The gap between two curves

Where the rulings run out established the reach of a single curved crease. The straight lines that rule the surface either side of it are not parallel; they converge; and the paper exists only as far as the first place two of them cross — at sin β / (β′ + κ), checked against an intersection of neighbouring rulings on four crease curves at five ruling angles to a worst relative gap of nine ten-thousandths.

That rung read the bound outward: a curved fold has an edge nobody drew, beyond which there is no surface. This one reads it inward, and the reading is more restrictive.

A design is never one crease #

A curved crease has no straight-crease theory behind it: the flat-folding conditions are about creases meeting at points and a curved crease has neither. Concentric circular pleating is the oldest curved-crease exercise there is and the one every account of the subject opens with: draw a family of circles, alternate mountain and valley, and the sheet buckles into a saddle nobody creased into it.

It is a family, and that changes what the bound applies to. The paper between two neighbouring circles belongs to both of them — it is ruled from the inner one and from the outer one — so it exists only if both can reach it.

The outer one can reach further, because reach grows with radius. So the binding constraint is always the inner crease of a pair, and the construction is limited from the inside rather than from the outside.

The arithmetic, which is one line #

For a circle of radius R the curvature is 1 / R, so the reach is R sin β — where β is the angle the rulings leave the crease at. Two circles a distance w apart need w of surface between them, so

R sin β ≥ w, that is R ≥ w / sin β.

Below that radius the construction has no surface at all. A concentric pleat pattern cannot be continued to the centre: there is a disc in the middle whose radius is set by the pleat spacing and the ruling angle and by nothing else.

The site solves for it rather than quoting it. The reach is measured off the envelope at a series of radii and the crossing with w is bracketed and bisected, so the number comes out of the same machinery that produced the reach bound rather than out of the formula that explains it. At w = 0.06 and β = 0.9 the solve gives 0.0766 and the formula gives 0.0766.

Why the inner crease is the binding one #

The direction of the constraint deserves a paragraph, because reading the bound the wrong way round is the natural mistake and it gives an answer that is comfortable and false.

Reach grows with radius: a circle of radius 0.4 reaches 0.4 sin β and a circle of radius 0.1 reaches a quarter of that. So in any pair of neighbouring circles the outer one has surface to spare and the inner one does not, and the gap between them is limited by the inner.

Walk inward and the reaches shrink while the gap stays the same. There is therefore a first pair whose inner member cannot cover the gap, and every pair inside it fails worse. The construction dies at a radius and not at an edge, and the dying is monotone: once it has failed it does not recover.

Read outward instead and the bound looks harmless. The outermost circle reaches further than any gap, the sheet’s edge arrives before the envelope does, and a designer concludes that the bound is about very large sheets. It is about very small circles.

What a bigger sheet buys #

A sheet of outer radius 0.3 at that spacing takes four circles and leaves a hole 25.5 per cent of its radius. A sheet of radius 0.6 takes nine circles and leaves a hole 12.8 per cent of its radius — the same hole, in absolute terms, now a smaller share of a larger sheet.

That is the sharpest statement of the result. The hole is not a proportion of the design; it is an absolute size, fixed by two numbers neither of which is the sheet’s. Making the paper bigger buys more circles and does not shrink the middle by a millimetre.

The hole costs a fixed number of circles #

There is a better unit for the loss than a length, and it comes from counting the circles a sheet takes rather than measuring the disc it cannot use.

A sheet of outer radius $R$ pleated at spacing $w$ carries circles from $R$ inward down to $w/\sin\beta$, so the count is

$$\frac{R}{w} - \frac{1}{\sin\beta}$$

At $R = 0.3$, $w = 0.06$ and $\beta = 0.9$ that is $5 - 1.28 = 3.7$, and the sheet takes four. At $R = 0.6$ it is $10 - 1.28 = 8.7$, and the sheet takes nine. Both are the essay’s own figures.

The first term grows with the sheet and the second does not. So the hole is not merely a fixed length; it is a fixed number of pleats, subtracted from however many the paper would otherwise hold.

Which is never fewer than one #

Written that way the limit has a floor, and the floor is the cleanest statement in the essay.

