A panel is not the unit of depth

Three kinds of pile maps the depth of paper over each printed pattern’s folded footprint and sorts the shelf into three kinds. A uniform pile is as deep everywhere as it is anywhere. An island is deep in one patch and shallow around it. A graded pile is deepest on a sliver and nearly as deep almost everywhere else, and the Miura — the pattern that gets built — is graded, reaching its sixteen layers in three separate strips with most of the rest at twelve or eight.

It ends by naming what a graded pile asks for, which is an allowance that varies across the sheet, and by noticing that the Miura’s pile makes that request smaller than it sounds. The pile takes four depths and nothing between them, so a graded allowance for the Miura need not vary continuously. It needs four thicknesses of panel, placed by a map with four colours — provided the four regions can be assigned to whole panels, so that no panel spans two depths.

That proviso is checkable, and the check comes back no. Not one of the Miura’s panels sits over a single depth.

What a panel-by-panel allowance needs #

Every technique for folding a panel with real depth works on panels. Getting thickness round a corner sorts them by what each gives away, and the ones that keep the pattern’s motion — offset panels, tapered panels, panels thinned toward a hinge — all assign something to a panel: a thickness, an offset of its hinge lines, a slope across its width. The unit of the design is the panel because the panel is the unit that is cut, laminated and hinged.

A graded allowance built that way would give each panel the allowance its depth of pile demands. That is a well-defined assignment exactly when every panel’s folded image lies over one depth: then each panel has a depth, and the map with four colours is a colouring of the panels. If a panel’s folded image lies over two depths, the panel has no single depth to be given, and whatever allowance it receives is too much over part of it or too little.

So the question is a count. Fold the pattern flat, lay each panel’s folded image on the depth map, and count the depths beneath it.

The Miura’s panels each cross three or four depths #

The count is made on the same kind of map three kinds of pile drew: the folded footprint sampled on a fine grid, each sample carrying the number of panels over it. A sample within a hair of any panel’s edge is skipped, because a point on the edge of a step reads the pile on both sides of it at once and would make every panel look as though it crossed a depth.

On the Miura every panel crosses at least three. Sixteen of its twenty-four panels lie over three depths and eight over all four. The second figure shades one of the eight. A third of the panel lies over four layers, a third over eight, a quarter over twelve and the remaining tenth over sixteen, and the boundaries between those regions run straight across the middle of the panel.

The panel in the third figure is one of the sixteen that lie over three depths. It never reaches the shallowest, four layers: nineteen per cent of it lies over sixteen layers, forty-three over twelve and thirty-eight over eight. The Miura’s panels come in three kinds, eight of each, and even the kind that sits deepest on average has less than a third of its area over sixteen layers.

Put against the map of the whole footprint, the reason is visible. The deep strips of the Miura’s pile do not follow the panels’ outlines; they follow the outlines of the panels stacked above and below, which are offset from one another by the zigzag. The steps in the pile are where some other panel’s edge lies, so a step can land anywhere on a panel that the folding lays another panel’s edge across.

The tapered corrugation is worse #

The tapered corrugation is the other graded pile on the shelf, and on it the count is total: every one of its twenty-eight panels lies over all four depths.

It is also the more lopsided. The fifth figure shades a typical panel: under one per cent of it at four layers, eighty-seven per cent at eight, twelve at twelve and one at sixteen. A panel given the allowance for eight layers would be right over most of its area and short over an eighth, and a panel given the allowance for sixteen would be over-provided by a factor of two across nearly all of it.

That is the case where a panel-by-panel allowance does least harm by being wrong — most of each panel is at one depth — and also the case where it can never be right. Every panel has a deep sliver somewhere on it, and a technique that ignores the slivers ignores exactly the places the pile essay found hardest.

An island has one panel that fits #

The island patterns fall between. On the square twist ninety-four per cent of the paper is in panels crossing more than one depth, on the hexagon twist ninety-two and a half, on the fold-and-cut triangle ninety. But on each there is a panel that does not cross: on the square twist it is the central square, which sits entirely over the island’s nine layers.

