The rim lies over less

The boundary of a sheet is normally discussed as an absence. None of the four conditions this subject checks is defined at a vertex on the edge of the paper, because the sectors there come in a line rather than in a ring; the outline of the folded object is mostly crease rather than raw edge; a crease that reaches an edge is a letter a hand can still change. Every one of those is a statement about the pattern.

There is a reading of the boundary in the folded object as well, and it is a number rather than a description. Sort a folded sheet’s panels by whether one of their own sides is a raw edge of the paper. Count, for each panel, how many of the others it shares ground with. The two groups do not come out the same.

The measurement #

A folded state places every panel somewhere in the plane. Two panels share ground when their interiors overlap — not when they touch along an edge, which is what two panels joined at a crease do and is not a stacking relation at all. The count of panels one panel shares ground with is its degree, and it is the quantity every ordering rule about that panel is written against: a panel that lies over nothing is a panel nothing can say anything about.

Whether a panel carries a raw edge is read off the planarised pattern, not off the drawing. An edge is raw when it is the sheet’s own boundary — which is why the same routine gives the right answer on a sheet with a hole in it, where the hole’s rim is boundary too and there is nothing special about the square outside it.

Across the eight printed patterns the two groups run like this.

Six patterns have both kinds of panel and on all six the rim is lower. The seventh has one middle panel with the same degree as its rim, and the two with nothing in the middle at all are the two with a single interior vertex — on those, every panel of the sheet touches the outside.

The differences are small. Three panels on the Miura, two on the hexagon twist, 2.4 on the Yoshimura, 6.7 on the waterbomb. What makes them worth reporting is the sign, which is the same seven times out of seven and would have been a coin toss if the boundary meant nothing to the pile.

Why it happens, and why it is not obvious #

The mechanism is the one a folder would guess and it is worth stating carefully, because the obvious version of it is wrong.

The obvious version is that the rim of the sheet becomes the rim of the folded object, so rim panels sit at the outside of the footprint where there is less paper. That is false on most of these patterns. A folded Miura brings the sheet’s own edges into the middle of the footprint repeatedly; a waterbomb tessellation folds its rim inward. Where a panel lands has almost nothing to do with where it started.

The true version is about area. A panel touching the raw edge of the sheet is, on every pattern here, a panel that got cut off by the edge — a half-cell of the tessellation rather than a whole one, a triangle where the interior has parallelograms. It is smaller, and a smaller panel covers less ground and therefore overlaps fewer of the others wherever it lands.

That makes the finding a fact about how tessellations meet a sheet, rather than a fact about folding. The way to see the difference is to look for a pattern where the rim panels are not smaller, and the square twist is one: its eight rim panels and its one central panel all lie over all eight of the others, because on that pattern every panel shares ground with every panel. Degree eight against degree eight, exactly, and the effect vanishes.

The one panel in the middle #

Two of the printed patterns have exactly one panel away from the sheet’s edge, and they are the two twists. That makes them the cleanest test available, because the “average over the middle” is a single number about a single named piece of paper rather than an average over a crowd.

The square twist’s middle panel is the square that gives the pattern its name — the polygon that rotates as the sheet closes — and it lies over eight of the other eight panels. So do all eight of its rim panels. The pattern collapses to a single pile, everything is over everything, and there is no room for the rim to be at a disadvantage.

The hexagon twist does not collapse that far. Its central hexagon lies over twelve of the other twelve panels; its twelve rim panels lie over ten each on average. Sixty-six of the seventy-eight possible pairs share ground, which leaves twelve pairs that never meet, and every one of those twelve has a rim panel in it.

That is the effect in its smallest possible form: the pairs that fail to meet are pairs of rim panels, and a bigger pattern has more of them. It is also the reason the effect is small in absolute terms. Nothing is pushing rim panels apart; they are simply the panels that can miss, because they are the ones the sheet’s edge made smaller.

