The tapered corrugation
Fold it
The sheet is on the printed page, at the size it says.
Printing this page gives the page, and adds one more: the pattern alone, at 160 mm across, on a sheet of its own. Cut a square that size, transfer or trace the lines, and fold. The size is repeated in the corner of the sheet, because a printer set to fit to page rescales silently and there has to be some way to notice.
Mountain and valley are distinguished by dash as well as by colour, so the sheet survives the monochrome laser printer it will almost certainly come out of.
What it is
| Provenance | Generated here, from Kawasaki's condition |
|---|---|
| Creases | 26 mountain, 19 valley |
| Interior vertices | 18, every one checked |
| Panels | 28, read off the pattern |
| Folding to do | about 1243 mm of crease at this size |
| Printed sheet | 160 mm across |
What was checked
Four theorems at every interior vertex, and the faces two ways.
- Developability — the sectors around each of the 18 interior vertices sum to a full turn, so the sheet was flat before it was creased.
- Kawasaki — alternating sectors sum to a straight angle at each of them.
- Maekawa — mountains and valleys differ by exactly two.
- Big-little-big — no strictly smallest sector is flanked by two creases of the same letter.
- The faces — 28 of them, found by walking the planarised graph, and checked against Euler's formula and against the area they cover. A face walk that goes the wrong way round or merges two faces usually still satisfies Euler; it does not conserve area.
None of this decides whether the whole sheet folds flat, which is NP-hard in general. Every local condition holds. That is a different and weaker statement, and it is the one being made.
Take it away
The field's own interchange format, so the pattern is reusable outside this site.
tapered-corrugation.fold — 40 vertices, 67 edges, 28 faces, 5596 bytes. It opens in ORIPA, Rabbit Ear and the rest of the FOLD ecosystem.
The export is short because this repository never converts anything: FOLD's
vertices_coords, edges_vertices and edges_assignment
have been the in-memory representation of a pattern here since the site's first phase. What
the file adds is the metadata that makes it openable, and the faces where they can be read.
Coordinates are in sheet widths, and the file says what one unit measures on paper.
What is argued with it
Essays that call leaf-corrugation — read off the figure index rather than listed by hand.
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.
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 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.
Where the paper stops
Every flat-folding theorem is a statement about a full turn of paper, so a vertex at the edge of the sheet is subject to none of them. Cutting a patch out of a pattern removes conditions rather than preserving them, and a small enough patch has almost none left.
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.
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.
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.
Sixty-four rules, sixteen fold
The Miura fold's letters are usually given as a recipe: rows one way, columns changing at every row. Write down every rule of that shape — the letter on a crease depending only on which row and which column it is in — and there are sixty-four. Sixteen fold flat. They are exactly the ones whose columns change at every row, the row letters do not matter at all, and every one of the forty-eight refusals is the counting theorem's alone.
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.
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.