Origami & Folding
what one sheet can do
Who found it, and when

How many wedges the paper allows

A k-pointed folk star is folded into 2k equal wedges and cut once, so the scissors go through 2k thicknesses of the sheet. The geometry is indifferent to k and the paper is not: at a stack a pair of scissors will shear cleanly in one pass, ordinary copier paper takes a three-pointed star and nothing more, and the five-pointed one everybody knows needs washi or thinner.

Assumes The star that was cut before it was proved and The paper had to arrive first.

18 min read 5 figures Paper is not idealOne sheet, no cuts

The star that was cut before it was proved takes the traditional five-pointed star apart. Fold a sheet into ten wedges, make one straight cut, and a regular five-pointed star falls out; the trick is at least two centuries old and the theorem that any straight-line drawing can be released by one cut is of 1998.

The construction there is drawn for five points and it is not restricted to five. The generator will produce a three-pointed star, a seven-pointed one or a twelve-pointed one from the same argument: the wedge angle is π over the number of points, the cut angle fixes the inner radius by the sine rule, and the regularity comes out rather than being assumed.

So the geometry has nothing to say about which orders exist. The paper does.

How many wedges the paper allowsFor each order of folk star, the stack the scissors have to cut through — twice the number of points, in thicknesses of the sheet — and whether each paper allows it in one clean pass. The geometry of the construction says nothing about this: it will draw a star of any order. What stops the repertoire is the substrate, and where it stops is arithmetic on a measured thickness.the stack under a single cuta clean cut taken as 0.6 mm of stackstarnewsprintcopier paperkamiwashifoil-backed tissueunryu tissue3 points6 layers390 µm600 µm420 µm240 µm156 µm108 µm4 points8 layers520 µm800 µm560 µm320 µm208 µm144 µm5 points10 layers650 µm1.0 mm700 µm400 µm260 µm180 µm6 points12 layers780 µm1.2 mm840 µm480 µm312 µm216 µm8 points16 layers1.0 mm1.6 mm1.1 mm640 µm416 µm288 µm10 points20 layers1.3 mm2.0 mm1.4 mm800 µm520 µm360 µm12 points24 layers1.6 mm2.4 mm1.7 mm960 µm624 µm432 µm16 points32 layers2.1 mm3.2 mm2.2 mm1.3 mm832 µm576 µm20 points40 layers2.6 mm4.0 mm2.8 mm1.6 mm1.0 mm720 µma k-pointed star is folded into 2k wedges, so the scissors pass through 2k layers · a clean single cut is taken here as 0.6 mm of stack
Fig. 1 For each order of folk star, the stack the scissors have to cut through — twice the number of points, in thicknesses of the sheet — and whether each paper allows it in one clean pass. The boundary is between three points and four, and it moves with the paper rather than with the construction.

Twice the points

The arithmetic is one line and the whole rung follows from it.

A k-pointed star has k-fold rotational symmetry and a mirror through each point, so the sheet is folded into 2k congruent wedges — a sector of 2π⁄2k, with the paper reflected back and forth across the folds. The single cut passes through every one of them, which is what makes the copies exact rather than merely similar.

So the layer count under the scissors is 2k. A five-pointed star is ten layers; a six-pointed one is twelve; a twelve-pointed one is twenty-four.

The symmetry order and the stack are the same number doubled. That is not a rule of thumb and it does not depend on how the wedges are made: any construction producing a k-fold symmetric outline from one cut has to bring 2k thicknesses to the blade, because the cut has to serve every copy of the outline.

What a pair of scissors will take

A single clean cut through a stack is a physical limit and it is not a sharp one. Taking it as six tenths of a millimetre is a working figure for household scissors and a hand — enough for eight sheets of copier paper if they are pressed, fewer if they are not, and the cut degrades before it fails: the blade wanders, the layers slip against each other, and the outline comes out ragged rather than absent.

Set that against the papers this field measures and the boundary is startlingly low.

