The taper decides nothing

A leaf packs by corrugating: folded inside a bud, it takes a zigzag whose amplitude falls toward the margin, and the taper is the whole design. A corrugation of constant width would not fit the bud’s shape, would not open evenly, and would leave the margin creased where a leaf’s margin is smooth. A leaf ends its pattern rather than cutting one out of a larger one, and the ending is the taper going to zero.

So the taper is what the pattern is for, and it seemed a reasonable guess that it would be what a pattern’s letters depend on too — creases of different lengths round the same circuit, pulling in different directions.

It does nothing at all.

Four profiles, one number #

The measurement is a sweep over width profiles. The pattern’s columns can be given any positive widths, so: the printed taper, a perfectly even set, a strong taper narrowing from a fifth of the sheet to a twentieth, and a one-sided ramp with no symmetry in it. Four rows in every case.

Every one of them gives a hundred and seventy-four consistent letterings of two hundred. Not close — the same number.

That is the second dial in this collection to come out flat on this question. The twist angle changes nothing either — thirteen consistent letterings of a hundred and twenty at every turn from a shallow one to a full radian — and for the same reason. The question is combinatorial, the dial is geometric, and the two do not meet.

The tangle is the same size too. On the letterings that do contradict themselves, the set of panels lying on some chain is a third of the sheet, at every profile. So it is not that the taper trades a smaller failure rate for a worse failure; nothing moves.

What the invariance settles elsewhere #

The flatness of this dial closes a question another essay left open with the wrong guess, and closing it is worth more than the invariance itself.

The corrugation that agrees with itself ranks the printed patterns by how often a drawn lettering is self-consistent, and the tapered corrugation comes last at eighty-seven per cent — below the Yoshimura’s ninety-five, on fewer chains: eighteen against twenty-two. That essay offers a guess: the taper is what costs it.

The sweep here rules that out. A hundred and seventy-four of two hundred at the printed taper, at even widths, at a taper of four to one, and at a one-sided ramp with no symmetry at all. Whatever costs the corrugation its place in the ranking, the widths are not it.

Which leaves a rate to explain #

Putting the two measurements together turns the puzzle into a number.

Model each of a pattern’s chains as closing with probability $\varepsilon$, independently, so the consistent share is $(1-\varepsilon)^{V}$. The tapered corrugation is 87 per cent at eighteen chains, giving $\varepsilon = 0.77$ per cent. The Yoshimura is 95 per cent at twenty-two, giving $\varepsilon = 0.22$ per cent.

So the corrugation’s chains close three and a half times as readily as the Yoshimura’s, and that ratio — not the chain count, and now certainly not the taper — is the whole of the discrepancy.

What is left to explain is a per-chain rate on a pattern whose widths do not matter. That points at the only things the sweep did not vary: the number of rows, the zigzag angle, and the arrangement of the vertices. The first two are as adjustable as the widths were and have not been swept; the third is what the pattern is.

So the next measurement is obvious and cheap: hold the widths fixed and sweep the row count and the zigzag instead. If either moves the number, the cause is geometric after all and the invariance found here is about widths specifically. If neither does, the rate is a property of the vertex arrangement, and the corrugation’s place in the ranking is settled by which creases meet where — which is the same answer the twist patches gave, arrived at by elimination rather than by argument.

Which is not obvious and could have gone otherwise #

It would be easy to shrug at that as though it were inevitable, and it is not.

The four conditions at a vertex are not invariant to the widths. The big-little-big lemma is a statement about the smallest sector, and changing a column’s width changes the sectors and therefore which crease pairs the lemma constrains. Kawasaki is an angle condition and depends on the shape throughout. So the set of admissible letterings could perfectly well have changed size, and the share of it that is self-consistent could have moved with it.

It does not, and the reason is that neither the panel graph nor the arrows depend on the widths. The panels are the same regions in the same arrangement; the crease joining any two of them is the same crease; and the arrow it contributes depends on the letter and on the orientation, both of which are combinatorial. The picture the layer test reads is untouched by anything the taper does.

What the widths do change #

Saying the taper does nothing to this question invites the reading that the taper does nothing, which would be absurd, so it is worth listing what it does.

