A disc as deep as the degree

A vertex creases the paper twice charged each crease for the ground near its ends where its band overlaps a neighbour’s. A crease is not a line on real paper but a band some six sheet thicknesses wide — about 0.6 mm on copier paper — and two bands meeting at an angle $\theta$ share the paper out to $w/\sin\theta$ along each. Add that reach at both ends of a crease and a crease no longer than the sum has no stretch of its own; on a 170 mm sheet of copier paper that happens to the Miura at 135 cells a side.

That argument measured the overlap as a length, along the creases. It left the obvious next question standing: what does the overlap look like as an area, how deep does it go, and how much of a sheet is it?

The answer has a part that surprised nobody who had drawn it and a part that is not visible from the length argument at all. The surprising part is that the deepest ground at a vertex does not depend on the vertex’s angles, only on its degree.

Every band contains the point #

A point of paper is creased as many times as there are bands it lies in, a band being the ground within half a width of a crease. At a vertex every crease starts at the point, so every band contains the point, and a small enough neighbourhood of a vertex of degree $d$ is creased $d$ times over.

How large is that neighbourhood? Take a point at distance $r$ from the vertex, in some direction $\varphi$. Its distance from a crease leaving at angle $\psi$ is $r\,|\sin(\varphi - \psi)|$ if the crease leaves within a right angle of $\varphi$, and $r$ itself if it leaves further round, because then the nearest point of that crease is the vertex. For the point to lie in every band it must be within $w/2$ of every crease. If $r \le w/2$ that is automatic. If $r > w/2$, every crease has to leave within a right angle of $\varphi$ — which puts all of them inside an open half-plane, and leaves a gap between two neighbouring creases wider than a half-turn.

A flat-foldable vertex never has such a gap. Kawasaki’s condition makes the alternate sectors sum to a half-turn each, so no single sector can exceed one. So at every flat-foldable vertex the ground creased as many times as the vertex has creases is exactly the disc of radius $w/2$:

$$A_{d} = \frac{\pi w^2}{4}$$

whatever the degree, whatever the sectors. The sampling agrees, to its own resolution, on both vertices drawn: 0.786 band widths squared at the Miura’s degree-four vertex, 0.787 at the preliminary base’s degree-eight centre, against $\pi/4 = 0.785$.

That has a consequence which is easy to state and easy to miss. A preliminary base’s centre is creased eight times over a disc the same size as the Miura’s four-times disc. The degree sets how deep the deepest ground is and has no say in how wide it is.

What the angles decide #

The angles decide everything around the disc. Between two creases at a sector $\theta$ below a right angle, the two bands run on together for $w/\sin\theta$ beyond the point, and the ground between them is creased twice all the way out. At or above a right angle the bands part at the disc’s rim.

The printed patterns carry eight kinds of interior vertex, and every one was sampled on its own, with its creases extended well past anything its overlap reaches. Every one has the same deepest disc — 0.782 to 0.787 of a band width squared, the spread being the sampling. The doubled ground round it is what differs.

At a right-angled vertex of degree four, the square twist’s and the waterbomb’s, the ground creased at least twice is 0.99 band widths squared: the disc plus four small corner pieces where each pair of neighbouring bands overlaps before parting. The Miura’s vertex, with sectors of 69.9 and 110.1 degrees at a slant of 0.35, carries 1.05; the tapered corrugation’s, at 65.9, carries 1.09; the hexagon twist’s, at 60, carries 1.14. At degree six the Yoshimura’s 60-degree sectors carry 1.74 and the waterbomb’s 45-degree ones 2.02. The preliminary base’s centre, eight creases at 45 degrees, carries 2.86 — nearly three times the right-angled vertex.

So the angles cost width and the degree costs depth, and a vertex with many creases at narrow angles pays both: the base’s centre is eight deep over its disc and at least two deep over three times the area a square grid’s vertex creases twice. That is the quantitative form of a thing every folder of a traditional base notices, that its centre is where the paper gives out first.

The centre of a preliminary base, in millimetres #

The preliminary base is the pattern this argument was always going to arrive at, because it is where the folding length sits most heavily: a traditional base carries five times its share of crease length in the middle few per cent of the sheet, and all of that length meets at one point.

On copier paper the base’s centre is creased eight times over a disc 0.6 mm across — 0.28 mm² — and at least twice over a patch of 1.03 mm², shaped as an eight-pointed star whose points run 0.85 mm out along each crease, which is $w/\sin 45^\circ$. The star is the whole of the base’s doubled ground, since its other creases meet only the rim. Every other square millimetre of a 150 mm sheet is creased once or not at all.

