Three kinds of pile

The pile, not the panel counts something every technique for thick folding leaves out. A technique is drawn and priced at one crease between two panels; a folded model has two layers nowhere except at its last fold, and the printed patterns reach eight, sixteen, thirty-two and sixty. The length a thick panel has to find at a crease is not the published allowance but that allowance multiplied by the pile, fifty-nine times over at the deepest.

It closes by naming the measurement that would make the count useful. Where the deep region is. A pile of sixty at one corner is a local problem, a pile of thirty everywhere is a global one, and the depth map that would tell them apart already exists — the count summarised it to a single number.

Unsummarised, the maps sort the shelf cleanly, and the sorting says more about what a thick design has to do than the depth alone did.

Two numbers from each map #

A pattern is folded flat in the zero-thickness model, and its folded footprint is sampled on a fine grid — up to twelve thousand points — with the number of panels lying over each point counted. That is the depth map the pile essay reduced to its maximum.

Two further numbers come off it. The first is the share of the footprint at the deepest count: how much of the folded model is as deep as its worst point. The second is the share at least half as deep as the deepest count: how much of the model is in the same order of difficulty as the worst point.

Those two shares separate three situations that the maximum cannot. When both are near one, the pile is the whole model. When both are small and close together, the deep region is a patch with shallow paper round it. When the first is small and the second large, the model is deepest on a sliver and nearly as deep over most of its area.

Uniform: the pile is the pattern #

Three of the eight printed patterns have piles that fill their footprints.

The preliminary base is eight layers over 99.7% of its footprint, and all of it is at least half as deep. The Yoshimura pattern is sixty layers over all of its footprint — every sampled point is sixty deep. The waterbomb tessellation is thirty-two layers over 96.6%, and nothing is shallower than half.

For these the depth really is the whole description. A thick-panel technique applied to them faces the deepest pile at every crease, because every crease is in it, and the allowance the pile essay priced — the published two-layer allowance multiplied by the pile — applies uniformly across the model.

That is a severe problem and a simple one. It needs one allowance, sized once, and the difficulty is that the one allowance is large. It is also the case the tube that gets built is closest to: a pleated tube folds its whole surface into one uniform stack, and the stack’s depth is the number the design lives or dies on.

Island: the deep region is a patch #

Three patterns pile deep in one region and shallowly everywhere else.

The square twist is nine layers over 17.4% of its footprint, and 17.5% is at least half as deep — so almost nothing is between shallow and deepest. The hexagon twist is seven layers over 24.5%, with the same 24.5% at least half as deep. The fold-and-cut triangle is seven layers over 2.9%, with 3.0% at least half as deep.

On these the deep region has a hard edge. Inside it the model is at its deepest; outside it the model is two or three layers thick, and there is almost no ground between.

For a thick design an island is a local problem in the plainest sense. The deep region can be treated differently from the rest — thinner panels there, a recess, a relief cut, a hole where paper was cheap — and the rest of the model can use the ordinary two-layer technique it was drawn with. The twists’ islands sit where a square that turns turns: the central polygon and the pleats that meet it overlap there and nowhere else.

Graded: deepest on a sliver, deep nearly everywhere #

Two patterns are neither.

The Miura fold piles sixteen layers deep over 11.9% of its footprint, but 75.5% is at least eight layers deep. The tapered corrugation piles sixteen deep over only 0.9%, and 87.9% is at least eight deep. On both, the deepest count is reached on a sliver of the model and a depth comparable to it covers most of it.

And the deepest region is not even one region. On the Miura it is three separate strips; on the tapered corrugation, four. A uniform pile is one piece of the footprint by definition, and each twist’s island is one piece; a graded pile reaches its maximum in several places with only slightly shallower paper between them.

That is the hard case. A graded pile cannot be solved with one allowance, because sizing for the deepest count over-provides across most of the model and sizing for the typical depth under-provides at the slivers. It cannot be solved locally either, because there is no region to treat separately — the deep places are several and the paper between them is nearly as deep. A graded pile asks for an allowance that varies across the sheet.

