Where the length sits

A pattern’s crease count is what its notation records and its length is what an evening costs, and the two rank the printed patterns in different orders — the preliminary base has eight creases and 724 mm of folding, the square twist twelve creases and 704 mm.

Length is still a total, and a total says nothing about where the work is. A metre of creasing spread evenly over a sheet and a metre concentrated into a square centimetre are the same number and not the same afternoon.

Why bands, and why from the edge #

Three choices were made before any number came out, and each of them could have been made differently.

Distance from the sheet’s edge is the coordinate, rather than distance from the centre. On a square the two are not the same — the corner is further from the centre than the middle of a side is, and equally far from the edge — and the edge is the right one here because it is what the paper has. A crease knows where the paper stops; it does not know where the middle is.

Five bands is the coarsest division that separates the shapes of answer below. Four merges the two central bands on the traditional bases, which is where their whole signature is; eight starts to show the lattice of the grids rather than the design, and the numbers become a report on where the rows happened to fall.

Length rather than count is the quantity binned, because length is what an evening costs and because a count cannot be divided between bands at all: a crease that runs from the rim to the middle belongs to every band it crosses, in the proportion of its length that lies in each.

The bands are not equal #

Cut a square into five bands by distance from its edge, each a fifth of the half-width deep. The bands are equal in depth and nothing like equal in area: the outermost holds 36% of the paper, then 28%, 20%, 12%, and the innermost holds 4%.

That is the whole reason the measurement needs stating carefully. A pattern with a fifth of its creasing in each band sounds even and is not: a fifth of the length in 4% of the paper is five times the density of a fifth of it in 36%. So each band’s share of the length is divided by its share of the paper, and the figure reports that ratio. One means the folding is spread as the paper is.

Three shapes of answer #

The eight profiles fall into three groups, and the groups are the three things a crease pattern can be.

A base concentrates at a point. The preliminary base runs 0.55, 0.72, 0.99, 1.68 and 4.95 from the rim inward: nearly five times its share of the folding in the middle twenty-fifth of the sheet. That is what a base is — four lines through the centre of a square — and it is why folding one is a fight in the middle and easy at the edges.

A fold-and-cut pattern is worse. The triangle runs 0.00, 0.20, 1.31, 3.43, 6.80. There is no folding at all in the outer third of the sheet, and nearly seven times its share in the middle. The skeleton lives where the shape is, and the shape is drawn in the middle of the paper because that is where a shape fits.

A tessellation is flat. The waterbomb runs 0.89, 0.93, 1.26, 0.95, 1.35 — every band within about a third of an even spread, which is exactly what “a material made of creases” should look like. The Yoshimura is nearly as flat at 0.79 to 1.78.

And a twist has a hole in it. The hexagon twist runs 0.60, 0.87, 1.24, 2.44 and then 0.00: the innermost band carries no crease at all, because the middle of a twist unit is the twist polygon, and the inside of that polygon is uncreased paper. The square twist does the same thing less sharply, at 0.95, because its polygon is smaller relative to the sheet.

The band areas are the odd numbers #

The four areas quoted above are not measurements; they are exact, and the exact form says something about the instrument.

A square of half-width 1 with the outer $d$ removed is a square of half-width $1-d$, so the paper inside band $j$ of $n$ is $(1 - j/n)^2$ of the whole. Differencing gives band $j$ a share of

$$\frac{2(n-j)+1}{n^2}.$$

At $n = 5$ that is $9, 7, 5, 3, 1$ over $25$ — the odd numbers, which is the schoolroom decomposition of a square read from the outside in. 36%, 28%, 20%, 12%, 4%, as quoted.

Two consequences follow with no measurement at all. The innermost band is always $1/n^2$ of the sheet, so its area share falls quadratically as the division is refined and the ratios reported there become correspondingly volatile — which is the real reason eight bands “start to report where the rows fell”. And the ratio scale has a ceiling of $n^2$: a pattern with every crease in the innermost band would score 25 and no pattern can score more.

That calibrates numbers the essay otherwise leaves floating. The fold-and-cut triangle’s 6.80 is 27% of the most a five-band profile can report, and the preliminary base’s 4.95 is 20%. They are large and they are nowhere near the top of the scale.

The five numbers are four #

There is also an identity, and it is worth checking rather than asserting, because it is the test these profiles could fail.

Each band’s ratio is its share of the length divided by its share of the paper, and the length shares sum to one. So the ratios, weighted by the band areas, must sum to exactly one.

The preliminary base: $0.36(0.55) + 0.28(0.72) + 0.20(0.99) + 0.12(1.68) + 0.04(4.95) = 0.997$. The fold-and-cut triangle: $1.002$. The waterbomb: $1.001$. Three patterns of three different kinds, agreeing to a third of a per cent — which is rounding in the published figures and nothing else.