How much of a sheet is stacked at all #

The same census answers a question the strip case could not pose. A folded strip’s segments all lie along one line, so every pair of them either overlaps or is separated by a crease. A folded sheet’s panels can miss each other entirely, and a pair that misses is invisible to every ordering rule there is.

The three small patterns are complete: every panel lies over every other, which is what a sheet that collapses to a point looks like. The Yoshimura is nearly complete for a different reason — it folds into a long thin object and stacks almost everything on almost everything. The Miura, at eighty-three per cent, is the printed pattern furthest from complete after the waterbomb and the tapered corrugation, and it is the one that folds into a flat rectangle rather than into a point or a line.

These Miura numbers replace ones this essay first printed — 13.0 against 15.0 for the rim, and 164 pairs, fifty-nine per cent. The overlap test behind them missed every pair of Miura panels that folding slides along a shared slanted edge, a blind spot one choice with eleven answers draws and repairs; no other printed pattern lost a pair to it, and the rim still lies lower on the Miura, by a little more than it did.

So “how much of a sheet is stacked” is not a measure of how tightly it folds. It is a measure of the shape it folds into, and the three answers here are a point, a line and a rectangle.

The deepest point #

The other number in the same sampling is how many panels lie over the deepest point of the footprint, and it separates the shelf differently again.

The depths themselves are already priced, as the length a thick panel has to find. What the rim census adds is the share: on the preliminary base, the square twist and the fold-and-cut triangle, the deepest point carries every panel the sheet has — eight of eight, nine of nine, seven of seven — while on the Miura it is sixteen of twenty-four and on the waterbomb tessellation thirty-two of fifty-two.

The averages tell the other half, and they are the number the depth alone cannot give. The fold-and-cut triangle averages 1.19 layers over its footprint: seven panels deep at one point and one panel deep almost everywhere else, which is exactly what a construction designed to bring one line of the paper together should look like. The Yoshimura averages sixty against a deepest of sixty, so it is the same depth everywhere. The square twist averages three against a deepest of nine.

Both numbers come from sampling the folded state on a grid, and the sampling is checked rather than trusted: the layer count integrated over the footprint has to come back as the area of the sheet, because that is where all the paper is. A sampling fine enough to get that right is fine enough to report a maximum.

What grows and what does not #

Grow one tessellation and the two quantities move in opposite directions.

A two-by-two patch has four panels and all six of its pairs share ground. A three-by-two has all fifteen, a three-by-three all thirty-six, a four-by-three all sixty-six and a four-by-four all hundred and twenty. Only the printed six-by-four falls short, at 228 of 276. The share stays at one while the patch is short in either direction and falls away only once it is long in both.

A long enough sheet of the same pattern is less stacked, not more. That is worth holding next to what the letters do over the same growth, which is the exact opposite: a bigger patch has proportionally more of its creases settled when it was drawn. The pattern gets more decided and the pile gets less connected, and the two facts are about the same sheet.

There is one more reading of the rim numbers and it belongs with the others rather than in a caveat. Because the rim panels are smaller, they are also the panels most likely to be entirely covered by something else in the folded object — a small panel under a large one contributes nothing to what a reader sees. So the parts of a folded model that came from the outside of the sheet are, on average, both the least stacked and the least visible, which is not a contradiction: lying over few things and having few things above are different facts, and only the first is what a degree counts.

What the rim does not decide #

Three things, and the first is the one most likely to be read into the table above.

It does not decide the order. A rim panel lies over fewer of the others; it does not lie lower. On every printed pattern whose ordering the search can finish, the order is unique and it is decided by the crease letters, and no panel is placed high or low because of where it started on the sheet.

It does not survive a change of scale. The effect here is an area effect: rim panels are cut-off cells and cut-off cells are smaller. Draw a tessellation whose boundary happens to fall on cell edges — a Miura patch trimmed exactly at a column — and the rim panels are full cells, the same size as everything else, and the difference should vanish. That is a prediction this measurement makes and does not test, and it is stated as one.

