The outline is mostly crease

Where the paper stops is about what the sheet’s edge does to the conditions: a vertex on the boundary is subject to none of the flat-folding theorems, because every one of them is a statement about a full turn of paper and a boundary vertex has less than one.

This is the same edge asked a different question. Not what it does to the theorems — what happens to it. Fold the sheet and the boundary goes somewhere; the folded object has an outline; and the two are much less related than they look.

What “exposed edge” means here #

The measurement needs stating precisely, because “outline” is ambiguous once there are layers.

A flat folded state is a set of panels, each carried from the flat pattern by its own isometry, lying on top of one another in the plane. Take any edge of any panel and any point along it. Step a little to each side, perpendicular to the edge, and count the panels covering each side. If one side has none, the paper stops there and that piece of edge is exposed: it is the boundary of the paper as seen from that layer.

Every panel edge is then charged to one of two accounts, according to whether it lies along one of the pattern’s own boundary edges or along a crease.

This counts every layer, not just the topmost. A stack of five layers whose edges are all flush contributes five exposed edges rather than one, which is the right thing when the question is what the paper is doing and would be the wrong thing if the question were the length of the silhouette. Both are legitimate; this essay measures the first and says so.

The answers #

A preliminary base — the square folded to a point through the diagonals and the mid-lines — exposes 13.66 units of edge, of which 29.3% is raw sheet. That is the highest figure of the five, and it is a base whose four corners all arrive at the same place.

A square twist exposes 5.09 units, 26.7% raw.

A Miura fold, five by four, exposes 50.58 units, 20.8% raw.

A Yoshimura pattern, six by five, exposes 30.00 units, of which 6.7% is raw.

A waterbomb tessellation exposes 22.63 units and none of it is the sheet’s edge. Every exposed edge of that folded state is a crease.

Why the raw edge is always losing #

The reason is arithmetic and it is worth doing, because it explains why the answer is never close to a half.

A sheet has one boundary and the folding does not lengthen it. Creases, on the other hand, are added: a pattern with n creases has 2n panel edges along them, since each crease bounds two panels, and every one of those edges is a candidate for being exposed. So the raw edge is a fixed budget being spent against a crease budget that grows with the pattern’s density.

Two things then decide the split. The first is how much crease there is, which is why the Yoshimura and the waterbomb — dense tessellations — are at the bottom of the table and the preliminary base, which has eight creases, is at the top. The second is where the raw edge lands: an edge that ends up buried under other layers contributes nothing.

The Yoshimura’s raw edge is buried almost entirely, because the pattern closes into a tube and the sheet’s two long edges come together and lie against one another. The waterbomb’s raw edge is buried completely.

What this changes about looking at a folded object #

The everyday reading of a folded model is that it is the sheet, rearranged, and that its silhouette is the sheet’s silhouette after the rearranging. The measurement says the silhouette is mostly something the sheet did not have before it was folded.

That matters most where the outline is doing design work. A uniaxial base’s flaps are bounded by creases, not by the paper’s edge, except at their tips; a tessellation’s outline is entirely a consequence of where its pleats land. So a designer shaping a silhouette is shaping creases, and the sheet’s own outline — the thing they chose when they chose a square — contributes a diminishing fraction as the design gets denser.

It also explains a small practical fact about folding from a printed pattern. The edges of the sheet are the one part of the geometry a folder can see and align by touch, and they are exactly the part that disappears first. After a few steps almost every alignment is crease-to-crease, and the error that accumulates is the error of aligning things that were themselves placed by earlier folds.

The preliminary base is the interesting row #

The base with the largest raw fraction is worth a paragraph, because it is the one that most contradicts the eye.

Fold a square into a preliminary base and the object in the hand is a small square with a point at one corner. Its four raw edges have all come together: they lie along the two edges of the finished square that meet at the open corner, stacked four deep. So the raw edge is not buried at all — it is the most visible feature of the object, the place a folder grips to open the base — and it still accounts for less than a third of the exposed edge, because the eight creases contribute so much more.

