Most of a patch is edge

A crease pattern is usually described from the middle outward. The vertex conditions are about interior vertices; the folded state is about panels; the tessellation is about a unit that repeats. The edge of the paper enters those descriptions as a qualification — the place four conditions stop applying, the place a crease may legitimately end — and qualifications sound small.

Counted, it is not small. Over the eight patterns this collection prints at true scale, the share of vertices sitting on the edge of the sheet rather than inside it runs from 39% to 91%. Over the tessellation patches, which are the largest patterns drawn here and the ones meant to show a repeating structure, it is between 34% and 44%. There is no pattern in this collection in which the boundary is a margin.

The arithmetic that makes it inevitable #

A repeating pattern of units n across on a square holds about n² units, of which about 4n touch the rim. The share cut is therefore about 4/n, which goes to nothing — eventually. The word doing the work is eventually.

At n = 3 the share is more than everything, which is to say every unit touches the rim. At n = 6 it is two thirds. At n = 20 it is a fifth, and a twenty-across twist tessellation on a 150 mm sheet gives units 7.5 mm across, with pleats a millimetre or two wide — past what most paper will take a crease at, and far past what a page can show. The asymptotic statement is true and the sizes at which it becomes true are not sizes anybody folds.

The measured version wanders around the arithmetic rather than following it, because a lattice landing on a square is not a smooth thing: at 0.34 of the sheet a square grid happens to fit nine units exactly and cuts none, and at 0.28 it cuts twelve of twenty-one. The trend is down and the individual numbers are about how the lattice fell.

Pattern by pattern #

The eight printed patterns spread across the whole range, and the spread is informative rather than noisy.

The fold-and-cut triangle is 91% boundary: one interior vertex, the node of the straight skeleton, and ten vertices on the rim, most of them the feet of perpendiculars. The preliminary base is 89%: one interior vertex at the centre of the square and eight where the creases reach the edges and corners. Both are patterns whose whole content is one point and the lines running out of it, so almost everything in them is where the lines stop.

The square twist is 75% and the hexagon twist 73%, for a different reason: a single twist unit is a small ring of interior vertices with long pleats running off it, and the pleats end on the rim. The Miura is 57%, the tapered corrugation 55%, the Yoshimura 51%: grids filling their sheets, where the interior grows with the area and the rim with the side. The waterbomb tessellation is 39%, the lowest, because it packs the most interior vertices per unit area of anything printed here.

The tessellation patches sit between 34% and 44% — lower than every printed pattern but two, and still a third. The rhombille’s clipped patch, at a hundred and ninety vertices the largest crease pattern in the collection, is 34%. Nothing here gets below it.

Why the boundary is one unit deep #

The share does not fall faster because a boundary cannot be thicker than the thing it bounds.

Take any repeating pattern and any convex sheet. A unit is cut by the paper’s edge if the edge passes through it, so the cut units are those within one unit’s width of the rim — a band one unit deep, whose area is the perimeter times the unit width, against a total area that is the side squared. The share is therefore the unit width times the perimeter over the area, which for a square of side s and units of width w is 4w/s — exactly the 4/n above, arrived at from geometry rather than from counting.

Three consequences follow, and only the first is obvious.

A bigger sheet at the same unit size helps in proportion, and only in proportion: doubling the sheet halves the share. A rounder sheet helps a little, because the perimeter-to-area ratio of a disc is lower than a square’s by about a tenth, which is real and small. And a sheet with a hole in it is worse, because a hole adds perimeter without adding area — cheap paper by the flap-packing measure and expensive paper by this one, which is a genuine tension between two ways of valuing the same square.

Three ways a crease can meet the rim #

A crease reaching the edge of the paper is doing one of three things, and only two of them are legitimate.

It ends there, because the construction put its endpoint on the rim. Every crease of the preliminary base does this; so does every crease of a Miura drawn to fill its sheet.

