A corrugation that tapers, and the direction it is allowed to taper in
leaf-corrugation 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
widths: [0.09, 0.15, 0.2, 0.22, 0.2, 0.15, 0.09], rows: 4, mm: 160
widths: [0.03, 0.1, 0.22, 0.3, 0.22, 0.1, 0.03], rows: 4, mm: 160
widths: [0.12, 0.15, 0.18, 0.2, 0.18, 0.15, 0.12], rows: 4, mm: 160
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 pattern carries 26 mountains and 19 valleys ×7
- 18 interior vertices, every one past developability, Kawasaki, Maekawa and big-little-big ×6
- developability holds at every interior vertex — its sectors close to 360° — 18 checked ×6
- Kawasaki holds at every interior vertex — the alternating sums of the sectors agree — 18 checked ×6
- Maekawa holds at every interior vertex — mountains and valleys differ by exactly two — 18 checked ×6
- the big-little-big lemma holds at every interior vertex — no strictly smallest sector is flanked by two creases of one assignment — 18 checked ×6
- the corrugation packs to 11.2% of its flat footprint at this zigzag angle ×6
- the columns taper from 0.22 to 0.09, a factor of 2.44 ×5
- the tapered corrugation is put past all four theorems before it is drawn ×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 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.
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.
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 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 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.
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.
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.
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.
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.
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