Every facet is a layer

The rung below established something about a sheet folded at random: every interior vertex it ends up with satisfies every local condition in the subject, because a sheet that has been folded flat is flat folded. That settles what kind of object a crumple’s crease pattern is. It says nothing about how much of one there is.

Counting turns out to be the more surprising half.

Three counts, two of them equal #

Take a sheet folded k times. Three quantities can be counted, and they are counted by three different pieces of machinery that share no code.

Creases: the segments the simulation recorded, one per layer per fold, in the sheet’s own coordinates.

Facets: the faces of the flattened crease pattern, found by planarising the segments so that crossings become vertices and then walking the faces — a routine written for the FOLD export and for the honest count of what a pattern is, which knows nothing about folding.

Layers: the pieces the stack is in when the folding stops, counted off the simulation’s own state.

On a sheet with 138 recorded creases: 139 facets, 139 layers. On one with 69 creases: 70 and 70. On one with 3: 4 and 4. Over seven seeds and every fold count from two to thirteen, without exception, the layer count and the facet count are equal, and both are one more than the crease count.

That contrast is the identity’s other side and it is worth pausing on. Both patterns have the same number of straight creases on the same square. One of them cuts the sheet into 83 pieces and the other into more than two thousand, and the difference is entirely in where the creases are allowed to be. A fold cannot put a crease across a piece of sheet it is not currently touching, so its creases are confined to one facet at a time; a pen has no such restriction and slices everything.

Why, in one paragraph #

The identity is exact rather than approximate and the reason is short enough to give in full.

A fold takes a line and turns everything on one side of it. Each layer that the line crosses is cut in two: one piece stays, one piece reflects. So that fold increases the layer count by exactly one for each layer it crossed — and it records exactly one crease for each layer it crossed, since the crease is where the line met that layer, carried back to the sheet.

So creases and layers rise in lockstep. The sheet starts as one layer with no creases, and the identity follows.

The facets need one more sentence. Each recorded crease is a segment across the piece of sheet that one layer occupied, running from one boundary of that piece to the other. A segment that cuts a face from edge to edge divides it into two, so each crease adds exactly one facet as well.

The consequence is a bijection and not just an arithmetic coincidence. Each layer of the stack is one facet of the flattened pattern, carried to where it lies by its own isometry. Unfold a stack of 139 layers and there are 139 pieces of paper on the table, joined at their edges.

The control says the same thing arithmetically #

The contrast with the drawn sheet is reported as a ratio — eighty-three facets against two thousand and ninety-eight — and there is an exact identity underneath it that says what the ratio is made of.

For any arrangement of segments in a square, the number of faces is one, plus one for each segment, plus one for each place two segments cross. A segment that runs from boundary to boundary of a face divides that face in two and adds one; a segment that also crosses an existing one divides two faces and adds two, and so on.

Put the drawn control’s numbers in. Two thousand and ninety-eight faces, eighty-two segments: one plus eighty-two plus X is 2,098, so X is 2,015. Eighty-two lines drawn at random across a square cross one another in two thousand places, which is well over half of the three thousand three hundred and twenty-one pairs available.

Now put the crumple’s numbers in. Eighty-three faces, eighty-two segments: one plus eighty-two plus X is 83, so X is nought. The recorded creases of a folded sheet cross one another nowhere at all.

Which is what confines a fold to one facet #

That is the identity’s mechanism written as arithmetic, and it sharpens the sentence about a fold being unable to reach a piece of sheet it is not touching.

A crease recorded on one layer runs across the piece of sheet that layer occupies, from one boundary of that piece to the other — and the boundaries of that piece are the sheet’s own edge and earlier creases. So a recorded crease ends on an earlier crease rather than passing through it. Its endpoints sit on other creases and its interior meets none of them.

Which reconciles with the fact that every interior vertex of a crumple has degree four, and the reconciliation is the whole picture. At a point where a new crease ends on an old one, the old crease continues through — two arms — and two new creases arrive there, one from the layer on each side of the old crease, because a single fold writes on every layer it crosses and adjacent layers share that boundary. Two plus two is four.

So the fold’s mark on the unfolded sheet is one bent polyline, and the bends are exactly where it meets earlier creases. Counted as segments it crosses nothing; counted as a polyline it passes through the vertices it makes. The identity counts segments, and segments made by folding cannot cross — which is the whole of why facets and creases stay one apart, and why a pen, which is under no such restriction, cuts the same square into twenty-five times as many pieces.

What this is not #

It would be easy to read the identity as the conservation law this site already has, and it is a different statement.

The paper is all still there says that the folded object’s footprint multiplied by the average number of layers over it is the area of the sheet. That is a statement about areas, it holds for every flat folded state of every pattern, and it is how the site’s shrink factors are checked.

This is a statement about counts, and it is not true of every pattern. A Miura fold has far more facets than its stack has layers, because most of its panels lie beside one another rather than on top. What makes the identity hold for a crumple is the all-layers fold: every fold takes the entire stack, so the pieces never spread out sideways, and the pattern’s facets and the stack’s layers stay in step.

So the identity is a property of a folding process, not of a folded state. A pattern that could have been produced by all-layers folds satisfies it; the same pattern reached another way need not.

How the counts actually grow #

The growth is worth looking at because the intuitive answer — doubling — is right on average and wrong in detail.

Fold a sheet through the middle of the whole stack and every layer is crossed, so the layer count doubles. Fold it near an edge and only a few layers are crossed, and it barely moves. A random line through the current outline does something in between, and the sequence is correspondingly ragged.