  • Copier paper, a hundred microns. Three points is six layers and six hundred microns, which just fits. Four points is eight hundred and does not.
  • Newsprint, sixty-five microns. Four points is five hundred and twenty and fits; five points is six hundred and fifty and does not.
  • Kami, seventy microns. Four points fits at five hundred and sixty; five does not.
  • Washi, forty microns. Six points fits at four hundred and eighty; eight points is six hundred and forty and does not.
  • Foil-backed tissue, twenty-six microns. Ten points fits at five hundred and twenty; twelve does not.
  • Unryu tissue, eighteen microns. Sixteen points fits at five hundred and seventy-six; twenty is seven hundred and twenty and does not.

The five-pointed star everybody knows is not cuttable in one pass in ordinary paper. It needs washi or thinner, or it needs a cut made in stages, or it needs the ragged version.

How many wedges the paper allowsFor each order of folk star, the stack the scissors have to cut through — twice the number of points, in thicknesses of the sheet — and whether each paper allows it in one clean pass. The geometry of the construction says nothing about this: it will draw a star of any order. What stops the repertoire is the substrate, and where it stops is arithmetic on a measured thickness.the stack under a single cuta clean cut taken as 0.9 mm of stackstarnewsprintcopier paperkamiwashifoil-backed tissueunryu tissue3 points6 layers390 µm600 µm420 µm240 µm156 µm108 µm4 points8 layers520 µm800 µm560 µm320 µm208 µm144 µm5 points10 layers650 µm1.0 mm700 µm400 µm260 µm180 µm6 points12 layers780 µm1.2 mm840 µm480 µm312 µm216 µm8 points16 layers1.0 mm1.6 mm1.1 mm640 µm416 µm288 µm12 points24 layers1.6 mm2.4 mm1.7 mm960 µm624 µm432 µm20 points40 layers2.6 mm4.0 mm2.8 mm1.6 mm1.0 mm720 µm30 points60 layers3.9 mm6.0 mm4.2 mm2.4 mm1.6 mm1.1 mma k-pointed star is folded into 2k wedges, so the scissors pass through 2k layers · a clean single cut is taken here as 0.9 mm of stack
Fig. 2 The same table with a more generous view of a clean cut, at nine tenths of a millimetre. Every paper gains an order or two and the shape of the boundary does not change — it moves, and the thinnest papers still reach several times as far as the thickest.

What the cut angle does, which is not this

The construction has a second parameter and it is worth separating from the point count, because it varies the star’s shape without touching the stack at all.

The wedge angle is fixed by the number of points — π over k, half of the 2π⁄k sector. The cut angle is free within a range, and it decides where the cut meets the wedge’s two edges: steeper cuts give a star with a narrow waist and shallower ones give a fat, nearly polygonal shape. The inner radius comes out of the sine rule as R·sin(cut − wedge) ⁄ sin(cut + wedge), and it is an output rather than a choice.

None of that changes the layer count. Every cut passes through all 2k wedges whatever angle it makes, so the whole family of star shapes at a given point count is equally expensive to cut.

Which means the substrate bound is a bound on the symmetry order alone, and a folder up against it has no shape adjustment available. The one thing that would help — a shallower cut, meeting less paper — does not, because the cut is through the stack rather than along it.

That is worth knowing because it is the sort of relief a craft usually has. Most material bounds can be traded against something: a thinner region, a different sequence, a slightly different shape. This one cannot, and the only variables are the paper and the tool.

What that says about the repertoire

The folk repertoire of one-cut stars is small and its members are small-numbered. Five points is the famous one; four and six appear; the higher orders are not folk objects at all.

Two explanations are available and they are testable against each other.

Aesthetic. Low-order stars are the ones people wanted. Five points is the shape of a star as drawn, six is the shape of a snowflake, and nobody wanted a sixteen-pointed one.

Material. Low-order stars are the ones the paper allowed. The construction is indifferent, the scissors are not, and the repertoire stops where the stack does.

The arithmetic above does not decide between them and it does something more useful: it says the second explanation is sufficient, which it was not obviously going to be. If the stack bound had allowed twenty points in ordinary paper, the material explanation would be dead and the aesthetic one would be all there is. It allows three, and the repertoire’s ceiling is inside the material’s.

That is as far as a computation can take it, and the field’s rule says so. Whether anybody in any tradition ever tried an eight-pointed star and gave up is a documentary question, and this collection’s record has nothing in it about scissors.

