Origami & Folding
what one sheet can do
Concept

Leaf folding — where it appears

The packing of a leaf inside a bud by folding. Its patterns are shaped by the container, by the leaf's own growth and by the requirement that opening needs no muscle, which is a different set of constraints from any designer's.

Named by 11 essays across one field — each of them below, with the objects they name alongside it.

18 interior vertices26 mountains · 19 valleyscolumns taper 2.44 : 1packs to 11.2% of flatmountainvalleyraw edgethe taper is in the columns, because Kawasaki does not mention their widthtapering the rows instead puts the alternating sums at 186.4° and 173.6°

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.

biology · Leaf folding 051015202500.511.522.5number of foldsbundle radiusthe bud, radius 1.248121624tightest at 12 foldsbud radius 1.2 · leaf area 340 · layer thickness 0.12

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.

biology · Leaf folding 00.20.40.60.811.21.402468fold angle (radians)exposed spanθ = 0.06θ = 0.75θ = 1.449 panels · one parameter · the span rises at every step, so nothing has to reverse

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.

biology · Leaf folding 0204060801001200510152025foldssurface heldpeak at 64 foldsbox of side 1 · sheet thickness 0.01 · most surface at 64 folds · 128 folds fills the box with sheet alone

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.

biology · Convergence the bar is the deepest point of the pile, in layersthe average is what an area calculation would use, and it is about half of it2 rows814 panels · average 4.15 · 1.93 times it3 rows1221 panels · average 6.23 · 1.93 times it4 rows1628 panels · average 8.31 · 1.93 times it5 rows2035 panels · average 10.38 · 1.93 times itthe footprint does not grow as rows are added; the depth does, and the ratio does not

Twice as thick where it is thickest

A folded leaf's thickness is quoted as an area calculation: so much lamina, so much footprint, so many layers on average. The average is not what has to fit in the bud. Sampling the folded state of corrugated leaf patterns of two to five rows gives a deepest point of eight, twelve, sixteen and twenty layers against averages of 4.15, 6.23, 8.31 and 10.39 — a ratio of 1.926 that does not move at all.

biology · Leaf folding 18 interior vertices26 mountains · 19 valleyscolumns taper 2.44 : 1packs to 11.2% of flatmountainvalleyraw edgethe taper is in the columns, because Kawasaki does not mention their widthtapering the rows instead puts the alternating sums at 186.4° and 173.6°

A leaf ends its pattern

A hornbeam leaf's corrugation does not stop at the margin by being cut off: the pleats narrow until there is nothing left of them, and the margin is where the pattern reaches zero rather than where it was interrupted. The same is available to a drawn pattern and costs nothing — a corrugation tapered by a factor of eighty-two across its columns folds with a Kawasaki residual of 4×10⁻¹⁶, exactly as an untapered one does, because the column widths never enter the condition.

biology · Leaf folding 18 interior vertices26 mountains · 19 valleyscolumns taper 2.44 : 1packs to 11.2% of flatmountainvalleyraw edgethe taper is in the columns, because Kawasaki does not mention their widthtapering the rows instead puts the alternating sums at 186.4° and 173.6°

The taper decides nothing

A leaf's corrugation narrows toward its margin, and the taper is what the pattern is for. It has no effect whatever on how often the pattern's letters agree with themselves: four width profiles from perfectly even to strongly tapered give a hundred and seventy-four consistent letterings of two hundred, identically. What moves the number is the count of rows, and on that measure a leaf tracks a Miura rather than the corrugation it most resembles.

biology · Leaf folding the bar is how many repeating rules fold, and it is the same bar six timesthe same corrugation redrawn at six geometries, every one of them swept in fullas printed1648 refused, all by the count · 38 of them close a circle of foureven columns1648 refused, all by the count · 38 of them close a circle of foura one-sided ramp1648 refused, all by the count · 38 of them close a circle of foura violent taper1648 refused, all by the count · 38 of them close a circle of foursix taller rows1648 refused, all by the count · 38 of them close a circle of foura steeper zigzag1648 refused, all by the count · 38 of them close a circle of fourno vertex condition reads a column width, a row height or a row count, so none of them can move the table

The leaf's rules are the Miura's

A plicate leaf packs into a corrugation that is broad in the middle and narrow at both ends. Enumerate every repeating mountain-valley rule it admits and the table is the Miura fold's table, rule for rule, number for number — and it stays that table when the leaf is redrawn with even columns, a violent taper, taller rows or a steeper zigzag. The plant's geometry cannot reach its own letters.

biology · Leaf folding 18 interior vertices26 mountains · 19 valleyscolumns taper 2.44 : 1packs to 11.2% of flatmountainvalleyraw edgethe taper is in the columns, because Kawasaki does not mention their widthtapering the rows instead puts the alternating sums at 186.4° and 173.6°

The plant's pattern is not a hard case

A hornbeam leaf packs into its bud by corrugating, and the pattern it uses gives up a consistent lettering at nine, twelve, fifteen, eighteen, twenty and twenty-four steps on nine, twelve, fifteen, eighteen, twenty and twenty-four panels. Nothing about the plant's problem is combinatorially difficult, and saying so is worth as much as finding a case that is.

biology · Leaf folding the tapered leaf: nodes against panels01020one a panel0 panels24every vertex of this family keeps 8 labellings

Nothing grown was cut out of anything

A leaf's corrugation costs twelve steps on twelve panels, sixteen on sixteen, twenty on twenty, twenty-four on twenty-four — exactly one per panel at every geometry, which is the most any pattern here costs. A tessellation patch costs half that, and the reason is that somebody cut it out of something. A leaf's creases stop at the margin because the plant stopped there.

biology · Leaf folding 0204060801001200510152025foldssurface heldpeak at 64 foldsbox of side 1 · sheet thickness 0.01 · most surface at 64 folds · 128 folds fills the box with sheet alone

Nothing grown has a seam

A gut is a tube and a leaf is a disc, and neither was made by joining anything. The sheets this collection builds by identifying a rectangle's edges are the same objects a body grows, reached by an operation no organism performs — and the difference shows up in where the boundary is and in what has to close.

biology · Surface in a volume

Named alongside it

The objects these essays reach for when they reach for this one.

CorrugationAssignmentCrease patternUnit cellGrowthPacking ratioBoundaryConstraint propagationConvergenceLayer orderingSearch costTaper

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