Paper through paper

A rigid folding is checked at a point. The four panels around a vertex have to close up on the sphere; each panel has to keep every distance inside it; each crease has to stay a straight line about which two panels turn. Those are the conditions, and a pattern that meets all of them at every vertex, all the way along a motion, is what the subject means by rigidly foldable.

Every one of those sentences is about a neighbourhood. None of them mentions a second part of the sheet, and none of them could — the paper on the far side of the sheet is not in the neighbourhood.

So there is a thing a pattern can do that satisfies every condition and is nonetheless impossible in metal: it can put one panel where another one already is.

What a local condition can see #

The conditions are worth restating in the form that makes their blindness obvious, because in their usual form it is invisible.

The closure condition says that the four sectors around one vertex, drawn as arcs on a unit sphere, form a closed spherical polygon at every instant. It is an equation in the fold angles at that vertex and nothing else. The isometry condition says that the distance between two points of one panel is the distance it was in the flat pattern. It is a statement about pairs of points inside a single panel. The hinge condition says a crease stays a line, which is a statement about two panels that share it.

Not one of them has a second panel in it that is not adjacent. The whole apparatus is a set of equations indexed by vertices and panels, and a pattern that satisfies them has been certified locally and only locally. Rigid-foldability is a much stronger requirement than folding flat, and it is still not a statement about solidity. That is the same shape of limitation this site has already priced on the flat-folding side, where a checker made of local conditions is the right instrument and is not a decision procedure.

The same shape of gap turns up in flat folding, where it has been named for decades. Every vertex of a crease pattern can satisfy Kawasaki and Maekawa and the sheet still not fold, because the layers have to be ordered and the ordering is not local. That is the flat-folding version of exactly this argument, and the rigid version is less often said out loud.

A witness with nothing in it #

The sharpest possible witness is not a clever pattern. It is the least clever one available.

Crease a strip along lines parallel to its short edge. Every crease runs from one edge of the sheet to the other, so there is no interior vertex anywhere on the paper — the sheet is almost entirely boundary, and the closure condition has nothing to be evaluated at. The isometry condition holds by construction, since the panels are laid down at their own widths. The hinge condition holds because a crease that reaches both edges is a line and stays one.

Turn every crease by the same angle and the strip curls. Its cross-section is a walk that turns by that angle at each step and lays a panel down along the new direction, which is to say the cross-section is part of a regular polygon whose exterior angle is the crease angle. A regular polygon of n edges closes when its exterior angle reaches 360°/n. A strip of n panels has n edges in its cross-section, so it laps itself at 360°/n — one division, from the turning alone, with no test of any kind in it.

That is a prediction and it is worth having, because it is made before any geometry is looked at and it can be wrong.

Where the paper meets itself, and where it was said it would #

The measurement is separate and knows nothing about the prediction. Each panel of the folded strip is a quadrilateral in space; each is triangulated; each pair that does not share a crease is asked whether either one’s edges cross the other’s triangles. The crease angle is scanned upward in half-degree steps and the first angle at which any pair is found is recorded.

Six strips, six angles. Four panels first overlap at 90.5° a crease against a predicted 90°; six at 60° against 60°; eight at 45°; ten at 36°; twelve at 30°; sixteen at 22.5°. The largest disagreement anywhere is half a degree, and half a degree is the resolution of the scan rather than a discrepancy in the geometry.

The relation is worth reading rather than merely checking. A longer strip meets itself sooner. More paper is not more room; it is more turning per unit of crease angle, and the sheet with the most panels needs the least fold at each of them to close the circle. That is the opposite of the intuition a folder brings from paper, where a longer strip is a floppier one.

The total turning is one revolution, always #

The relation 360°/$n$ has a reading that removes the paradox in “a longer strip meets itself sooner”, and it comes from multiplying rather than dividing.

A strip of $n$ panels has $n$ creases in its cross-section, each turned by the threshold angle. Multiply:

$$n \times \frac{360°}{n} = 360°$$

Whatever the strip’s length, the total turning available before it laps itself is exactly one revolution.