Fold into 10, cut onceThe traditional method, which is not the theorem. The sheet is folded into equal wedges about a point, one straight cut is made, and the shape that falls out has the symmetry the folding imposed. It gives a regular star for nothing and it gives nothing at all for a shape without that symmetry.10 layers, one cutwhat the fold decides5 points, 10 cornerscut at 54° to the foldwaist 0.309 of the pointregular, and checkedequal radii to 1e-12equal turning to 1e-12the symmetry is the method — a shape without it is not reachable thisway, and that is what 1998 changed
Fig. 3 The construction itself at five points, with the star it releases built from the fold geometry and checked for regularity rather than assumed. Nothing in it is harder at twelve points than at five, which is what makes the boundary a fact about the paper.

Why this is the loosest of the substrate bounds

This field now has three arguments from the substrate and they bind at very different places, which is worth setting out because it says which one is doing the work.

The layer bound on a folded model. A design with sixty-four layers at its thickest point needs paper thin enough for that stack to stay workable, which for copier paper means about thirty layers. That is a real limit on complex design and it is the ladder below this one.

The size bound. A model at a stated finished size needs a sheet larger by the square root of its layer count, which for deep designs is more paper than the largest standard sheet.

The cutting bound. A one-cut star of k points needs a stack of 2k thicknesses to shear cleanly, which for copier paper is six layers.

Six is a much smaller number than thirty, and the reason is that cutting is a harder operation than folding — a cut is surgery on the sheet where a fold is not. A fold has to bend a stack; a cut has to sever it, and the tool cannot be pressed against the fold line to help. So the cutting bound bites at about a fifth of the layer count the folding bound does, in the same paper.

That is worth stating because it inverts an expectation. The folk cut-outs look like the simple end of this subject and the complex models look like the demanding end, and on the substrate the cut-outs are the more demanding of the two by a factor of five.

How many wedges the paper allowsFor each order of folk star, the stack the scissors have to cut through — twice the number of points, in thicknesses of the sheet — and whether each paper allows it in one clean pass. The geometry of the construction says nothing about this: it will draw a star of any order. What stops the repertoire is the substrate, and where it stops is arithmetic on a measured thickness.the stack under a single cuta clean cut taken as 0.6 mm of stackstarcopier paperwashiunryu tissue3 points6 layers600 µm240 µm108 µm5 points10 layers1.0 mm400 µm180 µm6 points12 layers1.2 mm480 µm216 µm8 points16 layers1.6 mm640 µm288 µm12 points24 layers2.4 mm960 µm432 µm20 points40 layers4.0 mm1.6 mm720 µma k-pointed star is folded into 2k wedges, so the scissors pass through 2k layers · a clean single cut is taken here as 0.6 mm of stack
Fig. 4 Three papers spanning the range, at the orders a repertoire actually contains and two beyond it. The thickest reaches three points and the thinnest sixteen, and the five-pointed star sits between them — which is why it is the order a tradition with good paper would have and a tradition without it would not.

The theorem does not have this problem, which is the interesting part

One straight cut is the general result: any straight-line drawing whatever can be released from a folded sheet by a single cut. It is of 1998, it is constructive, and it is a much stronger statement than anything the folk method makes.

The folk method has a symmetry in it and the theorem does not. A wedge construction produces an outline with k-fold symmetry because the folding imposes it: the cut is copied 2k times and every copy is congruent. The general construction folds along the straight skeleton of the desired outline, and the outline need have no symmetry at all.

So the layer count is a different quantity in the two. In the folk method the layers are the symmetry order and there are 2k of them everywhere. In the general construction the layers are however many the skeleton produces at each point of the cut, which varies along the cut and depends on the shape.

That difference has a consequence for what the stack bound means. A five-pointed star by the folk method is ten layers, uniformly, along the whole cut. A five-pointed star by the general construction is a different number of layers at different places, and the maximum along the cut is what the scissors are up against.

Which of the two is easier to cut is not obvious and is computable, and this collection has the machinery for it — the fold-and-cut construction is built here from the straight skeleton with its perpendiculars and assignment verified. A layer census along the cut, for the same outline by both methods, would settle it, and it has not been made.