One run, fold by fold, adds 1, 2, 3, 3, 4, 10, 17, 5, 9, 15, 33, 12, 24 creases. The tenth fold added fifteen and the eleventh added thirty-three; the twelfth added twelve. Averaged over five seeds it comes out at roughly a doubling every two folds — 9 creases at four folds, 18 at six, 42 at eight, 78 at ten, 141 at twelve — which is a slower exponential than the ideal one and for an obvious reason: a randomly placed line usually misses most of an untidy stack.

The facets get smaller and never smoother #

Since a fold only ever cuts faces in two, the facets can only get smaller, and they do: the median facet falls from about a quarter of the sheet at two folds to four thousandths of it at twelve.

That has a consequence worth stating, because it is the one that matches the hand. Crumpling never smooths anything. There is no operation in the process that removes a crease or merges two facets — every fold subdivides — so a sheet that has been worked is a sheet with strictly more structure in it than before, at every step, without exception. A crumpled ball opened out and crumpled again is not re-crumpled; it is further crumpled.

Paper agrees with the model here more than it usually does. A sheet that has been crumpled and flattened is easier to crumple again along the same lines, and repeated crumpling produces a sheet that eventually behaves like a fabric — which is the fibres giving way rather than the geometry, but the geometry is going the same direction.

The vertex count is the third curve, and it is not tied to anything #

Creases, facets and layers move together. The fourth quantity — how many interior vertices the pattern has — does not, and the reason says something about what a crumple is.

An interior vertex appears where two creases cross. A fold’s crease crosses whatever earlier creases happen to run through the layer it lands on, and that is a number between zero and several: a fold onto a fresh corner of the stack may cross nothing, while a fold across a well-worked region crosses a dozen. So the vertex count is a fact about where the folds went in a way the other three are not.

Empirically the vertices arrive at roughly three-quarters of the crease count and the ratio climbs slowly: 0.6 vertices per crease at four folds, 0.7 at eight, 0.8 at twelve. That climb is the sheet running out of unworked area — as the facets shrink, a new crease is more and more likely to land somewhere that already has creases in it.

What a hand does differently #

Every fold in this model goes through the entire stack, and a person crumpling paper does nothing of the sort. They crush a region; some layers buckle, others do not; the sheet takes a set of creases that are not all made along one line.

The identity is the place where that difference shows up as a number rather than as a caveat. Under a partial fold a crease is still recorded on the layer it was made on, so the facet count still rises by one — but the layer count rises only if that layer was actually cut, and a fold that catches three layers out of forty adds three layers and three creases while a fold through all forty adds forty of each. The lockstep survives; what changes is how fast the counts climb, and therefore how deep a real crumple gets for a given amount of work.

A hand is somewhere between the models and it is not between them cleanly. A crush is many folds at once along lines that are not straight, and the result is a sheet whose creases are curved and whose vertices are cones. Everything in this essay is a count of straight segments, which is a model of that and not a description of it.

Where the model stops #

Layers are counted and not ordered. The identity says how many pieces the stack is in. It says nothing about which is above which, and the ordering is where the hard part of this subject lives. A stack of 139 layers has an order, the all-layers fold determines it, and none of that is in the count.

Every fold takes the whole stack. A hand crumpling paper folds some layers and not others, and the model that folds only some would break the lockstep immediately: a crease made on one layer of many still adds one facet, but it adds one layer only if that layer was crossed, and a partial fold crosses fewer. The identity is therefore a statement about a particular machine, and it is the strongest of the three the site models.

The sheet has no thickness. A hundred and thirty-nine layers of eighty-gram paper is a centimetre of material, and folding it is not possible after about seven doublings — which is a result this site has already measured and which puts every count above twelve folds firmly in the realm of the geometric model rather than of anything a hand could produce.

A crease is drawn as a line. The facets of a real crumple are bounded by ridges with curvature in them, so the count of “facets” in a physical sheet is a judgement rather than a number, and where the judgement is made changes the answer.

The identity is a check as well as a fact #

Because three separate pieces of machinery produce the three counts, their agreeing is a test rather than a restatement — and it is a test that has caught things.

The facet count comes from a face walk over a planarised pattern, and a face walk is easy to get subtly wrong: a traversal that goes round the wrong way, or that merges two faces because a vertex was missed, still returns some set of cycles. The layer count comes from the fold simulation and would be wrong if a split ever produced a degenerate sliver that was quietly dropped. The crease count comes from the recorded segments and would be wrong if two collinear creases from different folds were merged into one.

Any one of those three failures breaks the identity, and none of them breaks the picture. So a figure that draws a crumple and prints its three counts is checking its own machinery every time it is built, which is the habit this site runs on applied to counting rather than to geometry.

What the identity is good for #

It converts a hard measurement into an easy one.

Counting the layers of a crumpled sheet means either taking it apart or looking at it edge-on and hoping, and neither is practical past a few dozen. Counting the facets of the flattened sheet is a matter of looking at it flat, which is exactly the state a crumple is easiest to inspect in. If the sheet was folded rather than crushed, the second count answers the first.

That is a small thing, and it is the shape of thing this subject is made of. The shrink factor of a corrugation is its average layer count, read the other way up; the number of ways a strip folds is a sum over its assignments; the depth of a crumpled stack is its facet count. In each case a quantity that seems to belong to the folded object turns out to be legible on the flat sheet, and in each case the reason is that folding is an isometry and loses nothing but position.

Where the ladder goes next #

The identity holds for the all-layers fold and the interesting question is which weaker machines keep it. A machine that folds a contiguous block of layers at the top of the stack — the some-layers model — adds fewer layers than it does creases, so the two counts must diverge, and by how much is a measurement nobody has taken.

The other direction is the ordering. Every crumple in this essay has a layer order, produced by the folding and thrown away by the counting, and the orders a random folding produces are a population nobody has looked at: how many of the theoretically possible stackings a random crumple actually visits, and whether the answer is a vanishing fraction, is a question about the same simulation asked one level deeper.