That is the clearest statement of the general result. Even when the raw edge is exactly where a folder is looking and nothing is hiding it, it is outnumbered. The creases win on count before they win on placement.

The measurement, and the resolution it depends on #

The step used to decide “nothing on the other side” is a parameter, and a measurement that depends on its own resolution is not a measurement.

It is swept. At steps of three thousandths, one thousandth and three ten-thousandths of the sheet, the raw fraction moves by less than a fiftieth of a percentage point on four of the five patterns and by 0.2 points on the Yoshimura, where a slightly coarse step counts a nearly-flush edge as exposed. The figures are drawn at one thousandth, and the sweep is asserted rather than described: a generator whose answer moved across the sweep would refuse to draw.

Where the model stops #

This is not the perimeter of the silhouette. Summing over layers counts a flush stack of five edges five times, and the outline a photograph shows counts it once. The two questions differ by exactly the multiplicity of the stack, which is itself a measured quantity and could be divided out — it has not been, because the question here is about the paper rather than about the picture.

The sheet has no thickness. Real layers are offset by their own thickness, so an edge that is flush in the model is slightly proud or slightly recessed in a real model, and the outline a camera sees is the outermost of them. What thickness does to a folded object is measured elsewhere and none of it is here.

Nothing here is about layer order. Whether an edge is exposed depends only on whether other panels cover it, which is a question about position and not about which panel is on top. The ordering is a separate and much harder object, and a folded state’s outline is the same whatever order its layers are in.

The folded states here are computed, not posed. Each panel is placed by composing reflections along the pattern’s own connectivity, and every panel reachable a second way is a second opinion — the two agreeing is Kawasaki arriving from a direction the placement never computes. A pattern whose folded state did not close would not reach this measurement at all.

Five patterns is not a survey. The five here are the ones the site’s folded-state machinery handles, and they were not selected for their outlines. A pattern designed to present its raw edge — a model whose outline is deliberately the paper’s, which is a real design choice, especially for coloured paper — would sit at the other end and there is none in the library.

The two accounts, in one sentence each #

It is worth separating what each number in the table depends on, because they are not the same kind of quantity.

The exposed edge total depends on how much crease the pattern has and on how the layers stack, and it grows quickly: the Miura’s fifty units against the twist’s five. That number is about density.

The raw fraction depends on where the sheet’s edge ends up, and it is almost independent of density: the preliminary base with eight creases and the square twist with twelve are at 29.3% and 26.7%, which is close, and the two dense tessellations are at 6.7% and 0.0%, which is not close to either. That number is about the pattern’s plan for its own boundary.

What it takes to bury a raw edge #

The essay calls the raw fraction a fact about the pattern’s plan for its own boundary, and the plan can be stated as a condition rather than left as a description.

A piece of raw edge is buried exactly when some panel lies over it. So burying the sheet’s boundary requires the pattern to bring paper on top of the rim — not merely to bring the rim inward, and not merely to stack the rim against itself.

That distinction is the whole of the table’s spread, and the two extremes make it plainly.

The preliminary base brings all four of its raw edges together and stacks them four deep along two edges of the finished square. Nothing covers them: they are stacked face to face, all at the same place, and the outermost of them is the outside of the object. Bringing edges together is not burying them, and the base’s raw fraction is the highest in the table for exactly that reason.

The Yoshimura closes into a tube, so the sheet’s two long edges arrive at one another and the paper on either side of the seam laps past. That is paper over the rim, and 93.3 per cent of the object’s exposed edge is crease as a result. The waterbomb tessellation does it everywhere, which is why its raw fraction is nought.

So the criterion is: a pattern buries its boundary to the extent that it folds material across the rim rather than up to it. Collapsing toward a point does not do it however tightly the collapse goes; closing into a tube or corrugating past the edge does.