It changes the shape of the folded bundle, which is the whole point: a corrugation of constant width folds to a block and a tapered one folds to a wedge, and a bud is a wedge. It changes how much paper the pattern asks for, since a narrow column takes less than a wide one. It changes the layer count at each place across the leaf, which is what decides how thick the packed bundle is at its thickest.

And it changes the vertex conditions’ work. The sectors at each interior vertex are set by the widths and the zigzag angle together, so a change of profile changes which crease pairs the big-little-big lemma constrains and how close each vertex is to failing Kawasaki. A profile with a column much narrower than its neighbours produces a vertex with a strictly smallest sector where an even profile has none.

So the widths are doing a great deal and none of it reaches the arrows. That is a cleaner separation than one usually gets between a pattern’s geometry and its combinatorics, and it is why the invariance is worth reporting rather than assuming.

What does move it #

The row count, and it moves it a long way.

Two rows: fourteen panels, six chains, two hundred of two hundred consistent. Three rows: twenty-one panels, twelve chains, a hundred and eighty-one. Four: twenty-eight panels, eighteen chains, a hundred and seventy-four. Five: thirty-five panels, twenty-four chains, a hundred and fifty-eight. Six: forty-two panels, thirty chains, a hundred and twenty-six.

The independent closed chains of panels are what a lettering has to keep from closing, and the count is the pattern’s interior vertices — six per row added. So the leaf falls along the same axis as everything else here, and it falls steeply: a bud packing six rows of corrugation has a pattern whose admissible letterings work about five times in eight.

It is worth noticing what the row count is, physically. A row of the corrugation is one pleat of the zigzag — one out-and-back of the paper — so the number of rows is how many times the leaf is folded across its length, and a bud packing a longer leaf into the same space needs more of them.

Twice as thick where it is thickest measures what that costs in layers, and the cost there is linear in the rows. The cost here is not linear: the consistent share goes 100, 90, 87, 79, 63 per cent as the rows go two to six, which is a curve bending downward. Adding a pleat is cheap in thickness and increasingly expensive in the letters.

A leaf is a grid, not a strip #

The comparison worth making is with the two corrugations already measured, because it puts the leaf in the right family.

The Yoshimura at twenty-two chains is a hundred and ninety of two hundred consistent; at thirty-three chains it is a hundred and eighty-six. It barely falls.

The Miura at twenty chains is a hundred and sixty-two; at thirty-five, a hundred and twenty-eight.

The leaf at twenty-four chains is a hundred and fifty-eight; at thirty, a hundred and twenty-six. It is the Miura’s curve, not the Yoshimura’s.

The reason is structural. A Yoshimura’s chains are arranged in rows that overlap one another sparingly, so a combination of chains reaching across several rows has to pass through the few panels they share. A Miura’s are a grid, interlocking in both directions, so combinations are available everywhere.

A leaf’s corrugation is a grid: horizontal creases across every row and vertical creases down every column, with the zigzag shift between rows. Its chains interlock in two directions and it behaves accordingly.

The measurement that was expected instead #

Two guesses preceded this and both were wrong in the same direction, which is worth recording because the direction is a bias rather than an accident.

The first was that a strong taper would be worse: creases of very different lengths meeting at a vertex, an amplitude going to nothing at the margin, and generally more for the letters to reconcile. The second was that the printed profile — a symmetric taper narrowing from the middle to both margins — would sit between an even profile and a strong one.

Both guesses treat the pattern’s difficulty as a matter of how extreme its geometry is, and the layer question does not read geometry at all. A generation of intuition about crease patterns comes from the vertex conditions, which do read geometry and which fail exactly where the angles get extreme, and that intuition transfers badly.

The other corrugations in the same family #

Three more patterns here are corrugations of one kind or another and it is worth placing them, since the row-against-grid distinction does real work.

The waterbomb tessellation is a grid of cells with both diagonals, and at twenty-five chains it is a hundred and eighty-seven of two hundred — above both the Miura and the leaf at comparable counts. Its chains are arranged in a grid of units rather than of vertices, and there is no account here of why that helps.

The tapered corrugation on the printed shelf is this same leaf pattern, and its eighty-seven per cent is the same measurement quoted under another name.