The cost is $1/\sin\beta$ circles, and $\sin\beta$ can never exceed one. So

the hole costs at least one circle, at every spacing, on every sheet, at every ruling angle.

At a steep ruling of 1.5 radians the cost is 1.003 circles — as close to the floor as the geometry allows. At 0.9 it is 1.28. At a shallow 0.4 it is 2.57, and at 0.3 it is 3.4.

So the two levers a designer has do not move a length, they move a count between about one and three, and the count is what a pleat pattern is actually made of.

And it says what a design is really losing #

That reframes the sheet-size result rather than restating it. A larger sheet gains circles at one per $w$ of radius and loses the same one-to-three at the middle — so the share lost falls, exactly as the essay says, but the thing lost is unchanged: one to three pleats, always the innermost, always the tightest.

For a designer that is a more actionable statement than a percentage. A pattern of five circles has lost a fifth to a quarter of its rings before anything is folded; a pattern of thirty has lost a tenth of that share and the identical rings. The innermost pleat a curved-crease design can actually carry is the second or third one drawn, and the first one or two exist only on paper.

Both levers, and they pull the same way #

A designer has two things to move. Spacing: halving w halves the limit, so a pleat twice as fine has a hole half the size. Ruling angle: the limit is w / sin β, so rulings nearer to perpendicular reach further and the hole shrinks toward w itself.

Neither escapes. As β approaches a right angle the limit approaches w and stops — a pleat can never be continued inside a radius equal to its own spacing, however it is folded. And the ruling angle is not a free parameter anyway: Huffman’s relation ties it to the crease’s curvature and the fold angle, so a designer choosing β has chosen how hard the fold is.

At the shallow end the limit grows without bound. At β = 0.4 with a spacing of a tenth of the sheet’s radius, the hole is 25.7 per cent of it — a quarter of the design gone.

When the crease is not a circle #

A circle’s reach is proportional to its own radius, so the limit is a clean ratio and the dead region is a disc. On any other curve the curvature varies along the crease, so the reach does too, and the pleat dies at whichever point runs out first.

Measured on a wave at a median spacing, thirty of sixty sampled positions cannot carry the pleat. That the reach varies along a non-circular crease is the earlier rung’s own measurement — an ellipse, a parabola and a wave all reach the same share of their own tightest radius — and what this rung adds is what the variation does to a family. The dead region is a set of arcs rather than a disc, so a curved-crease design on a non-circular family fails in patches, and the patches are where the crease is tightest.

That is worse news for a designer than the circular case, because a hole in the middle can be planned around — cut it out, or put something there — and a set of arcs cannot.

The numbers, laid out #

The band is small enough to give in full, and reading it is the fastest way to see how the two levers behave. Each entry is the inner limit as a fraction of a sheet of radius 0.45.

Two things are visible immediately. Each column is exactly proportional to the spacing — double w and every limit doubles — which is the formula’s linearity showing up in a solve that never used it. And each row falls toward the spacing itself and stops: the last column is only a little above w, and no amount of steepening the rulings takes it below.

The worst cell is the design a beginner draws: a coarse pleat, shallowly folded, on a small sheet. It loses more than half its radius.

Which theorem was checked, and how #

Four things, and the third is the one that would be easy to get away with.

The reach is re-measured rather than inherited. Nine radius-and-angle pairs, envelope against closed form, agreeing to a fiftieth of a per cent — because everything in this rung stands on that number and quoting it would be quoting a result rather than checking one.

The inner limit is solved off the same machinery and required to agree with the formula. Two routes to one number, one of them a bisection on a measured curve and the other a division.

The sheet-size independence is checked as an equality between two computations rather than argued from the algebra: the same spacing on two sheets must return limits equal to a part in a million, and the larger sheet must take more circles. Either half alone would be satisfied by a bug.

And the degenerate cases must come back unbounded. A straight crease has no curvature, so its rulings are parallel, nothing crosses, and the reach is infinite — which is why the straight half of this subject never met any of this. A spacing wider than any circle can reach at a given angle must return no limit inside the sheet rather than returning the bottom of its own search bracket.