Which costs paper, and says so #

That criterion has a price attached and the price is the reason nobody buries a boundary by accident.

Paper lying over the rim is paper doing nothing else. It is not a flap, it is not structure, and it is not visible — it is a lap, and its area is area the design has spent on tidiness. So a pattern with a low raw fraction has paid for it, and the payment shows up in the same ledger every other use of the sheet is charged to.

That explains a small fact about the traditional repertoire, which is full of models whose raw edges are conspicuous. A classical base is designed to spend its paper on flaps, so it brings the rim inward and leaves it showing; the folder is then told to tuck it, which is a step performed afterwards on a model that did not plan for it. A tessellation is not designed to spend paper on flaps at all, so lapping the rim costs it nothing it wanted, and its boundary vanishes without anybody deciding it should.

The design reading follows directly. A model that must hide its raw edge — because the paper is coloured on one side, or because the edge would read as an error — has to buy a lap, and the lap has to be in the pattern rather than added at the end. That is a constraint on the design and it is one the two accounts above can price before anything is folded.

What the conservation law does not say #

This site already has a strong statement about a folded sheet’s material: the paper is all still there, so the footprint times the average layer count is the sheet’s area, exactly.

That is about area and it has no boundary analogue. The sheet’s boundary is not conserved in any useful sense — it does not disappear, but where it goes is not determined by anything the area argument knows, and the fraction of it that stays visible is a fact about the particular pattern rather than about folding. A Yoshimura and a preliminary base fold from the same square, conserve the same area, and expose 6.7% and 29.3% of their edge respectively.

So there are two conservation-shaped questions about a folded sheet and only one of them has an answer. Area is conserved and computable. The boundary is preserved and unpredictable, and the only way to find out where it went is to fold the thing and look.

The asymmetry has a reason. Area is additive over the panels and folding is an isometry, so the sum survives every rearrangement — nothing about where the panels went enters the argument. The boundary’s fate is entirely a question of where the panels went, because a piece of raw edge is exposed or buried according to what is lying on top of it, and that is the layer structure rather than the geometry. The one quantity that a conservation argument reaches is exactly the one that does not care about position, which is why there is a second question and no second law.

What a folder already knows about this #

None of this is news to anybody who folds, and it is worth saying what form they know it in.

A folder describing a model says the raw edge goes here as a step in an instruction, precisely because the raw edge is one of the few features that can be named unambiguously — everything else is the crease made three steps ago. The tradition’s vocabulary has words for the paper’s edge and almost none for individual creases, which is a notation problem this site has looked at from the other side.

So the raw edge is disproportionately important to making a model and disproportionately unimportant to looking at one. Those two facts sit oddly together and they are both consequences of the same measurement: there is very little raw edge, so it is a useful landmark and a small part of the finished object.

Why nobody had the number #

The measurement is easy once a folded state is available as an object, and a folded state has only been available as an object on this site since the first breadth phase. Before that, a folded model was a photograph or a thing in a hand, and neither of those can be asked which of its edges are crease.

That is the ordinary reason a quantity goes unmeasured — not that anybody doubted it, but that asking required a folded state with every panel placed and every edge identified, which is a considerable amount of machinery for a question nobody had phrased. The traditional literature contains the observation in qualitative form all over the place: instructions say this raw edge should now be hidden, which is the same fact stated as a step rather than as a fraction.

Where the ladder goes next #

The immediate continuation is the silhouette proper — the outline a photograph shows, which is the exposed edge divided by its multiplicity, and which is what anybody designing a shape is actually shaping. It is computable from the same folded state with one more step, and it would let the question “what shape is this model” be answered from the crease pattern.

The other direction is the design use. If almost all of a folded object’s edge is crease, then the creases near the outside of the fold do a job the creases in the middle do not, and a design could be graded by which of its creases reach the boundary. That is a ranking of creases by visibility, it is computable by exactly the test above, and no design method the subject has considers it at all.