Four panels at 90°, ten at 36°, sixteen at 22.5° — every row of the measurement is one full turn distributed differently. So more panels does not buy less room; it buys the same room in finer steps, and the per-crease figure falls because the total is fixed.

Which is what the cross-section is doing #

That also says what the threshold is, rather than merely where it is.

At one full revolution the cross-section is a closed regular polygon: the strip has rolled into a ring whose circumference is its own length. That is why the chord jumps from nothing to a whole panel width rather than growing — the panels do not begin to clip, they arrive on top of one another, because the walk has come back to where it started.

The dip at 40° on the ten-panel strip is the same statement one crease later, and reading it that way makes it unsurprising: nine creases at 40° is 360°, so the tenth panel lands on the first exactly, coplanar and touching. The essay’s careful defence of that zero is a defence of the geometry doing precisely what one revolution means.

And it removes the roll from the catalogue #

There is a consequence for the packing comparisons this collection makes elsewhere, and it is a sharp one.

A roll is credited with the best packing of any geometry here — one over the number of turns, improving without limit as the sheet is wound tighter. Winding requires the cross-section to spiral, which is to say to turn through more than one revolution.

A rigid-panel assembly cannot. At exactly one revolution the material arrives back on itself, and every further degree drives a panel through a panel. So a roll of rigid panels has a maximum of one turn, and its packing ratio is bounded at whatever one turn achieves rather than improving with more.

That explains an absence the packing essays note and do not account for: the roll packs best and nothing is built with it. Its advantage is entirely in the turns past the first, and those are exactly the turns a solid material may not take. Paper manages because paper is thin enough that a spiral’s layers clear each other; panels of any real depth do not, and the geometry above says the limit is not thickness-dependent at all — it is one revolution, at zero thickness.

Nothing, and then a whole panel #

The threshold is not a place where a quantity becomes large. It is a place where a quantity that was exactly zero becomes exactly one panel wide.

Below 36° a crease, no pair of panels on the ten-panel strip shares anything at all and the measured chord is zero. At 36° the first and last panels arrive in the same place and the chord is 2.00 panel widths, which is the strip’s full extent along the creases: they do not clip one another, they coincide.

There is one angle in the sweep where the chord returns to zero and it is the most interesting point on the axis. At 40° a crease the nine creases have turned the cross-section through exactly 360°, so the tenth panel lands on the first rather than through it — two coplanar panels touching along their whole area. The measurement reports zero, correctly, because contact is not penetration and a test that blurred the two would be a worse test. The dip is not noise in the figure; it is the figure being right about a case that looks like the failure and is not.

Nothing on the sheet changes at 36°. No sector angle moves, no panel deforms, no hinge stops being a hinge. Every local condition holds on both sides of that line with the same slack it had before, because there is nothing for them to hold at. What changes is which panels are in the same place, and that is not a local quantity.

Which claim was checked, and how #

A measurement that has never refused anything has established nothing, so the collision test is handed two cases with known answers before it is allowed to report on the strip.

The first is a strip that has not been folded. Nothing in the triangle test knows what flat means, and a flat strip must come back with no overlapping pair at all — an overlap there would mean the test was finding collisions in a sheet lying on a table, and every number after it would be noise.

The second is the opposite failure and the more dangerous one. A test that never fires is indistinguishable from a pattern that never collides, so two unit squares are driven squarely through one another’s middles by hand. The chord they share is exactly 2 by inspection, and the test is required to find one pair and to measure 2 to within a part in a thousand million. It does.

Then the prediction. The scan and the division share no code and no intermediate quantity: one triangulates polygons in space and the other divides 360 by an integer. They are required to agree on three strip lengths rather than one, because agreement at a single size is a coincidence and agreement across a family is a relation. A disagreement would have meant the collision was not arriving where the turning says, and the honest conclusion would have been that the cross-section is not the polygon it was claimed to be.

The refusals are what make the rest of the essay evidence rather than illustration, and each of them is a case the machinery could have failed and did not.

The pattern that does not collide #

If every rigidly folded pattern lapped itself the finding would be about rigid folding, and it is not. It is about the tests.