The reason it is worth making is that it would say something about why the folk method survived. If the general construction puts fewer layers under the blade for the same star, then the folk method is a worse way of doing something a theorem does better, and it survived because it is describable in a sentence. If it puts more, the folk method is materially superior for symmetric shapes and its survival needs no explanation at all.

A cut in stages is a different bound

The whole rung takes “one clean cut” as the operation, and that is the operation the folk method is described by — fold into wedges, one cut — so it is the right thing to price. It is worth noticing what happens when the constraint is relaxed, because the relaxation is available and cheap.

Cutting a thick stack in two passes halves the layers per pass if the stack is split, and does not if it is not. Splitting a folded wedge stack means unfolding part of it, cutting, and refolding — which loses the guarantee that every copy is congruent, since the second pass has to be aligned to the first by eye.

So cutting in stages trades a material bound for an accuracy one. The single cut is what makes the copies exact, and a folder who cuts twice has a star whose points are as similar as their alignment was.

That is the same trade this collection keeps meeting in another form: an operation performed on a folded stack is exact across all the copies because they are one piece of paper, and any operation that touches them separately is only as good as the registration. It is why the folk method’s regularity is a theorem and not a skill, and it is why relaxing the single cut is more expensive than it looks.

Fold into 12, cut onceThe traditional method, which is not the theorem. The sheet is folded into equal wedges about a point, one straight cut is made, and the shape that falls out has the symmetry the folding imposed. It gives a regular star for nothing and it gives nothing at all for a shape without that symmetry.12 layers, one cutwhat the fold decides6 points, 12 cornerscut at 50° to the foldwaist 0.347 of the pointregular, and checkedequal radii to 1e-12equal turning to 1e-12the symmetry is the method — a shape without it is not reachable thisway, and that is what 1998 changed
Fig. 5 The construction at six points rather than five, which is twelve layers rather than ten. Nothing about the geometry is harder — the inner radius still comes out of the sine rule and the regularity is still checked — and the two extra layers are the whole of the difference.

Which theorem was checked and how

The wedge count is checked against the point count on every row. A k-pointed star must come out at 2k wedges, and a row that did not would be a slip in the only piece of arithmetic the rung has.

Both extremes must be present. Some order must be cuttable in every paper compared and some in none of them, or the table has no boundary in it and the figure refuses to draw.

The boundary must be ordered. The largest order every paper takes must be below the smallest order no paper takes, which catches a sort that has gone wrong.

And the star’s regularity is computed rather than assumed in the construction figure: the ten vertices are built from the fold geometry and then checked to alternate between exactly two radii with equal turning at every corner.

Where the model stops

The shear figure is a working number and not a measurement. Six tenths of a millimetre is what a household pair of scissors and a hand will take through cleanly, stated rather than measured, and the essay computes the boundary as a function of it — so a reader who prefers a different figure can carry it through and the boundary moves accordingly.

Cutting degrades rather than failing. The model treats the bound as a threshold and it is a slope: a stack a little over produces a ragged outline, one well over produces a torn one, and there is no single layer count at which the operation stops.

Layers are taken as lying flat. A folded wedge of many layers is a wedge whose outer layers are longer than its inner ones, so the stack is not uniform and the scissors meet more paper near the fold line than near the edge — which makes the bound tighter than the arithmetic says.

And the papers are the same six this field measures for the folding bounds. They span the range and none of them is a measurement made here; what is computed is the shape of the boundary, which is the same shape at any thickness.

What the picture cannot show

The table gives a stack and a verdict and cannot show the cut. Whether a stack of ten layers of washi yields a crisp outline or a slightly furry one is a judgement the figure has no way to represent, and the difference between those two is exactly what decides whether an order is in a repertoire.

Nor can it show the technique. A folder who cuts in two passes, or who scores first, or who uses a knife against a board rather than scissors, is operating under a different bound entirely — and every one of those is available and none is in the model.

The most conspicuous absence is any evidence about what anybody did. This is a computation about what a material permits, offered as a sufficient explanation for a pattern in a repertoire, and the repertoire is described from the general literature rather than measured. A sufficient explanation is not a demonstration, and this field’s whole discipline is in keeping the two apart.