Four depths and nothing between #

The maps shade three classes of depth, and the counts underneath the shading are sharper than it is.

The Miura’s footprint takes exactly four depths: sixteen layers over 11.9 per cent of it, twelve over 32.0, eight over 31.6 and four over 24.5. No point of it is any other depth. The tapered corrugation takes the same four depths, sixteen over 0.9 per cent, twelve over 19.5, eight over 67.6 and four over 12.1. A graded pile on these two patterns is not a slope. It is a staircase in steps of four layers, and a technique for either has four cases to provide for, not sixteen.

The twists show what an island’s edge is in the same terms. The square twist is one, two or three layers deep over 33.7, 34.8 and 14.0 per cent of its footprint, and then nothing until eight layers, over a tenth of a per cent, and nine, over 17.4. The hexagon twist is one, two or three layers deep and then seven, with nothing between. The shoreline the square twist’s map shows is a jump from three layers to nine with no ground at four, five, six or seven.

The fold-and-cut triangle is the extreme island. Ninety-six and a half per cent of its footprint is a single layer, the paper outside the folded outline, and 2.9 per cent is seven layers deep, with a trace at three and five between them.

What sizing the allowance to the pile would save #

Suppose a thick-panel technique could size its allowance point by point to the pile at that point, rather than once to the deepest pile. What it provides is then proportional to the mean depth over the footprint rather than to the maximum, and the ratio of the two says what sizing locally is worth.

For the uniform piles it is worth nothing. The preliminary base averages 7.99 layers against its deepest eight, the Yoshimura sixty against sixty, the waterbomb tessellation 31.05 against thirty-two. Local sizing would provide between 97 and 100 per cent of what uniform sizing does, which is what the pile is the pattern means as a number.

For the islands it is worth the most. The square twist averages 3.02 layers against nine, so local sizing needs 34 per cent of the uniform allowance; the hexagon twist 3.26 against seven, 47 per cent; the fold-and-cut triangle 1.19 against seven, 17 per cent. And the saving is cheap to collect, because the deep region is one patch with a hard edge: treat the island and leave the rest alone.

For the graded piles it is worth about half. The Miura averages 9.25 layers against sixteen, 58 per cent; the tapered corrugation 8.37 against sixteen, 52 per cent. The saving is nearly as large as an island’s, and it is available only to a technique that follows the staircase across the whole footprint.

Put in the pile essay’s own terms, the Miura’s thick panels have to find fifteen times the two-layer allowance at its deepest pile and a little over eight times it on average over the footprint. The kinds rank twice: by how much local sizing is worth, and by how hard it is to collect. The Miura is where both rankings are unfavourable together — a saving worth having, collectable only by the hardest kind of technique.

The pattern that gets built is graded #

The consequence for the patterns engineering actually uses is the part worth dwelling on.

The worst rate on the shelf found that the Miura, the pattern that gets built, converts creasing into compaction worse than anything else on the shelf, and that what it buys with the difference is a single degree of freedom rather than smallness. The pile census adds a second property to the same pattern: its stack is graded, which is the one kind of pile that no single allowance fits and no local relief solves.

So the pattern chosen for deployable hardware is, of the eight printed here, one of the two that pose the hardest thickness problem — not the deepest, but the least uniform. Depth is the number the pile essay counted; uniformity is the number a thick design has to plan around, and on uniformity the Miura is near the bottom of the shelf.

That does not argue against the Miura. It argues that the depth figure published for a pattern is not the figure a thick design needs, and that for the pattern that gets built the missing figure is unfavourable.

The bookbinder’s graded pile #

The graded pile has a close relative in a trade that has been compensating for it for centuries.

A saddle-stitched booklet is a stack of folded sheets nested one inside the next and stapled through the fold. Each sheet wraps the ones inside it, so the outer sheets have further to go round the fold, and their pages stick out less far at the fore-edge than the inner ones: the stack creeps, by roughly the paper’s thickness for every sheet outside it. The creep formula for piles, $\pi(k - 1)t$ for $k$ layers, is that arithmetic.