It is cut there, because the construction drew a longer crease and the paper ran out. That is what clipping a tessellation out of the plane does, and it is legitimate for the same reason the first is: a crease that reaches the paper’s edge has a fold the paper can make.

It stops short of the rim, in the middle of the sheet, with nothing at its end. That is not legitimate. The material has to turn about the line, and past the end of the line there is nothing to turn about.

The third is a drawing fault and the collection has produced it: the honeycomb’s assembled patch carries two such creases. The fix for it — extend the crease to the rim — is what produced the crossings that made four patches unfoldable, which is the whole reason the boundary is worth counting rather than qualifying.

The rim is where the constructions are decided #

Everything difficult about the patterns here happens at the boundary, and the reason is that the interior is determined and the rim is not.

Inside a tessellation the construction has no choices: the polygon at a vertex is forced by the tiling’s own angles, the pleat between two polygons is forced by the pair, and the letters are propagated. At the rim, the same construction is asked what to do about a unit whose neighbour is not there, and no rule inside the pattern answers that.

The same is true of the other families. A Miura fills its sheet because the sheet was chosen to fit a whole number of cells. The waterbomb tessellation’s mask is a repeating rule and the rule is applied to a patch whose edge cells have fewer creases than the rest. A tapered corrugation is tapered because of what its two ends have to do. In every case the pattern in the middle is a consequence and the pattern at the edge is a decision.

What the boundary does to the checks #

Four conditions are asked at every interior vertex and none of them is defined at a vertex on the edge, because the sectors there come in a line rather than a ring. That is answerable and is answered — a boundary vertex is a strip, and the one-dimensional solver decides it exactly — but the answering is a separate machine from the one every figure runs.

So a pattern that is 44% boundary is a pattern where nearly half the vertices are outside the checks the figures actually perform, and are inside a second check that has to be invoked deliberately. On the printed shelf the count is a hundred and five vertices on the edge against ninety-two inside it: more than half, and the majority of them carrying a single crease, where nothing is being decided at all.

What the boundary does to the numbers #

Several measurements this collection reports are averages over a pattern, and a pattern that is a third boundary is a pattern whose averages are a third about the boundary.

The layer counts. A panel carrying a raw edge lies over fewer of the others than one that does not — 18.0 against 21.0 on a Miura, 31.0 against 37.7 on a waterbomb — so the mean layer count of a patch is pulled down by however much of it is rim. On a small patch that is most of the pattern.

The crease length. A cut crease is shorter than a whole one, so the total folding length of a patch is not the length of its units times how many there are. Clipping the triangular patch from the plane rather than assembling it adds forty creases and 1.2 sheet widths of folding, all of it at the rim.

The shrink. What a corrugation costs is computed over the units that sit wholly on the paper, deliberately, because a cut unit does not take up its full pleat. That choice makes the number a property of the tessellation rather than of the picture — and it means the figure and the number are about different regions of the same sheet, which is worth saying out loud.

A folder’s sheet is mostly rim too #

None of this is an artefact of drawing patterns small. It is what folding paper is like.

A sheet of 150 mm paper folded into a twist tessellation of nine units has units 50 mm across, and every one of them touches the edge. The traditional bases are worse: a preliminary base is one interior vertex and eight boundary ones, and a bird base built on it adds creases that nearly all reach the paper’s edge. The fold-and-cut triangle is one interior vertex against ten on the rim — 91%, the highest here — and it is a pattern the whole subject is proud of.

What that means for a reader with paper is that the boundary is not the part where the interesting behaviour has faded out. It is where most of the creases end, where the hand starts every fold, and where the alignment is checked: a folder brings an edge to an edge, or a corner to a crease, and both of those are boundary features.

The sheet’s shape is part of the pattern #

If the boundary is a third of what is drawn, then the shape of the boundary is a third of the design, and this collection has said so from the other direction: the square is a choice rather than a given, and a corner of it is worth four times what the middle is when the question is how much flap the paper carries.