Put that way the centre’s trouble is not its area but its depth. One square millimetre in twenty-two thousand is nothing; eight creases stacked on a disc the width of a pencil line is a great deal. The sheet has a thickness is the account of what happens when real layers pile up at a fold, and the centre of a base is the extreme case of it, reached at the very first vertex most people ever fold. That a model’s centre is where the paper wears through, bulges or refuses to lie flat is folding lore; the disc says that none of the base’s angles could have spread the damage, since the deepest ground is the same disc at any angles whatever, and only a lower degree would have made it shallower.

That also marks the difference from the bound the length argument drew. The shortest crease is not a crease found creases on printed patches shorter than a wavelength of light, which is a statement about a pattern’s drawing. The preliminary base has no short creases at all — every one runs from the centre to the rim, 75 mm or more — and it is still the most deeply creased point on the shelf. Depth is not a property of length.

The printed shelf, creased twice almost nowhere #

At the sizes the patterns are printed, none of this is visible.

On copier paper a band is 0.6 mm wide, so a band width squared is 0.36 mm². The Yoshimura pattern’s twenty-two interior vertices crease 13.8 mm² at least twice, the waterbomb tessellation’s twenty-five crease 12.3, the Miura’s fifteen crease 5.7, and the preliminary base’s single centre creases one square millimetre. The largest share of any printed sheet creased twice is 0.066 per cent.

Counting each vertex on its own is right here and the shelf is the regime it is right in. At printed size the nearest two vertices on any pattern are tens of band widths apart, so their doubled ground never meets and the sheet’s total is the sum of its vertices’. The total is small because a sheet printed at 170 mm has few vertices and a band is small against the space between them — nothing about any single vertex is small.

That is the reason the length argument found its bound so far from printed practice. The density a paper allows put the densest printed pattern a factor of nineteen below the paper’s limit, and the doubled ground says the same thing in area: a pattern is fine enough to notice its vertices only when there are many of them in each square centimetre.

The share grows as the square of the fineness #

The doubled ground at each vertex is a fixed area in band widths squared, and a Miura has one interior vertex a cell. So while the bands are thin against a cell, the share of the sheet creased twice is that fixed area over the cell’s area:

$$\text{share} \approx a_2 \left(\frac{w}{\ell}\right)^2$$

with $\ell$ the cell’s side and $a_2$ about 1.05 at the Miura’s vertex. Halve the cell and the share quadruples. That is not how a crease pattern’s other costs grow: its total crease length per unit area, which how much line is on the paper measured, grows only as the first power of the fineness, since a finer pattern has more creases but each is shorter.

The sampled sheet shows the square law, and shows it holding further than it had any right to. A block of interior cells is sampled with every crease near it, so neighbouring vertices’ doubled ground merges where it would on the paper rather than being added twice. At sixteen cells a side the share creased twice is 0.37 per cent; at thirty-two, 1.46; at sixty-four, 5.3 — each doubling close to a factor of four. The ratio of share to $(w/\ell)^2$ stays between 1.03 and 1.12 all the way to two hundred cells, because even there a Miura’s vertices are more than a band apart and their fringes barely touch. What does turn over is the ground creased exactly once, which peaks near half the sheet and then falls as the doubled ground takes it.

At 135 cells a side, the size where the length argument said a crease first becomes all overlap, 24.3 per cent of the sheet is creased at least twice and about 18 per cent four times over. By two hundred cells, past the bound, more of the sheet is creased twice than once.

That is what the length bound was measuring from the other side. A crease that is overlap from end to end is a crease whose own ground has been used up by its vertices, and the area picture says how much of the rest of the sheet has gone the same way by then: a quarter. The bound is not a line crossed by one unlucky crease on an otherwise sound sheet. It arrives when the sheet as a whole is a quarter doubled.

The same quarter on every paper #

The last figure is the one that looks like a coincidence and is not.

Thinner paper has narrower bands and carries a finer Miura before any crease is overlap from end to end: 135 cells a side on copier paper, 193 on kami, 337 on washi, 519 on foil-backed tissue. The cells shrink from 1.26 mm to 0.33. At each paper’s own finest Miura, 24.3 per cent of the sheet is creased at least twice — the same number, to the digit, on all four.