That is the island’s advantage restated. Three kinds of pile said an island can be treated locally, and the local treatment it has in mind turns out to be almost exactly one panel: the central polygon a twist turns, which is where the pleats all overlap. The pleats themselves cross the island’s shoreline and cannot be given one depth each.

Uniform piles stack their panels exactly #

The uniform piles are the patterns on which the count is trivially satisfied, and the reason is the useful part.

On the Yoshimura every one of the sixty-five panels lies over sixty layers, all of it. On the waterbomb tessellation every one of fifty-two lies over thirty-two, and on the preliminary base every one of eight over eight. The seventh figure shades a Yoshimura panel and the shading is a single colour.

A uniform pile is uniform because the pattern folds its panels exactly onto one another: every panel’s folded image is the whole footprint, or coincides with every other panel’s. Then no edge of one panel lies across the middle of another, and there is nowhere for a step to fall. The patterns on which a panel is the unit of depth are the patterns whose panels stack without stagger.

That is also the arrangement the thick-panel techniques are drawn on. A cross-section of two panels meeting at a crease, or a stack of identical plates hinged along one edge, is a uniform pile in miniature, and a technique that works panel by panel there has no reason to meet the problem this count describes.

Steps parallel to the sides #

If the allowance cannot be assigned panel by panel, it has to vary within a panel, and what shape that variation takes decides whether a panel can still be built.

On the Miura, every edge of every folded panel runs in one of three directions: along the rows, and along the two directions of the zigzag. The steps in the pile are edges of other panels, so the boundaries between depths inside any panel also run in only those three directions — parallel to the panel’s own sides. A graded allowance for a Miura panel is therefore not a smooth taper but a stepped plate whose steps are parallel to its edges, with three or four levels, which is something that can be made by laminating offset plates of one thickness.

That has a precedent a folder may not expect. In brickwork laid in a running bond, each course is offset from the one below by half a brick, so that no vertical joint runs through two courses; every brick therefore has the joints of the bricks above and below it crossing its top and bottom faces. The Miura’s folded panels are staggered the same way, by the zigzag rather than by half a brick, and for the same geometric reason every panel carries the edges of its neighbours across its middle. In a wall the stagger is what makes the bond strong; in a folded Miura it is what makes a panel’s depth vary.

The tapered corrugation’s panels are not parallelograms of one shape, but the same count holds for them: their folded edges also run in three directions, and the steps inside each panel are parallel to one of its sides.

A panel sized for its own worst point #

There is a weaker way to build a graded allowance panel by panel, and it is the one a designer would reach for once a panel turns out to cross several depths: give each panel the allowance for the deepest pile anywhere under it. No panel is then under-provided. The question is whether it saves anything over giving every panel the allowance for the deepest pile on the sheet.

It saves nothing, on any pattern on the shelf. The eighth figure counts, for each printed pattern, how many of its panels reach the pattern’s deepest pile somewhere under them, and the answer is every one. All twenty-four of the Miura’s panels have some part of themselves over sixteen layers, all twenty-eight of the tapered corrugation’s, all nine of the square twist’s over nine. So a panel sized for its own worst point is sized for the sheet’s worst point, and an allowance chosen panel by panel costs exactly what a single allowance for the whole sheet costs.

Sizing to the pile at every point is what saves, and the figure prices it. On the Miura it needs 67.3 per cent of the single allowance; on the tapered corrugation 55.1; on the twists 64.5 and 68.8; on the fold-and-cut triangle 29.7, because nearly all of its paper is shallow and a deep sliver of it is on every panel. The uniform piles need the full hundred, because on them the point and the panel and the sheet are the same depth.

The saving is not spread evenly across the Miura’s panels, and the drawn panels are enough to say how uneven. The Miura’s twenty-four panels are congruent, so each has the same area and each carries the same weight in the total. The panel in the second figure — a third over four layers, a third over eight, a quarter over twelve, a tenth over sixteen — has a mean depth of about 8.4 layers, so sizing it point by point needs about fifty-two per cent of its own sixteen-layer allowance. The panel in the third figure, spread over eight, twelve and sixteen, has a mean of about 11.2 and needs seventy. The panels come in three kinds of eight, and the three together need 67.3 per cent, which leaves the third kind needing about eighty — the deepest of the three, with only a fifth of its allowance to give back. The same pattern offers one kind of panel half its allowance and another a fifth, which is one more thing a single figure for the whole sheet conceals.