The Miura fold, driven through its motion and tested at every step with the same triangle test, never has a single overlapping pair. Not at one fold state, not at any of them, and not by a small margin: the worst chord anywhere in the motion is zero. The panels that touch are the ones sharing a crease, which is what sharing a crease means.

So the local conditions are blind rather than wrong. A pattern that passes them may be perfectly solid, and the ordinary case is that it is. What the conditions do not do is tell anyone which case they are in, and the difference between a certificate and a silence is the whole of the practical problem: an assembly that jams is discovered on a bench, at the point where somebody has already cut the panels.

What the measurement is not saying #

There is a much larger question standing next to this one and this essay deliberately does not enter it.

Given that a folded position is blocked, the natural next question is what a mechanism’s positions look like taken all together, and how a motion may or may not join two of them. That question has its own vocabulary and its own owner: it belongs in this fleet to machinekinematics.xyz, under its ground on serial chains and the open arm, and a reader who wants it should go there. Rigid origami is entitled to say that its patterns are linkages, because they are; it is not entitled to borrow that site’s argument, and nothing above counts anything or asks whether a blocked position has another way in.

What this essay has is narrower and can be stated in one line: a penetration, measured in panel widths, over the fold parameter. It is a length on an axis, it is zero or it is not, and where it stops being zero was predicted by a division.

Where the cross-section stops #

Three idealisations are doing work in every number above, and each is false in the direction that makes a real assembly worse rather than better.

The panels have area and no depth. Two panels a millimetre thick meet before two surfaces do, so every threshold quoted here is the latest possible angle rather than the first — and where the depth goes is a subject of its own, with its own arithmetic and its own costs. Restoring the thickness the pattern never had is the central problem of turning any of this into hardware, and it moves every collision earlier.

Contact is treated as the only thing that stops a fold. That is a geometric limit and the last one anything reaches: a real sheet is not an ideal one, and something will have bound, buckled or scraped long before two panels arrive in the same place.

And the picture is a cross-section. It shows the strip end-on, so it cannot show where along the creases two panels meet, and it cannot show a case in which panels overlap in one region of the sheet and clear one another in the next. The measurement is not a cross-section — every pair of quadrilaterals is tested in full — but the drawing is, and a reader who takes the drawing for the measurement will believe the finding is one-dimensional when it is not.

Who noticed it, and when #

Nobody discovered this in the sense of proving a theorem, and that is characteristic of the gap rather than of the field’s carelessness.

The closure conditions at a vertex are nineteenth- and twentieth-century spherical trigonometry, arriving in this subject through engineers with structures to deliver — Koryo Miura’s corrugation work in the 1970s and 1980s, and the deployable literature that grew around it. The conditions were written down as equations to be solved, and a solver that solves them is a solver for the positions of the panels rather than for their occupancy.

When rigid-folding simulators appeared in the 2000s, self-intersection was handled the way it is handled in graphics: as a separate pass, run or not run according to what the model was for. Tomohiro Tachi’s rigid-origami work from 2009 onward is explicit that the kinematic constraints and the collision test are different computations, and that the second is much the more expensive.

That separation is exactly right as engineering and exactly what makes the gap easy to forget. A pattern goes past the constraint solver, comes out with a motion, and is described as rigidly foldable — which it is. Whether anybody ran the other pass is not recorded in the phrase.

The strip is the part worth keeping, because it removes every other explanation. It is not a hard pattern, or a large one, or one that needed a search to find. It is the easy case in every other part of the subject — the one dimension where flat-foldability stops being intractable, where the layer orders can be enumerated, and where a folder’s intuition is reliable. That the sharpest example of a blind test is the pattern with nothing in it to be blind about is the finding, and it is a finding about the tests.

Where the ladder goes next #

The immediate direction is the vertex the strip does not have. Where four creases meet, the panels around them move together, and a collision between two of them is a collision between two things whose positions are not independent — which is a harder measurement and a more useful one.

The other direction is back toward the sheet, and toward what an assembly does about a condition it cannot check. A pattern that passes every local test still has to be built with room in it and still has to reach a folded position a motion can actually arrive at — two more places where the certificate a pattern carries is narrower than the sentence people say about it. What gets built is the small set that survives all of them, and it survives because somebody checked, not because the geometry promised.