Curves and material

A crease that curves

Bend a crease and the paper either side is forced into a shape nobody creased. The flat-folding theorems say nothing about it, because they are statements about straight creases meeting at a point.

The sculptors got there first

Curved-crease folding produced its best objects decades before anybody could compute one. The surfaces were made by hand, the ruling lines that determine them were not calculated until much later, and the mathematics has been catching up ever since.

What a flat sheet can become

A sheet that cannot stretch cannot become a sphere. That much belongs to differential geometry; what belongs to folding is the three ways round it — seams, curved creases, and a few percent of stretch — and what each one costs.

Four things that are not true

Zero thickness, no stretch, creases that are lines, and perfect memory. Every theorem on this site rests on all four, every one of them is false, and the interesting engineering is exactly where each fails.

Paper that stretches on purpose

Wet-folding breaks the assumption every theorem of flat folding rests on, deliberately. It does not repeal the geometry — it buys a few percent of strain, and a few percent of strain is worth about twenty degrees of sphere.

The crease has a radius

A fold does not go through a line. It goes round a small arc, and the arc uses more paper than the stack advances by — a fraction of a millimetre per crease, and several millimetres across a grid, which is why an ambitious tessellation comes out short.

One curve and one number

Folding cannot change how curved a crease is within the surface — that is what an isometry means. Everything a curved fold produces is the curvature it adds out of the surface, and one number controls all of it.

How many times can it be halved

The folklore says seven, and the folklore is a statement about one sheet of paper. What actually binds is arithmetic: every halving doubles the layers and the paper spent at the closed end grows as the square of the layer count, so the length needed for twelve folds is nearly a kilometre.

Where curved creases meet

A curved-crease design looks like a smooth object and its constraints are not smooth. They live at the finitely many points where creases cross, and at each of those the conditions are about the creases' tangent directions — the curvature does not appear in them at all.

What one cut buys

A fold moves paper about and cannot change how much of it surrounds a point. A cut can, and that one difference is the whole of what this site's founding rule is worth. Take a wedge out and the sheet closes into a cone; let one in and it has more paper than the plane will accept.

Bought with holes

A Miura-folded sheet gets wider as it is pulled, and by how much depends on its panels and on how far it happens to be folded. A sheet cut into squares joined at their corners does the same thing and holds the value at exactly minus one, everywhere in its motion — the same property, bought with different geometry, and paid for in holes.

A curve has no panels

A rigid folding is a finite list of flat pieces joined along lines. A curved crease has no such list, and refining one does not help: the kink at each joint falls as one over the segment count, and the total of the kinks does not fall at all, because it is a constant of the curve.

The creases a sheet gives itself

A crease pattern drawn at random satisfies the flat-folding condition at essentially none of its vertices. A sheet crumpled at random satisfies it at every single one, on every seed, at every size — and the reason is a tautology that is very easy to miss.

Every facet is a layer

Fold a sheet at random as many times as patience allows, then count three things: the creases it carries, the facets they cut it into, and the layers in the stack. The last two are the same number, always, and it is one more than the first — so how deep a crumpled sheet folds can be read off the flattened pattern without folding anything.

One cut short of falling apart

Everything a cut sheet can do is bought out of the material between the end of one cut and the start of the next. That material shrinks to nothing in a straight line as the cuts grow, and the sheet stays in one piece the whole way down — until the instant it does not, and then it is in six.

A cut that removes no paper

Cuts in this subject are graded. Take a wedge out and the angle at a point falls by exactly the wedge; take twice as much and it falls twice as far. A hole is not like that. Its effect on what the sheet can do is the same whether it is a tenth of the paper or a ten-thousandth, and it is the same because it is not a quantity at all.

The crease that stops in the middle

A sheet folded flat at random writes a crease pattern that satisfies every condition in the subject, everywhere. Leave one layer behind on each fold — one layer out of a dozen — and it stops writing crease patterns at all: the creases stop in the middle of the paper, and a crease with a loose end is a thing no flat folded sheet can have.

Where the rulings run out

A curved fold's surface is made of straight lines leaving the crease, and the lines are not parallel, so they cross. Past the first crossing there is no surface: two points of the paper have been sent to one point of space. The boundary is a curve nobody drew, no crease pattern shows it, and it sits at the sine of the ruling angle times the crease's own tightest radius — on every curve tried.