And it does not make the boundary special to the theorems. A vertex on the edge of the paper is still a vertex no condition applies to, which is a fact about sectors in a line, and nothing in the pile changes it. What the pile adds is that the boundary is visible after folding — a thing the pattern’s own theorems cannot see is legible in the object they were checking.

A pile says which panel lies over which; what generates it is a pair of non-crossing rules, one about a crease landing inside a panel and one about a crease landing on its boundary. How many of each a pattern produces is decided at the rim.

What a folder should take from it #

The edge of the paper is findable in the finished model. Not by looking for raw edges, which are mostly buried, but by noticing that the parts of the pile with fewest layers over them are disproportionately the parts that were at the outside of the sheet.

A pattern that stacks everything on everything has no rim in this sense at all. The preliminary base, the square twist and the fold-and-cut triangle are all of that kind, and all three are patterns whose folded state is a single pile.

And the share of a sheet that is stacked is a reading of its shape. A hundred per cent means it folded to a point. Ninety-nine means a line. Fifty-nine means a rectangle, which is what a deployable wants and is the reason the Miura is the pattern that gets built.

What a degree does not count #

The degree is deliberately one-sided, and it is worth being explicit about what that costs, because the tempting reading — that the rim matters less — is a stronger claim than the census supports.

Sharing ground is symmetric. If one panel lies over another then the second lies under the first, and the pair counts once in each of their degrees. So a low degree says a panel meets few others, in either direction together. What it cannot say is the thing the paragraph above reaches for: whether a panel finishes at the top of its pile or at the bottom. That is a question about the order, and the order is a different object — the same set of overlaps admits many stackings, and which one the sheet is allowed to take is what the ordering rules decide. Those rules speak only about pairs that overlap, which is why a pair that misses is invisible to them; they say nothing at all about how many pairs a panel is in.

There is a second thing the number hides, and it is the one that keeps the rim effect small. Degree counts panels and not area. A rim panel is a half-cell, so a pair of rim panels overlaps in a smaller region than a pair of interior panels does — and each pair counts one. A census weighted by the area of each overlap would separate the two groups further than this one does, and it would be answering a different question: how much paper is doubled, rather than how many pieces are involved. Both are worth having and only the second is a stacking relation.

What the census depends on #

Three things bound the finding, and none of them is hidden.

The overlaps are a property of the creases alone. This is the reassuring one and it is not obvious. Where each panel lands is decided by the folding map, which reflects the sheet across one crease after another — and a reflection does not ask which way the crease folds. So the planar image of a flat-folded sheet is fixed by the crease pattern, and the assignment and the layer ordering decide only what is stacked on what, not what is stacked at all. Every number in the tables above is therefore a property of the pattern rather than of the particular folded state it was read from, and a pattern with several valid orderings gives the same degrees and the same pair share on all of them. The depth numbers inherit that too: how many panels lie over a point is a count, and counts do not care about order.

Eight patterns are a shelf and not a sample. They were chosen to be foldable, printable and worth printing, which selects for patterns that collapse tidily and against patterns that sprawl. Nothing here licenses the sign of the difference on a pattern nobody has drawn. The honest statement of the result is the one the table makes: on every printed pattern that has panels of both kinds, the rim lies over less, six times out of six with a seventh tying, and with a mechanism — the edge cuts cells in half — that is visible in the areas rather than inferred from the counts.

The footprint is sampled. The layer counts come from a grid laid over the folded plane, so a feature narrower than the grid spacing can be missed. What makes that a bounded worry rather than an open one is the identity the sampling is checked against: integrate the layer count over the footprint and the answer has to come back as the area of the sheet, because that is where all the paper is. A grid too coarse to resolve a sliver loses area and fails the check. A grid that passes it has found the paper, and a maximum taken over a grid that has found all the paper is a maximum over the panels themselves.