What each geometry packs to
wing-packing 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
which: [corrugation, miura, fan, roll]
which: [corrugation, miura, fan, roll], folds: 8
which: [miura, roll], folds: 16
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 best and worst differ by 3.3× — the pattern decides the packing, not the material ×7
- a hinge that will not close past 6° caps the ratio at 10 for a corrugation and 92 for a Miura ×4
- a quoted ratio of 30 means 1.9° for a corrugation and 10.5° for a Miura — a factor of 5.5 in the quantity the material actually limits ×4
- the Miura keeps its 15 interior vertices at every sampled angle, so one pattern is being priced throughout ×4
- 4 geometries, each packing to a fraction of its deployed area computed from its own parameters ×2
- every geometry's ratio runs away as the fold shuts — at a hundredth of a radian they read 50 and 2500 — so an unbounded quantity is being quoted as though it described a pattern ×1
- the ratio is plotted against the angle rather than quoted at a state, so the reader can see which of the two the number is about ×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.
A leaf packs by corrugating
A corrugation is the cheapest fold there is — parallel creases, no interior vertex to think about — and a leaf that uses one has to taper it, because a leaf is broad in the middle. Which direction the taper is allowed to run turns out not to be a matter of taste.
A wing that folds into nothing
A beetle stows a wing longer than its body under a case a fraction of that length, and the ratio is the whole engineering problem. What a fold achieves is computable from the pattern alone, and the four geometries available are not close to each other.
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 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 organism is not the model
Every figure in this field draws a fold this repository computed. Not one of them measures a leaf, a wing or a gut. That is the rule the field was built to, and it is worth stating as a table rather than as a preamble — because a field about living things is where a computed geometry is most likely to be read as an observation.
The same corrugation in four places
A leaf, a wing, a crushed cylinder and a solar array arrive at nearly the same fold, and none of them copied any of the others. Convergence stories are cheap; this one is checkable, because the constraint that forces it can be computed rather than admired.
Every generator · The folding nobody designed field · The patterns a reader can fold