Generator

The opening has to run one way

A generator in the folding nobody designed library, called 44 times across 10 essays. Below: what it draws with nothing specified and with the settings the essays give it, what it checked while drawing, and everywhere it is used.

opening-monotone is one function. Everything below was drawn by it when the site was last published, at settings taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and when the generator changes, this page changes with it.

With nothing specified

`view: "memory", show: "reverse", held: 1.2`

`view: "memory", show: "signed"`

`folds: 12`

What it checked while it drew

Collected while it drew, not written by hand. The count is how many separate times that claim was put to the test.

the corrugation is sampled at 9 fold states between 0.06 and 1.44 radians ×5
every one of the 4 non-uniform rules reaches a strictly smaller ratio — the best of them gets to 9.00 against 12 ×4
the exposed span increases at every one of the 8 steps — the motion is monotone, so growth alone can drive it ×4
the mean layer count over each footprint reproduces that rule's ratio to 0.003 layers, sampled rather than assumed ×4
the span grows 16.5× from the packed state to the open one ×4
and clearance times holding force is 46.8992 at every one of them — what a finer fold saves in depth it pays in force, exactly ×3
at every span the force for 12 folds is exactly 2.00 times the force for 6 ×3
the force to hold the span is the slope of the energy the hinges store, to 2.6e-11 ×3
the uniform corrugation of 12 panels reaches a ratio of exactly 12, which is its layer count ×3
the span is S·sin θ for every fold count drawn — 4, 8, 16 panels sweep the same width at the same angle ×2
a corrugation creased to 0.4, 0.8, 1.6, 2.4 radians rests at 98%, 92%, 70%, 36% of its span with nothing holding it — every one of them part open, neither shut nor flat ×1
a hinge whose memory carries a sign gives one resting state and the flat sheet is no barrier; one whose reversed crease remembers the same size gives two, with the flat sheet between ×1
a ten per cent error in that holding time leaves 7.5% of the creased turn ×1
a wider hinge props the packed panels further apart, so the restraint it needs falls further below its own π limit as the radius grows — 98.0%, 95.2%, 90.9%, 83.7% ×1
and a sharper memory rests further in, monotonically — the remembered angle is the resting state and nothing else enters it ×1
and both resting places are below it, at the angle each branch was creased to ×1
and every partly open state lies inside a band of restraint exactly π⁄2 = 1.5708 wide, whatever the sheet, the fold count and the hinge ×1
and it sits at the stiffness-weighted mean of what they remember, 1.300 radians — disagreement averages rather than offering a choice ×1
and the barrier is exactly proportional to the fold count — 22.4 a fold at every one of 4, 8, 16 — because every hinge is turned through the same angle to flatten the sheet ×1
and the barrier to the mirror branch grows while the structure is held — 5.13 times its creased value after one relaxation time, 8.92 after 5 ×1
and the last of them is within 4.0 per cent of it, so the pull is going as one over the square root of the span that is left and the flat sheet is never reached ×1
and the release runs ahead of the span by at most 4.4 percentage points anywhere, though the force needed to stop it falls by half as much again from end to end ×1
and the shortfall is exactly the square of the turn each hinge still has at its own packed state ×1
and the whole family of held states lives between 65.3 per cent of the shut restraint and all of it — below that share the sheet is held nowhere and springs to flat. The floor sits just above 2 ⁄ π = 63.7%, because a hinge of radius 0.05 props the panels short of doubled-back ×1
creased to 0.8 and held at 1.2 the other way, the remembered turn reaches zero after 0.511 relaxation times, the logarithm of one plus their ratio ×1
each further decimal place of flatness costs more than the last by a factor that falls toward the square root of ten — 5.83, then 3.62, then 3.29 ×1
each weakening of the grip leaves the corrugation further out and never further in — the held state is one number, not a choice ×1
every corrugation that remembers a fold needs an outward pull near flat rather than a restraint — the sign of the force changes at the angle it was creased to ×1
every fold count drawn can hold the remembered angle of 1.6 radians, so the barrier is a comparison of counts rather than of what each can fold to ×1
folded at each material's own best count, ρ² times the holding force is the same number for every hinge radius — halve the radius and the spring is four times as stiff ×1
half the stored energy is gone by the time 45.7 per cent of the span is out ×1
held at 2.4 radians from a remembered 0.8, the torque the container supplies and the torque later needed to open the corrugation to 0.4 sum to the same total at every time — to 1e-16 — so holding moves the force rather than using it up ×1
hinges relaxing at 1 and 6 halve the container's load in 1.51, which one relaxation time of 2.17 would also do — but at 15 the mixed sheet still needs 3% of the total where that single time would leave 0.1% ×1
hinges resting at 0.4, 0.8, 1.6, 2.4 radians give the corrugation exactly one turning point across its whole range, not one per rest angle ×1
the band falls short of π⁄2 at every fold count drawn — 1.551, 1.532, 1.496 — because a hinge of radius 0.05 props the panels apart before they meet ×1
the barrier between a creased corrugation's two resting places is exactly the energy of the flat sheet, 179 — the flat sheet IS the transition between them ×1
the census returns one member at the stated ratio — every other spacing spends paper on a wider footprint instead ×1
the energy a packed corrugation holds rises with the fold count but not quite in proportion — per fold it falls 7.6 per cent from 4 folds to 16, because a finer corrugation is propped further from shut ×1
the restraint needed falls at every one of 2000 steps from shut to flat — a grip that is enough at one state is more than enough at every state after it ×1
the span is S·sin θ for every fold count drawn — 4, 8, 16, 32 panels sweep the same width at the same angle ×1
the span is S·sin θ for every fold count drawn — 6, 12 panels sweep the same width at the same angle ×1
while the sheet that remembers nothing needs holding in all the way to flat, which is the difference a rest angle makes ×1