The sheet remembers

Perfect memory is the fourth idealisation, and the least examined of the four. It is usually read as a complaint that paper will not lie flat again; the large half is the opposite. A sheet folded once is no longer blank, so folding a second model into it is folding the union of two patterns — and a union folds flat only where every new crease meets every old one at a right angle.

The fifth thing that is not true

Four idealisations underlie every theorem here and each has had an essay. There is a fifth and it has never been named, because it is invisible in exactly the way the others are not: paper has a grain, no theorem in the subject mentions a direction, and so nothing in the whole apparatus can tell a folder which way up to lay the pattern down.

The gap between two curves

The rulings leaving a curved crease are not parallel, so they cross, and the surface exists only as far as the first crossing. That bound is usually read as a limit on how far a design extends outward. It is not: the paper between two curved creases has to be reachable from both, so the bound bites hardest where the circles are smallest, and a concentric pleat has a hole in the middle that no sheet size removes.

A cut is a licence

What a cut buys is usually described in words — freedom, release, a shape a fold cannot reach. It can be counted, and the unit is vertices. Cutting one crease of a square twist turns two interior vertices into vertices no theorem applies to, and the share of letterings the pattern admits goes up by a factor of two for each vertex released: exactly, on every cut tried.

The decision a crumple has taken

A sheet crumpled at random satisfies every condition in the subject, because it just folded. It also wrote itself a lettering — one of very many the pattern admits — and it is now in a piece of that space it cannot leave: at six folds a crumpled sheet carries thirty-nine buried creases, which is half a million million million pieces, and every change it admits stays inside one of them.

A cut is not local

Cutting one crease of a square twist takes its letterings from sixteen mutually unreachable pieces to two. The cut crease is one of the four that were settled when the pattern was drawn — and it takes two others with it, because the vertices it releases were the far ends of those. Even a cut along a crease that was never settled quarters the count.

The crumple keeps its options

Every crease pattern this site prints has exactly one folded state and not one of its thirty-nine available rearrangements is legal. A sheet creased by folding it at random four times has an average of 2.16 folded states, one of them has nine, and six of three hundred and thirty-five rearrangements are legal. The sheet nobody designed is the one with room left in it.

How much line is on the paper

A crease pattern is described by its creases: how many, at what angles, in what arrangement. What a folder spends is length. The Yoshimura this site prints has eighty-six creases and 2,380 millimetres of folding on a sheet seventeen centimetres across; the Miura has thirty-eight creases and 1,049, and the fold-and-cut triangle has six and 258 — and the two counts do not rank the eight printed patterns the same way.

Where the length sits

A pattern's folding length is a total, and a total says nothing about where the work is. Divide each printed sheet into bands by distance from its own edge and the answer separates the patterns by kind: a traditional base carries five times its share of folding in the middle 4% of the paper, a tessellation carries between 0.9 and 1.4 everywhere, and a twist unit carries none at all at its centre. On all eight, the outermost band carries less than its share.

A cut that reaches the edge

A ring of paper with three creases running from its hole to its rim satisfies every condition the subject has — vacuously, because it has no interior vertex at all — and cannot be folded: its panels take no two colours and the two routes to one of them end up 1.75 sheet widths apart. One cut from the hole to the edge, crossing no crease and changing no letter, and it folds exactly. The cut removes an adjacency, which is the one thing neither a fold nor an edge can do.

The letters a crumple was given

A sheet creased by folding it and folding it again arrives with a mountain-valley labelling that cannot be wrong, because a folding produced it. Nothing about the pattern protects it: reletter the same creases and the share of labellings whose letters agree falls from every one of forty at eight panels to eleven of forty at forty-one. The foldability of a crumple is a fact about its history, not about its drawing.

One cut removes one arc

A crease pattern whose letters contradict themselves has, in principle, an obvious smallest repair: cut one crease and the statement it was making goes away. Cut every crease of four tessellation patches in turn — four hundred and seventy-four cuts — and sixteen of them leave a sheet whose panels still land anywhere at all. A cut gives the paper a freedom, and a sheet with a freedom in it has no folded state to order.

