The deepest point pays for the paper

Eighty layers and the sheet decides the rest put three bounds on one map. The crease floor fixes the thinnest paper that holds a fold at about thirty-eight microns; that paper’s stack runs out at eighty layers; and the largest sheet two arms can lift, about twelve hundred millimetres, falls away as the square of the finished size. The region under both ceilings has a corner at a hand’s width — 134 millimetres — below which the paper binds and above which the sheet does.

It then noticed the map was drawn for a design with one layer count, and a design has a distribution of them. The sheet is consumed by the mean: a sheet has a size as well showed the folded footprint times the mean layer count is the sheet’s area. The fold is stopped by the maximum: the paper had to arrive first bounds the thickest point of the stack, not its average. So the honest description of a design is a pair of numbers with a ratio between them, and the essay guessed where the ratio would fall — near one for a tessellation, large for a base with a deep spine and broad flat flaps — and said the ratio could be measured from what was already here.

It can, and the guess sorts the patterns wrongly.

The ratio, measured #

Every printed pattern is folded flat and its footprint is sampled on a grid. At each sample the number of panels over it is counted; the largest count is the deepest point, and the average over the footprint is the mean. The same census is what the paper is all still there checks by integrating the count back to the area of the sheet, so the mean is exact to the grid’s resolution.

The ratio of deepest to mean runs across the whole shelf. The preliminary base and the Yoshimura are at 1.00 — every point of their footprints carries the same number of layers, eight and sixty. The waterbomb tessellation is at 1.02, thirty-two at its deepest against 31.5 on average. The Miura is at 1.73, sixteen against 9.2; the tapered corrugation 1.92; the hexagon twist 2.15; the square twist 2.98, nine layers at its deepest against three on average. The fold-and-cut triangle is at 5.89: all seven of its panels lie over one small region, and the rest of its footprint is one layer thick.

Not bases against tessellations #

The guess was that tessellations sit near one and bases far from it. The shelf does not divide that way.

The preliminary base is a base, and its ratio is exactly one. It collapses its eight panels onto a single triangle, and every point of that triangle has all eight over it. There is no spine and no broad flap; the whole base is the spine. The two twists are tessellation units, and their ratios are among the largest on the shelf. The Miura, the tessellation most often folded, sits in the middle.

What does divide the shelf is the distinction which side arrives drew for a different reason: whether a pattern collapses toward a point or toward a shape. A pattern that folds every panel over every point of its footprint — the preliminary base, the Yoshimura, the waterbomb — has a uniform pile, and its ratio is one by definition. A pattern that keeps a footprint with distinct regions — the twists, whose folded state is a polygon with arms; the fold-and-cut triangle, which piles everything over one small triangle and leaves the rest single — has piles of different depths in different regions, and its ratio is how different they are. The Miura sits between them: its folded footprint is deep in some places and shallower in others, sixteen layers at most against nine on average.

That earlier essay found the same split in which colour a reader sees: a pattern collapsing toward a point shows one panel over nearly its whole face, and one keeping a shape shows a patchwork. The same property decides how evenly the paper is used. It is a property of the folded footprint’s structure and not of what the pattern is called.

How much of the footprint is at its deepest #

The ratio says how much deeper the deepest point is than the average. It does not say how much of the footprint is that deep, and for the paper’s limit that is the part that matters: the stack ceiling bounds the thickest place a folder has to press flat, and a thickest place the size of a pinhead is a different problem from one the size of the model.

The same layer map answers it. On the three patterns at ratio one, nearly the whole footprint is at the deepest depth — 99.8 per cent of the preliminary base’s, all of the Yoshimura’s, 98 per cent of the waterbomb’s. On the patterns above one and a half the deepest region is small: 24 per cent of the hexagon twist’s footprint, 17 per cent of the square twist’s, 12 per cent of the Miura’s, 2.9 per cent of the fold-and-cut triangle’s and 0.8 per cent of the tapered corrugation’s. The tapered corrugation reaches sixteen layers over less than a hundredth of its folded area, where its columns’ tapers meet, and averages eight.

That sharpens what the stack ceiling is a bound on. For the uniform patterns the deepest point is the object, and pressing it flat is pressing the whole model. For the corrugation it is a line where two tapers cross, and a folder working a thick paper could accept that line being slightly proud of the rest — a tolerance the bound, written as a single maximum, does not allow for. The ratio understates how favourable a high-ratio pattern is, because a small deep region is cheaper than its depth alone suggests.

