Nothing in a body folds on a line — the hinge bound generator
hinge-bound is one function. Everything below came out of it during this
build, at arguments 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.
At its defaults
view: "materials", rhos: [0.02, 0.05, 0.1, 0.2], sheet: 10
view: "materials", rhos: [0.01, 0.04, 0.16], sheet: 10
rho: 0.02, counts: [8, 16, 32, 64, 128, 256], sheet: 10, share: 0.5
What it checked while it drew
Collected by running this generator with a listener on the assertions, not written here. The count is how many separate times this build put that claim to the test.
- the hinge radius is 0.08 and the stack is priced at 6 fold counts ×6
- the best whole fold count sits within 0.40 of the closed form S ⁄ 2(π−2)ρ on all 4 materials ×5
- the hinges pass 50% of the sheet at 64 folds — the point past which a finer pattern buys nothing ×4
- 4 hinge radii, 4 different optimum fold counts — one pattern, and no shared best number of folds ×3
- the loss is exactly proportional to the fold count — 32× the folds is 32.0× the loss ×2
- at every material's own optimum exactly half the sheet has gone to hinges — the four disagree about the fold count and agree about that ×1
- ρ × k* is the same number on every material to 0.0e+0 — twice the hinge radius is exactly half the fold count ×1
Where it is called
Changing this generator changes every figure on this list, which is what makes the list worth publishing rather than keeping in a check script.
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.
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.
How much surface fits in a body
An organ whose whole job is to have area — a gut, a gill, a cortex — is solving a packing problem in reverse. Folding buys surface inside a fixed volume, and with a sheet of zero thickness it buys an unlimited amount. With a thickness the curve turns over and then falls to nothing.
Nothing in a body folds on a line
A crease in an organism is not a crease. It is a compliant region — a patch of thinner material that bends — and a region has a width. The width consumes surface in exact proportion to the number of folds, which puts a ceiling on how fine a pattern can usefully get.
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.
The number is the angle
Every packing ratio worked out so far is computed at a fold closed all the way, and a folded wing is not closed all the way. At zero thickness the ratio runs away as the fold shuts, so the size of a quoted number says how far the fold got and not what the pattern is — and the pattern contributes only an exponent, which makes the same quoted ratio mean two quite different angles depending on which geometry produced it.
The surface has to be supplied
The curve that turns over does so because the sheet's own thickness fills the box it is folding into. A surface in a body has to be reached as well as fitted, and the channel that reaches it takes depth out of the same box on exactly the same terms — so the best fold count and the surface it delivers both fall by the ratio of the sheet's thickness to the sheet and its supply together.
What a second deployment costs
Every folded structure this field builds deploys once. The reason is a power law: a hinge asked to survive more cycles cannot be as sharp, a blunter hinge takes more surface out of the sheet, and the fold count that packs best falls as the cycle count to a fatigue exponent. A structure required to work a thousand times packs thirty times worse than one required to work once, and the exponent decides how fast rather than whether.
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