Where it is called

Every figure listed here is drawn by the same rule, so a mistake in the rule would be in all of them at once, and so would a correction.

A crease held the other way

Hold a creased sheet on the far side of flat and its memory crosses zero after τ ln(1 + φ₀⁄φₕ), a time that contains no stiffness and no fold count. Asking what happens at that moment shows that two accounts of a corrugation's energy cannot both stand: a hinge whose memory can cross zero gives one parabola and no barrier at all. The barrier belongs to a degree-four vertex, whose two branches turn out to be at right angles in the space of its fold angles. Held on the other branch its wall dips and never falls, and its two resting states trade depth at nearly the same logarithm, with lengths in place of angles. Only the exact mirror takes every rest angle through zero together, which is the fastest way to erase a crease and the least forgiving.

Four finders, one option

Four unrelated lineages arriving at the same corrugation is read as evidence that the corrugation is good. It is at least as much evidence that there was nothing else to arrive at: how far a folded sheet shrinks is exactly its average layer count, so a lineage choosing a packing ratio is choosing a number of layers and nothing else — and the quantity that is genuinely free turns out to be almost uncorrelated with it.

Four materials, four optima

The convergence argument gets its pattern and stops there. A hinge has a radius, the radius takes a fixed length of surface out of every fold, and the fold count that gets the most packing out of a sheet is inversely proportional to it — so a leaf, a wing, a gut lining and a metal array agreeing on a corrugation still disagree by an order of magnitude about how many creases to put in one.

Holding a fold moves the force

A creased hinge held at an angle slowly comes to remember that angle, so a leaf or a wing packed in a bud for a season is gradually holding itself and the bud has less to do. The force is not used up in the process. The torque the container must supply falls as e^(−T⁄τ), the torque later needed to open the structure rises by exactly the same amount, and the two sum to the same total at every moment of the holding. Held for three relaxation times, a corrugation creased to 0.8 radians and packed to 2.4 needs 4 per cent of the total from its container and 96 per cent from whatever opens it — and the barrier to its mirror image has grown more than eightfold. A packing that lasts buys independence from its container with a harder unfolding.

How far open is a question about the grip

A corrugation of hinges that rest flat is loaded when it is shut, so it opens by itself and the force in the held-state calculation is a restraint rather than a drive. Followed from shut to flat that restraint only ever falls, and by exactly π over two — so every partly open state a structure can occupy is squeezed into a band a third wide, and a grip that weakens by a third leaves the sheet nine tenths open.

Opening with nothing to pull

A leaf is not opened by a hand, a hinge or a motor. It opens because it keeps growing — which puts a condition on the pattern that no folder ever has to satisfy, because a person can always push.

Springs that disagree do not offer a choice

A corrugation of hinges that remember different angles was expected to have more than one position in which nothing pushes. It has exactly one, at the stiffness-weighted mean of what they remember, because a sum of parabolas in one variable is a parabola. The second resting place comes from somewhere else entirely — the mirror pattern — and the flat sheet is the barrier between them, which is also why a creased sheet cannot be pulled flat at all.

The bud chooses the pattern

Nothing in the geometry of a corrugation says how many folds it should have. The container does: too few folds is a strip too wide to fit, too many is a stack too thick to fit, and the window that fits at all is narrow and has a best point in the middle of it.

The census returns one

The rung below this one asked for a census: every pattern reaching a stated packing ratio while opening from a single input, with its crease density. The census is makeable for the corrugations and it comes back with one member. A corrugation piles its panels over a footprint as wide as its longest panel, so its ratio is the total length divided by that longest one — and that equals the panel count only when every panel is the same.

The fold count sets the spring

A corrugation sweeps the same span at every fold count, and the count decides only how much room the zigzag needs while it does it — which was counted as a gain with nothing pushing back. Something does push back. Every hinge is a spring, a finer corrugation has proportionally more of them, and the force to hold a given span rises exactly as the clearance falls: the product of the two is the same number whatever the count, and at each material's own best count the spring goes as one over the square of the hinge radius.

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