Printers do not compensate creep with one allowance. They shift each page’s content by a different amount, progressively, according to how far that page’s sheet is from the centre of the stack — a practice known in print production as shingling. That is a graded allowance, varying continuously across the stack, and it exists because a booklet’s pile is graded: deepest at the centre fold, shallowing outward.

A Miura has the same problem spread over two dimensions and three separate deep strips, and the thick-panel techniques set out for one crease are the equivalent of a single allowance for every page. What the census suggests is that the Miura needs the equivalent of shingling.

What the maps cannot show #

The maps are the folded state of a sheet with no thickness, and a sheet with thickness does not fold to that state.

Thickness changes the pile it is being sized for. Panels with depth fold to a different arrangement from panels without, the layer count over each point is a property of that arrangement, and so the pile that prices a technique is itself a function of the technique. The pile essay named that iteration and did not solve it; the maps here are its first step, not its fixed point.

The maps do not say which layers are in a pile. A point sixteen layers deep has sixteen panels over it in some order, and whether they are sixteen different panels or a few panels wrapped several times changes what a technique has to do at the creases bounding them. Depth is a count and not a structure.

And the shares depend on the sampling. A sliver of 0.9% at a grid of 110 samples a side is a few dozen points, and a finer grid would move that number a little and the three-kind sorting not at all.

A depth that covers a tenth of a per cent is probably an edge rather than a region. The square twist’s eight layers over 0.1 per cent of its footprint, and the triangle’s three and five layers, are the kind of count a sample picks up where it falls on the boundary between two larger regions, and a finer grid would most likely thin them further. The depths themselves are exact — a layer count is a whole number at every point — so sampling can shrink or enlarge a region and cannot invent a depth between the steps.

The stack the census assumes #

Each pattern is folded flat in the model every printed pattern is verified in: panels of no thickness, layers exactly coincident. The count over a point is the number of panels whose folded image contains it.

The deepest pile is counted over the footprint, not over the sheet, so a point on the folded model is weighted by its area on the model rather than by how much paper went into it — which is the weighting a technique applied to the folded object meets.

The three kinds are defined by two thresholds. A pile is uniform when its deepest count covers at least nine tenths of the footprint, an island when the region at least half as deep is within two per cent of the deepest region, and graded otherwise. The eight patterns sit far from both thresholds, so the sorting does not depend on their exact values.

How the maps were read #

Every printed pattern is folded and sampled, and its deepest count, the share at that count and the share at half of it or more are read off the samples rather than estimated from the crease count.

The deepest region’s connected pieces are counted on the sampling grid, and the census requires every uniform pile to be one piece and every graded pile to be several, which is what the maps show.

All three kinds are required to be present on the shelf, so a sorting that put every pattern in one kind would not be drawn as a sorting.

Still open: an allowance that varies across the sheet #

The graded kind names what a thick design for a Miura would need, and one of the published thick-panel techniques already has the ingredient.

Tapered panels vary a panel’s thickness across its width, which lets the fold’s geometry be kept while the material thins toward the hinge. A taper chosen panel by panel, thicker where the pile is shallow and thinner where it is deep, is a graded allowance — shingling in two dimensions — and whether such a taper exists that keeps every vertex’s motion, as offset panels keep the hinge lines, is a question with a definite answer on a pattern as regular as the Miura.

The staircase makes that question smaller than it sounds. A Miura’s pile takes four depths, so a graded allowance for it need not vary continuously at all: it needs four thicknesses of panel, or four sizes of offset, placed according to a map with four colours. Whether those four regions can be assigned to whole panels — so that no panel spans two depths — is the first thing such a design would have to check.

The second continuation is the iteration. Fold the Miura with panels of real thickness, recount the pile, and see whether the graded structure survives or whether thickness itself evens the pile out — thickness has a sign, and a sign could plausibly make a graded pile more uneven rather than less.

The habit worth carrying is a question to put to any worst-case number. Ask how much of the object is at the worst case, and how much is near it. A maximum reached everywhere, a maximum reached in one place and a maximum approached almost everywhere are three different problems that report the same number.