Where the model stops #

The whole calculation is about where a developable surface exists, and that is a statement about an idealised sheet of zero thickness that does not stretch. A real sheet inside the limit does not vanish; it crumples, or it stretches a little, or it develops a small cone at the centre — all of which are things paper does and none of which this model contains.

So the honest reading of the hole is that it is where the smooth construction stops, and what a folder finds there is a mess rather than an absence. That is consistent with what a concentric-circle fold actually looks like: the middle is crushed, and the crushing is not a failure of technique.

The model also assumes the rulings are the ones the crease’s own geometry produces, which is exact for an isolated crease and is an approximation once several creases interact. A full treatment would solve the whole surface at once, and this one bounds it crease by crease.

What the picture cannot show #

The pattern figures draw a disc and shade it, and the shading is an absence — there is no surface there, so there is nothing to draw. A reader will read the shaded disc as a piece of the design, and it is the place where the design has stopped.

Nothing here shows the folded object. Every figure is of a flat pattern, and the saddle a concentric pleat folds into is exactly what a curved crease is for; it is left out because the argument is about the flat pattern’s own limits and drawing the folded state would suggest the limit was about the shape.

What a folder does with the middle #

The practice has three answers and each of them is a way of admitting the bound rather than beating it.

Cut it out. A concentric pleat with a hole punched in the middle is a common object, and the hole is usually described as a design choice. The arithmetic says it is a necessity with a computable radius, and that a hole smaller than w / sin β leaves paper that cannot exist.

Stop the creases short. Let the innermost circles be arcs rather than full circles, so that the region they would have enclosed is never bounded by a crease at all. That works and it changes the pattern into a different object — the arcs have ends, and a crease that stops in the middle is a vertex of odd degree with its own difficulties.

Let it crush. Fold it anyway and accept that the middle is a knot of paper. This is what most people do and it is why the effect is so familiar; what it costs is the smoothness the whole construction was for.

None of the three is a repair. The bound is about where a developable surface exists, and no technique makes a surface exist where the geometry says it does not.

The generalisation #

The useful statement is about local constraints that couple neighbours. The reach bound is a property of one crease: it says how far the surface beside that crease extends. Applied to a family, it becomes a constraint on the gaps, and a constraint on gaps in a nested family propagates inward and accumulates.

That shape recurs. A tolerance on a single joint becomes a constraint on a chain of joints; a bound on one panel’s size becomes a bound on how many panels fit; and in each case the interesting failure is at the end of the chain where the individual constraints are tightest, not where they were first derived.

The curved-crease case is unusually clean because the accumulation has a closed form: nothing has to be summed, because the binding constraint is always the innermost pair and the innermost pair’s condition is one inequality. That is why the answer is a radius rather than a count.

There is also a reading about what a curved crease is. A straight crease is a line and has no reach bound at all — which is why nothing in the straight half of this subject has ever needed one; a curved one carries a length scale — its radius of curvature — and everything about it is measured against that. A crease that curves forces the paper into a developable surface, and this rung is the price: the surface is bounded, and a family of them is bounded from the inside.

Who found it, and when #

The envelope of a curved crease’s rulings is classical differential geometry and the reach bound follows from it in a line. Curved-crease origami as a practice is much older than any treatment of it — Bauhaus students were folding concentric circles in the 1920s — and as a subject of study it is recent, with Huffman’s relation in the 1970s and a modern literature since about 2008.

The inner limit does not appear to be stated. That is a little surprising, because the phenomenon is unmissable: every concentric-circle fold has a ruined middle and every account of the technique mentions it, usually as a practical difficulty about handling small circles. What is new here is that it is not a difficulty of handling. It is where the surface stops existing, and the radius can be computed before the paper is cut.

Where the ladder goes next #

The immediate continuation is the non-circular case done properly. This rung measures where the reach falls below the spacing along a wave and counts the positions; characterising the dead set — its shape, whether it is connected, how it moves as the spacing changes — is the rung that would tell a designer where a general curved pleat fails.

The other direction is the interaction the model leaves out. Two creases close together do not each rule their own surface independently; the strip between them is one surface with two boundaries, and solving it as such would replace this rung’s bound with an exact answer. The bound would still be the right way to think about it, and the exact answer would say by how much it is conservative.