A family of maps #

The ratio enters the substrate map in one place. The stack ceiling bounds the deepest point: at most eighty layers of the thinnest paper. The sheet ceiling bounds the mean: a model finished at $f$ millimetres needs a sheet of $f\sqrt{L_{\text{mean}}}$, so a sheet of $S$ allows $L_{\text{mean}} \le (S/f)^2$. Written in terms of the deepest point $L_{\max} = \rho\,L_{\text{mean}}$, the sheet allows

$$L_{\max} \le \rho\left(\frac{S}{f}\right)^2.$$

The stack’s ceiling does not move and the sheet’s rises by the ratio. A design twice as deep at its deepest point as on average can reach the same maximum from half the paper per unit of footprint, because the sheet pays only for the mean.

The corner is where the two ceilings meet: $80 = \rho(S/f)^2$, so

$$f_{\text{corner}} = S\sqrt{\frac{\rho}{80}} = 134\,\sqrt{\rho}\ \text{mm}$$

at a twelve-hundred-millimetre sheet. A design with ratio one reaches the paper’s ceiling only up to 134 millimetres; a design with ratio two, up to 190; ratio three, 232; ratio six, 329. The corner moves out as the square root of the ratio, and so does the size at which the paper stops being what limits a design.

Each pattern’s own corner #

Read off the shelf, the corners run from a hand’s width to twice and a half that.

The preliminary base and the Yoshimura have their corners at 134 millimetres and the waterbomb at 135. The Miura’s is at 177, the tapered corrugation’s at 186, the hexagon twist’s at 197, the square twist’s at 232. The fold-and-cut triangle’s is at 326 millimetres — a model two and a half times the preliminary base’s size can still reach the paper’s full eighty layers, because only a small part of it is that deep.

That is the first correction the ratio makes. The earlier essay stated the corner as a single number, a hand’s width, and called the region above it the place where the vat and not the paper is the limit. That is true only of designs whose pile is uniform. For a design with a deep region and a shallow rest, the paper stays the binding constraint well above a hand’s width, and better paper keeps buying layers up to two and a half times the size the single map allowed.

The preferred size, corrected #

The earlier essay suggested a design criterion nobody uses: a designer given a sheet and a paper has a preferred finished size at which nothing is left over, and it wrote that size as $S/\sqrt{L_{\max}}$. With the ratio in hand the formula can be checked, and it is right only for $\rho = 1$.

A design at its preferred size uses its sheet exactly — $f\sqrt{L_{\text{mean}}} = S$ — and its paper exactly — $L_{\max} = 80$. Eliminating $L_{\text{mean}} = L_{\max}/\rho$ gives

$$f^{*} = S\sqrt{\frac{\rho}{L_{\max}}},$$

which is the earlier formula times $\sqrt{\rho}$. For the square twist that factor is 1.73 and for the fold-and-cut triangle 2.43; a designer using the uncorrected formula would fold the triangle at two fifths of the size its sheet and paper allow, and leave most of the sheet’s area as a margin cut away.

And the criterion now has content the uncorrected one lacked. A design is described by its ratio before it is folded — the ratio is a property of the crease pattern’s folded footprint, computable from the pattern — so the preferred size is computable before anything is drawn at scale, and it differs between designs by as much as their ratios’ square roots.

That figure is the one the ratio corrects most directly. Its curve charges the sheet for the thickest point’s layer count at every point of the model, which is right only when the pile is uniform; for a design whose deepest point is three times its mean, the sheet actually needed is the one the curve gives for a third of the layers — a sheet smaller by the square root of three in each direction.

A worked example #

The printed patterns are shallow — nine layers at most on the square twist, sixty on the Yoshimura — and the map is about designs that reach toward eighty. So the corners above read each printed pattern as a shape of depth profile: a design whose deepest point is eighty layers and whose ratio of deepest to mean is the printed pattern’s.

Take a folder with the thinnest paper that holds a crease and a sheet of twelve hundred millimetres. A design shaped like the preliminary base — every point as deep as the deepest — reaches eighty layers only if it is finished no larger than 134 millimetres, because every square millimetre of the model then holds eighty layers of sheet, and twelve hundred millimetres of sheet folded eighty deep covers a square 134 on a side. A design shaped like the square twist — eighty layers at its deepest, over about a sixth of its footprint, and twenty-seven on average — can be finished at 232 millimetres: the same paper and the same sheet give a model 1.73 times as wide, because most of that model is not eighty layers deep and the sheet is not asked to supply eighty layers under it.