Rare is not hard

Crumple a sheet deeper and the share of its labellings that agree with themselves falls from thirty-four in forty to eleven. The number of steps a search needs to find one of them does not move at all: it stays at about one per panel, with no backtracking, the whole way down. How often an answer turns up at random and how much work it takes to find one are different quantities, and a crumpled sheet is where they come apart.

The shortest crease is not a crease

A crease pattern's density is usually quoted as total crease length over sheet area, which treats a metre of folding as a metre whether it arrives as one long line or ten thousand short ones. Reading the lengths individually instead finds twelve creases on a printed patch that are shorter than a wavelength of light.

A crumple has no tail

The least structured crease pattern this collection can produce is a sheet folded at random and flattened. Its consistent letterings get rarer as it deepens — thirty-four of forty down to eleven — and finding one costs one step per panel from beginning to end, with no wrong guess anywhere. Disorder and difficulty turn out to be unrelated quantities.

A count is not a length

Cut a rectangle out of a tessellation and it reports fifty per cent more creases than the pattern has, then twenty-five, then seventeen — converging on the truth from above and never reaching it. The crease length per unit area it reports is exact at every size, because the two halves of a divided crease add back up. One measurement survives the cut and the other does not.

A cut is surgery

Two cuts that look identical on the paper do completely different things to the sheet. A slit run inward from the rim changes nothing at all; a closed cut in the middle removes a disc and leaves a sheet carrying a condition it did not have before. What separates them is not the length of the cut or how much paper it removes.

Two holes are two conditions

One hole in a sheet of paper gives one loop that cannot be shrunk and one parity to satisfy. Two holes give two, and they are independent: an arrangement of creases can satisfy the condition round one hole and fail the condition round the other, and the sheet refuses on the strength of the one it failed.

A crease with no vertex to belong to

Crease density is measured as length of line per area of paper, and everything else about a crease is measured at the vertex it runs into. A band of paper has creases that run from one edge to the other and meet nothing, so it has density and no vertices at all — and it still refuses to fold.

The sixth thing that is not true

Five idealisations underlie every theorem here and each has had an essay: no thickness, no stretch, creases that are lines, perfect memory, no grain. There is a sixth, it is more basic than any of them, and it is the one nobody has ever thought to name — the paper is a disc.

The cut that changes nothing

A slit goes right through the material and leaves the sheet exactly the object it was. A closed cut removes almost no paper and produces a different sheet with a condition it did not have. Kirigami is made almost entirely of the first kind, which is why every result about it survives the distinction untouched.

A tuck keeps what a gore cuts

A flat disc gathered into a spherical cap has more circumference than the cap, and a gore removes the excess while wet-folding stretches it away. A tuck folds it under, which keeps the sheet whole and turns the excess into thickness. At the rim of a gathered cap the paper is α ⁄ sin α sheets thick on average — π⁄2 for a hemisphere — and a simple tuck is three, so single tucks reach a cap of 130.6° before they run into one another. And because a sphere's circles fall short of a plane's as the cube of the radius, a tuck that follows the sphere widens as the cube too: its edges are curves.

A straight tuck is a cone point

A tuck with straight edges hides length in proportion to how far past its start it has gone, which is a cone's law and not a sphere's. Started at the centre, straight tucks make a cone. Started at several radii, they hide length in a broken line that follows a sphere's cubic, and the worst shortfall falls as the square of the number of starting radii: 36.9 per cent of the rim's hiding from one start, 12.3 from two, 3.3 from four, 0.8 from eight. Every start is three creases at a point, which is a vertex that cannot fold flat — and it is exactly where the gathered sheet's curvature goes.

Crowd the tucks toward the rim

Straight tucks started at several radii follow a sphere's hidden length in a broken line, and evenly spaced starts leave a worst shortfall that falls as the square of their number. Evenly spaced is not the best spacing. A sphere's hiding bends hardest near the rim, so the best starts crowd outward — on a hemisphere, four of them at 0.36, 0.60 and 0.80 of the radius — and leave 38 per cent less error than four evenly spaced. As the count grows the saving closes on 42 per cent, a limit set by the square root of how the sphere's hiding bends; on a shallow dish it approaches five ninths. To follow a hemisphere within one per cent takes six rings of tucks instead of eight, and within a tenth of a per cent eighteen instead of twenty-four.