The two measurements point the same way for different reasons. The flap-packing one values the rim because a disc centred at the edge needs only the part of it that is on the paper, so the edge is where a flap is cheap. This one counts the rim because that is where the units get cut and the constructions get decided. A corner is the extreme of both: it is the cheapest place to put a flap, and it is the place where two rims meet and two families of creases run into each other.

The two rim measurements are one number #

The tension named above — a hole makes paper cheap by one measure and expensive by another — is closer than a tension. The two measurements are the same functional applied with different lengths.

The boundary share derived here is the area of a band one unit deep against the whole sheet: $wP/A$, for unit width $w$, perimeter $P$ and area $A$. The flap-packing measure works out, for flaps short against the sheet, as a saving of $\tfrac{2}{3\pi}LP/A$ against a solid interior, for flap length $L$.

Both are a length times the perimeter-to-area ratio. One length is how wide a unit is and the other is how long a flap is; the geometry of the sheet enters both only through $P/A$, and nothing else about the outline matters to either.

So a sheet has a single number describing how exposed it is, and the two fields read it with opposite signs. A high $P/A$ means a flap finds a boundary easily, which is cheap; it also means a repeating unit finds one easily, which is a cut.

Which prices the hole exactly #

That makes the trade computable rather than rhetorical.

A unit square has $P/A = 4$. The comparison square of the same area as a square holding a 0.28 hole has $P/A = 3.84/0.9216 = 4.17$; the holed sheet has $5.12/0.9216 = 5.56$. The ratio is 1.33.

So cutting that hole raises the exposure by exactly a third, and both consequences scale by that same third: the boundary term in the flap price improves by a third, and the share of units the paper cuts through rises by a third. A designer who values the first is paying the second at a fixed exchange rate, and the rate is one.

Neither measurement is a reason to prefer a shape; the ratio between them is a constant. What decides is which length is larger — a flap in an ambitious design is a substantial fraction of the sheet, while a tessellation unit is a small one, so the same hole is worth much more to a packing than it costs a tiling.

That also disposes of the rounder-sheet suggestion in the same breath. A disc lowers $P/A$ by about eleven per cent against a square of equal area, so it cuts eleven per cent fewer units and makes flaps eleven per cent dearer. It is not a free improvement to either; it is a small move along one axis that both quantities sit on.

The page is the binding constraint #

There is a size at which the boundary really would be a margin, and the reason it never appears here is not the paper. It is the page.

A figure on this site is drawn a few hundred points across. A crease drawn at less than about a point is a grey smudge; two creases a point apart are one line. So a patch that is to be legible on a page can carry something like twenty units across at the very most, and twenty across is the size at which the share cut is still a fifth. Every patch drawn here is between three and nine across, because below that the pleats stop being distinguishable from the polygons they run between.

The same limit applies to the paper for a different reason. A twist pleat has to be wide enough for two folds and the paper between them; at 150 mm and twenty units across the pleats are under a millimetre, which ordinary paper will not hold as two distinct creases. A folder’s practical ceiling is not far above a page’s.

So the regime in which a tessellation is mostly interior is one that neither the reader nor the figure can reach, and the regime both of them work in is the one where a third of the pattern is rim. That is not a limitation to apologise for. It is the reason the boundary deserves a construction rather than a convention: at every size anybody will ever see, it is a third of the object.

What the count is not #

It is not a criticism of the patterns. A pattern that is mostly boundary is what a small sheet gives, and small sheets are what readers have. The number is a description, and its use is that it says which parts of a figure carry the argument.

It is not a single number. The share depends on the unit size, on how the lattice lands, on whether the pattern was assembled or clipped, and on whether a vertex of degree one counts — a crease ending on the rim makes a vertex there, and half the boundary vertices on the printed shelf are of that kind, where no condition of any sort has anything to decide.

And it does not say the interior is understood. The conditions at every interior vertex do not decide the sheet, and the patches whose boundary is now drawn correctly still have letterings that force a loop. Fixing the rim removed one obstruction; it did not make the middle easy.