That factor is decided by how the crease pattern distributes its layers, and it is known before the sheet is cut. It is also the whole of the difference between the two corners, since everything else on the map — paper, stack, sheet — is the same for both.

What a uniform pile costs #

The measurement also prices the patterns at ratio one, which the single map could not.

A pattern whose pile is uniform uses its paper’s thinness everywhere at once: every point of the preliminary base is eight layers, so eight layers is both what the stack must hold and what the sheet must supply per unit of footprint. Nothing is spare in either direction. A uniform pile is the most sheet-hungry way to reach a given depth: to reach eighty layers anywhere, it reaches eighty everywhere, and the sheet must supply eighty times its footprint.

The fold-and-cut triangle is the opposite. Its seven layers are over one small region and its footprint elsewhere is one layer; it reaches its maximum while its sheet supplies barely more than its footprint. That is why a pattern built for one cut rather than for a shape is the extreme: the star that was cut before it was proved folds a sheet into a thin wedge exactly so that the scissors meet many layers at one line and nowhere else.

So the ratio is a trade a designer makes, not a fault. High ratio buys depth where it is needed without paying for it everywhere; low ratio buys a uniform object, which is what a base is for, since every flap of the finished model will be folded from all of it.

What the ratio cannot say #

It is a statistic of a flat folded state. Every pattern here is measured as its zero-thickness flat fold, and a real model is folded further — a base is shaped into a subject, and shaping moves layers around. The ratio of the base is not the ratio of the crane made from it; it is the ratio at the moment the base is complete, which is when a folder decides what paper and size to use.

It uses the deepest point, and the deepest point is a single place. The stack ceiling is a bound on the thickest point a folder must press flat, and a model with one very deep point and a slightly less deep region beside it has almost the same constraint as one with a large deep region. A finer description would use the area at each depth, and the whole distribution is what which ceiling is binding would need to say more.

And eight patterns are eight patterns. The shelf printed here was chosen for other reasons; the patterns nobody owns described how it was assembled. Complex designs with many flaps, which are the designs the eighty-layer ceiling is about, are not on it, and the claim that they sit at high ratio is a prediction from the point-and-shape account, not a measurement.

The numbers the map is drawn from #

The paper is at the crease floor, a fibre and a half thick, about thirty-eight microns; the stack a folder can press flat is three millimetres; so the deepest point may be eighty layers. The sheet is twelve hundred millimetres, the middle of the bracket a sheet is as large as two arms found for a hand-lifted mould. Both are the earlier essay’s numbers, unchanged, so that the ratio is the only new thing on the map.

Depths are counted on a grid over each folded footprint, the same grid whose integral returns the sheet’s area; the deepest point is the largest count on it.

How the ratios were checked #

Every ratio is required to be at least one, since no footprint’s deepest point can be shallower than its mean, and the shelf is required to have patterns at one and a pattern above five, so a census that found every pattern alike would stop the figure rather than draw a family of maps with nothing to distinguish them. The mean depths are the same ones the paper is all still there checks against the sheet’s area.

Still open: the whole distribution #

The ratio is two numbers of a distribution, and the map uses exactly those two. The natural next measurement is the area of each footprint at every depth, not only at the deepest, from which both the sheet’s use and the paper’s limit could be read at once. The share at the deepest already shows that the corrugation’s and the triangle’s deep regions are slivers; the full distribution would say how a bound that tolerated a small proud region — a hundredth of the footprint a layer or two over the ceiling — would move each pattern’s corner, and whether such a tolerance is what folders of thick paper already practise without naming it.

The other direction is design. If the ratio decides the preferred size, a designer choosing between two layouts of the same subject — two of the circle packings a tree allows, say — could prefer the one with the higher ratio for a large model and the lower for a small one, and the choice would be made before any paper is cut. Whether two real layouts differ enough in ratio for that to change a decision is a measurement on designs that are not among the patterns printed here.

Sideways from here, the point-and-shape distinction has now decided two unrelated things — which colour a folded object shows, and how evenly it uses its paper — and a third, how deep the other side lies, turned on the same piles. A property of the folded footprint that keeps reappearing is worth a name of its own, and a measure of it — how far a pattern’s folded state is from collapsing onto a point — would be a single number to set beside the ratio.

The habit worth carrying is about constraints that read different statistics. When two limits bound the same quantity, check that they bound the same statistic of it. The sheet bounds a mean and the stack bounds a maximum, and treating a design as one number made the map correct for exactly the designs whose mean and maximum coincide.