The idealisation, named

The sheet is uniform, the folds are exact, the wedges are congruent, and the cut is a straight line through a flat stack.

The one that matters most is congruence. A hand-folded 2k-wedge stack is not congruent — the folds accumulate error, and an error at a crease is folded too, so a systematic bias grows in proportion to the number of folds. At sixteen wedges a small bias at each fold is a visibly irregular star before the scissors are anywhere near it.

That is a second bound on the symmetry order, it acts on the same variable, and it acts in the same direction. A high-order folk star is limited both by what the scissors will take and by whether the wedges are still congruent when they get there, and the second is not computed here at all.

Which makes the boundary this rung draws a ceiling in one respect only. The paper allows three points in copier paper; whether a hand could fold sixteen congruent wedges in unryu tissue is a different question with, presumably, a lower answer.

Where the ladder goes next

The other famous piece of folk cutting is the connected cranes, and it has a combinatorial question in it rather than a material one.

The 1797 book slits a square into a grid and leaves the cranes attached at the interior lattice points, each of which holds four cranes at once. Which subsets of those points leave the piece in one connected object? Which cranes can stay joined counts them by exhaustion and finds almost none: on a three-by-three every join is load-bearing, and the share that work falls as the piece grows.

Sideways from here, the cutting bound belongs beside what one cut buys, which prices a cut in the design currency rather than the material one. That ladder asks what a cut is worth; this one asks what it costs to make; and a folk repertoire is the intersection of the two.

The habit worth carrying is a small one about symmetry. A symmetry order that is achieved by folding is a layer count, so any operation performed on the folded stack costs in proportion to the order — and a construction that is indifferent to the order is not indifferent when a tool has to get through the result.

Where this goes next

10 layers, one cutwhat the fold decides5 points, 10 cornerscut at 54° to the foldwaist 0.309 of the pointregular, and checkedequal radii to 1e-12equal turning to 1e-12the symmetry is the method — a shape without it is not reachable thisway, and that is what 1998 changed The star that was cut before it was proved Fold a sheet into ten wedges, make one straight cut, and a regular five-pointed star falls out. The trick is at least two centuries old and the theorem that any straight-line drawing can be released by one cut is of 1998 — because the traditional method is not the theorem, and works only on shapes with the symmetry the folding imposes. a fold stops working when the stack reaches 3 mm8 layers16 layers32 layers64 layers128 layersnewsprint65 µm520 µm1.0 mm2.1 mm4.2 mm8.3 mmcopier paper100 µm800 µm1.6 mm3.2 mm6.4 mm12.8 mmkami70 µm560 µm1.1 mm2.2 mm4.5 mm9.0 mmwashi40 µm320 µm640 µm1.3 mm2.6 mm5.1 mmfoil-backed tissue26 µm208 µm416 µm832 µm1.7 mm3.3 mmunryu tissue18 µm144 µm288 µm576 µm1.2 mm2.3 mmthickness measured across the sheet; the smallest feature is a folder's working figurerather than a constant of nature The paper had to arrive first A model with sixty-four layers at its thickest point, folded in ordinary copier paper, is six and a half millimetres of stack. The layer count a design can reach is fixed by the substrate, not by the folder — so the elaborate tradition is downstream of a manufacturing achievement with its own dates. the cut line3 straight edgesthe pattern3 skeleton arcs3 perpendiculars1 interior vertexassignments that fold30 of 646 creases in allarcs one way, perpendicularsthe other: fails Maekawaequidistance off by 1.9e-16mountainvalleyevery node sits the same distance from each edge that formed it,which is why one fold can carry several edges onto the line at once One straight cut Any drawing made of straight lines can be folded so that the whole drawing lands on a single line, and one cut releases it. The construction is a shrinking process, and it explains itself the moment the shrinking is drawn.

What this makes readable

Essays that name this one as a prerequisite.

Named alongside this one

Essays reaching for the same objects. Nobody chose these; they are what the concept index makes visible.

What links here

Every essay whose body links to this one.

The objects this essay names

Each one links to every other essay that touches it.

The fold-and-cut theoremKirigamiLayer countSubstrateSymmetryThickness