Crowding outward costs almost nothing

Placing the tuck starts for equal error crowds them toward the rim, where the gathered paper is already at its thickest, and the obvious worry is that the accuracy is bought with depth. Measured, it is not: on a hemisphere the crowded placement's deepest start sits in 3.37 sheets against the even placement's 3.32, because a start is three sheets of its own and the gathering beneath it is only one and a half.

Three answers, one count

Seams, curved creases and a few per cent of stretch are the three ways round the sphere, and a tuck is a fourth. All four dispose of one quantity — the excess circumference a flat disc has over the sphere's circle — and all four dispose of it by dividing the circle. So the number of divisions needed is the same whichever answer is chosen: nineteen for a hemisphere in a material that gives two per cent, eight at five, four at ten. What differs is what a division costs, and one of the four runs out.

Where a ring of divisions belongs

A pattern that divides the circle everywhere as finely as its rim requires is over-divided for most of its radius, because the excess grows from nothing. Putting a ring of new divisions in wherever the residual would otherwise pass what the material takes gives seven rings on a hemisphere at five per cent of stretch, at 0.35, 0.50, 0.62, 0.72, 0.81, 0.90 and 0.98 of the way out — and the first of those sits where a completely different criterion put its first tuck start.

A length needs a scale

These essays measure crease length, and a crease length is a length in the pattern's own coordinates. Six of the eight printed patterns are built on a unit square, so their coordinates are sheet widths and the distinction never arises. Two are not — a Miura laid out as six cells of unit width spans 6.37 — and on those two the shelf multiplied by the printed size without dividing by the width. The Miura's folding length was reported as 6,679 millimetres and is 1,049, and the same pattern's printable sheet has carried the right number all along.

The density a paper allows

Every density these essays measure is a quotient a pattern hands over, and nothing has asked what the paper's own answer is. It has one: a crease occupies a band a few thicknesses across, so two creases closer than that are not two creases, and a sheet of a given thickness carries a largest density. Copier paper allows 1,667 metres of crease a square metre and the densest pattern on the printed shelf asks for 89 — a factor of nineteen below the worst paper's ceiling. The material is not what limits a crease pattern's density at any fineness anybody folds.

A paper limits spacing, not density

The density ceiling put a sheet's limit at parallel creases a crease-width apart, 1⁄w of crease a square metre, and claimed no arrangement carries more. Crossing families do: they meet at vertices, which is shared ground, and never come closer than w anywhere else. Measured over every pair of creases that do not share a vertex, density times closest spacing settles at about 1.9 for both the Miura and the waterbomb, so each can be folded nearly twice as fine as the density bound said — 262 cells a side on copier paper for the Miura rather than 139, and 141 for the waterbomb rather than 74.

A vertex creases the paper twice

Two creases that meet share ground near the point, and their bands overlap out along each of them to w⁄sin θ for a sector angle θ below a right angle. Add the overlap at both ends of a crease, and a crease no longer than that is overlap from end to end — a crease only in the drawing. That third bound binds before the spacing does: the finest Miura on copier paper is 135 cells a side by its vertices against 262 by its spacing, and the finest waterbomb 82 against 141. Both land within a tenth of the density bound, which had the wrong argument and nearly the right number.

Only one side can run out

A curved fold has two surfaces and every reach ever computed here has been one of them. The closed form's denominator is the crease's curvature plus the rate the ruling angle turns at, and crossing to the other surface negates both — so at any point of any crease at most one of the two surfaces can have its rulings converge. A crease that never changes the way it bends therefore has a surface with no such boundary at all, anywhere along it.

An angle that turns faster than the crease

Which of a curved fold's two surfaces runs out is decided by a sum of two rates — how fast the crease bends and how fast the ruling angle turns — and every measurement so far has set the second to zero. Let it turn and it carries the sign on its own: past a rate of exactly one, a crease of unchanging curvature bounds both of its surfaces, which no constant angle on that crease can do. Below that rate the turning costs reach without changing anything else.