And the Yoshimura is the outlier the whole comparison rests on: a hundred and eighty-six of two hundred at thirty-three chains, against the leaf’s hundred and twenty-six at thirty. Two corrugations, comparable size, sixty points apart.

What that means for a bud #

None of this is a difficulty a plant has, and saying why is the point of measuring it.

A leaf’s letters are not chosen. The corrugation forms as the leaf grows inside the bud, and which way each crease goes is set by which surface is on the outside of the fold at the moment it forms — a mechanical fact about the growing tissue, not a labelling anybody selects. So the plant is in the position of a sheet that folded itself: its lettering came from the folding, so it cannot contradict itself, and the share of admissible letterings that would have worked is irrelevant to it.

The bud chooses the pattern is the account of what does vary: the container’s shape decides which corrugation forms, and it decides it through geometry rather than through combinatorics.

Where the number does bite is on anybody copying the pattern: a folder or an engineer taking a leaf’s corrugation, redrawing it and lettering it themselves. Six rows and three of eight labellings that pass every published condition do not fold.

There is a second reason the plant is unaffected and it is worth stating because it is not the same as the first. A leaf does not fold flat. It folds into a bundle inside a bud, with the corrugation partly closed and the layers separated by their own thickness — so the flat folded state this whole measurement is about is an idealisation the leaf never occupies.

The organism is not the model is the standing caution and it applies with full force here. What the measurement describes is the crease pattern a leaf’s corrugation is, treated as a flat-foldable pattern, which is a legitimate object and is not the leaf.

The row rule, and where it comes from #

One detail of the construction decides everything the taper does not, and it is worth saying what it is.

The horizontal creases alternate by row — mountain on odd rows, valley on even — and the vertical creases alternate the other way. That is a rule about parity, applied to the whole pattern, and it is what makes the zigzag a zigzag: a row folded the same way as the row below it is not a corrugation, it is a bend.

So the pattern’s letters come from a rule with one bit in it, and the rule is fixed by what a corrugation has to be. That is the same shape as a Miura’s periodic lettering and it has the same consequence: the letters the construction assigns are consistent at every size, and the shares measured here are about labellings nobody would produce.

The two dials that do nothing, together #

Two of this collection’s constructions now have a geometric dial that leaves the combinatorial answer unmoved, and stated together they are more than a coincidence.

The twist angle changes every crease direction, every sector, every panel shape, and the twist patch’s consistent share is thirteen of a hundred and twenty at every value of it.

The leaf’s widths change every column, every sector and every panel, and the consistent share is a hundred and seventy-four of two hundred at every profile.

In both cases the panel graph is fixed by the construction’s combinatorial skeleton — which polygons are adjacent to which — and the dial moves only where the paper is, not what is next to what. So the invariance is not a surprise once it is stated, and it was not predicted before it was measured, which is the ordinary order of events here.

Why a plant would not have solved this anyway #

There is a temptation, when a folding property turns out to be fragile, to ask how the organism avoids it, and here the honest answer is that the question does not arise.

Natural selection acts on what happens. A leaf that unfolds is a leaf whose corrugation formed and opened; a corrugation that formed differently and did not open is not a variant that was tried and rejected, because the letters are not a variable the plant has. They are a consequence of which surface of the growing tissue is outermost at each fold, and that follows from the geometry of the bud.

Four finders, one option records the same shape of answer about a different property: four independent lineages arriving at the same corrugation, not because it was selected from alternatives but because it is what the constraints leave. The letters here are a stronger version — not selected from few alternatives, but not selected at all.

So the eight-in-eight-eighths this measurement reports is a fact about crease patterns rather than about leaves, and its audience is the person copying the pattern rather than the plant.

What would move it #

One thing would, and it is available: changing the rows’ parity rule — which letter the horizontal creases in each row take, and which the verticals.

The construction fixes it: horizontals alternate by row and verticals alternate by row the other way. That is one rule of the several the vertex conditions admit, and it is the one that produces the corrugation a leaf actually makes. Changing it changes the arrows directly, which is exactly what the widths cannot do.

Nothing here sweeps those rules the way the waterbomb’s five hundred and twelve were swept, and it is the obvious next measurement on this pattern. What can be said now is where it would have to look: not at the taper, which is the pattern’s whole biological content and is invisible to the question, but at the letters, which the plant never chooses and a copier always does.