Those numbers are larger than the ones three kinds of pile gave for the same saving — fifty-eight per cent for the Miura, fifty-two for the tapered corrugation — and the difference is the weighting, which is worth being clear about. That essay weighted each point of the folded footprint once. This count weights each point once for every panel over it, because each of those panels is paper that has to be built with its own allowance at that point. A deep point is paper sixteen times over, and counted as material it weighs sixteen times as much, so the deep strips count for more and the saving from sizing them locally comes out smaller. Weighted by the footprint, local sizing saves about two fifths of a Miura’s allowance; weighted by the paper that carries it, about a third. The paper weighting is the one a manufacturer pays.

Put together, the two counts say something stronger than that a panel crosses several depths. On every printed pattern with a step in its pile, every panel’s area reaches the pile’s deepest point, so there is no panel on which a smaller allowance is enough everywhere. The saving available from a graded pile is entirely inside panels. A design that keeps the panel as its unit keeps none of it.

What the maps assume #

Each pattern is folded flat with no thickness. The depth of pile is the number of panel images over a point in that state, and a sheet of real panels folds to a different state in which the stagger that produces the steps would itself shift — and thickness has a sign, so the shift need not be the same both ways. The count is a first step of the iteration the pile, not the panel named, not its answer.

A panel is taken as the folded image of one face of the crease pattern. A technique that splits a face into several plates, or joins two faces into one, has a different unit, and the count says only that the unit cannot be the face as drawn.

Samples at a step are discarded. A point within six thousandths of the footprint’s size of any panel edge is skipped when counting the depths under a panel, so that a panel is not charged with a depth it only touches along its boundary. With that margin removed, panels on the staggered patterns pick up slivers of their neighbours’ depths along their edges, which is a property of the sampling rather than of the pile. The margin is small enough that a sliver of a panel over a real depth — the tapered corrugation’s traces at four and sixteen layers — survives it.

What the count cannot show #

The count says a Miura panel crosses three or four depths and says nothing about whether a stepped panel keeps the pattern moving. Panels with somewhere to go keeps a thick Miura’s motion by placing each panel’s material on one side of its hinge lines, and a stepped plate has its material at several heights; whether offsets can still be chosen so that every vertex moves as it did is a separate geometric question, and the count does not touch it.

Nor does it say how much a stepped panel saves. Three kinds of pile priced sizing the allowance to the pile at every point at about fifty-eight per cent of sizing it to the deepest pile for the Miura. A stepped panel with four levels is exactly that local sizing, so the saving is available — but at the cost of making every panel a laminate, which is a manufacturing price in a different currency.

And it cannot say anything about which layers are in each step. A point twelve layers deep has twelve panel images stacked over it, and whether those include the panel’s own neighbours or panels from far across the sheet changes where a thick fold’s material has to go.

Still open: the stepped panel that keeps the motion #

The count turns the graded allowance into a precise object: on the Miura, a panel stepped in three or four levels, with steps parallel to its sides, placed so that each step’s height matches the depth of pile beneath it. Whether such panels can be offset so that every one of the Miura’s vertices keeps its motion is the question that decides whether the object can be built, and it is a question about whether the offset conditions still have a solution when a panel’s material is at several heights instead of one.

The other question runs in the opposite direction. Uniform piles are the ones on which panels are units of depth, and they are the patterns whose panels stack exactly — the Yoshimura, the waterbomb, the preliminary base. None of those three folds as a rigid mechanism driven by a single input, which is the property the Miura is built for, at the worst compaction rate on the shelf. Whether stagger is the price of a single freedom — whether a pattern can fold rigidly with one input and still stack its panels without offset — would say whether the Miura’s graded pile is an accident of one design or a property of every pattern that gets built for the same reason.

The habit worth carrying is a check on any assignment of a property to parts. Before giving each part a value, lay the parts on the map of the property and count how many values fall under each. A property that is constant on a map need not be constant on the pieces a design is made of, and the pieces are what gets built.