all three, on a four-by-four mesh at twenty angles, is a fraction of a second — less than the cost of drawing the picture the reader is looking at.

Where the ladder goes next #

The obvious continuation is the one this rung deliberately does not take: a solver that avoids collisions rather than reporting them, which means adding a condition on positions to a system of conditions on angles, and which is a different kind of problem.

Nearer at hand is what the collision means for a tolerance. If a mesh can be solid or not solid, then the surface of solutions that the joint solve leaves behind has a region on it that can be built and a region that cannot, and the boundary between them is a constraint nobody has written down. Which direction a manufacturing error points in turns out to matter for the closure as well, and that is the next rung.