The reason is that the length bound is a statement about a ratio, band width to cell size, and the doubled share is a function of the same ratio and nothing else. The paper sets the band; the bound sets the cell in proportion to it; the doubled ground is then fixed. A thinner paper does not make the finest pattern any less doubled. It makes it smaller.

That turns a practical question into a clean one. A folder choosing paper for a fine tessellation is choosing a scale, not a margin. Every paper runs out of room at the same proportion of band to cell, and at that proportion every paper’s sheet is a quarter creased twice; what differs between papers is only how many cells that proportion allows on a sheet of a given size.

What a band assumes #

A band is a strip of fixed width with sharp edges. The paper inside it is creased and the paper outside it is not. The crease has a radius measured what a real fold looks like end on — straight, a curved arc, straight again — and the width here is a stand-in for that arc’s footprint, six sheet thicknesses by the convention these essays have used. A real crease’s disturbance falls off rather than stopping, so the areas here are the areas inside a stated width, not the areas where anything physical changes.

Depth is a count, not a thickness. A point in four bands is ground four creases have each tried to bend. Whether real paper tolerates that — whether a Miura vertex on copier paper at 1.3 mm cells is a crisp point, a small dome or a tear — is a question about fibres, and nothing measured here answers it. The figures say how much of the sheet is asked to do it.

The vertices are the pattern’s own. Each vertex kind is sampled with its creases extended in straight lines to twelve half-band widths; on the printed patterns the real creases are much longer than that, so the isolated reading is exact for them. On the fine Miura the block sampling is used instead, and it is the only place neighbours matter.

And the disc result needs a flat-foldable vertex. A point where creases meet with a gap wider than a half-turn between two of them — a crease that bends, three pleats ending together on one side — has deepest ground larger than the disc, reaching out along the side where its creases crowd together: two creases at sixty degrees are creased twice over 0.96 band widths squared of common ground, three at sixty degrees apart three times over 0.80. No interior vertex of a pattern that folds flat can have such a gap.

Width and depth are different costs #

The unexpected connection in all of this is between two things that seem to have nothing to do with each other: Kawasaki’s condition, which is about whether a vertex folds flat at all, and the shape of the ground a real crease damages. The condition says no sector exceeds a half-turn; the band geometry turns that into the deepest ground is a disc. A theorem about angles fixes a fact about material.

It also separates two costs a pattern pays at its vertices that the length argument had to lump together. Depth is set by the degree alone — four at a Miura vertex, six at a Yoshimura’s, eight at a preliminary base’s centre — and it is paid over a disc whose size nothing about the pattern can change. Width is set by the angles — the doubled fringe grows with every narrow sector — and it is paid over ground the pattern could have spared by opening its sectors. The smallest sector decides whether a vertex folds; here the smallest sector decides how much paper round it is creased twice, and the two roles are not the same.

A designer who wants a fine pattern on a given paper therefore has two separate levers. Lower degree buys shallower damage and more of it survives; wider sectors buy narrower damage. The Miura is good at the second — 1.05 band widths squared against the waterbomb’s 2.02 — and fixed at degree four for the first, which is the lowest degree a genuine vertex can fold flat at.

Still open: whether the sheet or the paper gives out first #

The quarter is a statement about geometry and it invites a statement about material that has not been made. Is there a share of doubled ground past which a real sheet stops holding its creases? A crease on real paper is a zone of broken and rearranged fibres, and a sheet a quarter of which has been broken twice is a different material from the one it started as. If there is such a share, it would be a bound on fineness that is independent of the length argument and could bind before it — and it could be measured by folding a graded series of Miuras on one paper and recording where they stop holding a crisp fold.

The geometric half has a second question in it. What is the fringe area as a function of the sectors? The table reads it off samples for eight vertex kinds, and it should have a closed form in the cotangents of the half-sectors, from which the degree-eight vertex’s 2.86 and the right-angled vertex’s 0.99 would both follow. With that form, the share of a sheet doubled at any fineness would be a sum over a pattern’s vertex kinds, readable before anything is folded.

Sideways from here, a paper limits spacing, not density found that crossing families of creases can be packed nearly twice as densely as parallel ones because they meet at shared ground; this essay measures that shared ground, and a pattern whose vertices share ground very efficiently — many creases, wide sectors — would be the one to test that bound’s other limit against.

The habit worth carrying is about which quantity a condition constrains. When a theorem restricts angles, ask what else is shaped by those angles before assuming it is only about angles. Kawasaki was proved to decide flat-foldability, and it turns out also to fix the shape of the most damaged paper at every vertex that satisfies it.