So a profile is not five independent readings. It is four, and the fifth is arithmetic.

Which turns the rim result into a forced consequence rather than a coincidence. The outer band carries 36% of the weight in that sum, so a pattern scoring 0.63 there has spent only 0.227 of its budget on more than a third of the paper and must find the remaining 0.773 inside. A starved rim does not merely accompany a peaked middle; given the identity, it requires one. The finding is that every printed pattern starves its rim, and the arithmetic says the peak is then not a second observation.

The rim is starved on every one of them #

One number is the same across all eight patterns and it is the one this collection keeps finding.

The outermost band — 36% of the paper — carries less than its share of the folding in every printed pattern. The values run from 0.00 on the fold-and-cut triangle to 0.89 on the waterbomb, and the median is 0.63. Not one pattern reaches an even spread at the rim.

That is a third measurement of the same asymmetry. A panel carrying a raw edge lies over fewer of the others — the rim does not stack. The share of a patch’s units that the rim cuts never falls below a third at drawable sizes — the rim is most of a small pattern. And now: the rim carries less creasing than its area, on everything printed here.

The three are not the same fact and they point the same way. A crease pattern is a thing that happens in the middle of a sheet, and the outer third of the paper is mostly there to be held.

The flattest profiles in the collection #

The patterns that come closest to an even spread are not on the printed shelf at all. They are the tessellation patches, and they are flat for the reason a tessellation is supposed to be.

A square-grid twist patch runs 1.29, 0.48, 1.34, 0.88, 0.69 — lumpy, because nine units on a sheet is a coarse lattice and the bands cut across them. The triangular patch runs 1.05, 0.94, 1.01, 1.06, 0.81; the honeycomb 1.02, 0.91, 1.14, 0.84, 1.20; the rhombille 1.06, 0.97, 0.92, 1.08, 0.86. Everything between 0.81 and 1.20, which is flatter than any printed pattern manages.

And the rim behaves differently here, in a way that says something about the repair. Assembled unit by unit, the triangular patch’s rim band carries 0.94 of its share and the rhombille’s 0.84. Cut out of the plane instead, the same two carry 1.05 and 1.06 — because clipping draws the units at the edge as far as the paper goes rather than omitting them, and the creases it adds are all at the rim.

So the repair does something the folding-length total also reports — the triangular patch gains forty creases and 1.2 sheet widths — and the profile says where: entirely in the outermost band, which was the one part of the pattern that was short.

Why it is not simply the boundary effect #

A pattern’s creases have to end somewhere, and near the rim there is less room, so a sceptic could put the whole profile down to geometry rather than to design. Two of the eight say otherwise.

The Miura runs 0.43, 1.41, 0.76, 1.93, 1.68 — up and down rather than up. Its profile is a lattice artefact: the bands cut across rows of cells, so a band that happens to contain two rows of creases scores twice what a band containing one does. Nothing about the rim explains a value of 1.41 next to one of 0.76.

The tapered corrugation runs 0.61, 1.24, 1.40, 0.73, 1.63, and the dip at the fourth band is the taper doing exactly what it was designed to do: the columns are narrow at the ends and broad in the middle, so the creasing is deliberately unevenly distributed and the profile reports it.

So the profile is reading the design where there is one and the geometry where there is not. That is what makes it worth having: a pattern’s profile is a signature of the construction rather than of the sheet.

What this is not #

It is a radial profile, and a radial profile is one number per band. That is a strong compression of a two-dimensional thing, and it is worth being explicit about what it throws away.

It cannot see an asymmetry. A pattern with all its creasing along the left-hand edge and a pattern with the same length spread round all four rims give the same first band. Every printed pattern here is roughly symmetric, so the compression is nearly free; on a design with a hinge along one side it would not be.

It cannot see a gap. A band with a ratio of one might be evenly creased or might be half dense and half bare. The hexagon twist’s empty middle shows up only because the empty part happens to coincide with the innermost band.

And it is not a map. The layer map this collection computes for a folded state is a genuine two-dimensional picture of where the paper piles up, and it is a different quantity: layers are about the folded object and creases are about the flat sheet. The two are related and not the same — a band with little creasing can still carry many layers, because the layers arrived from somewhere else.

What a folder gets out of it #

The profile predicts three things a total cannot, and all three are things a reader meets in the first ten minutes.

Where the fold gets hard. Layers accumulate where creases are dense, so the band with the highest ratio is where the paper is thickest and least willing. On a base that is the centre, on a fold-and-cut pattern the centre, on a tessellation nowhere in particular.

Where accuracy matters. An error is folded too and a misplaced crease carries its error into every layer above it. In a band carrying five times its share of the folding, five times as much of the pattern depends on each millimetre.

And what to do first. A pattern with a flat profile can be folded in any order — the collapse is the same everywhere. A pattern with a sharp central peak has to be assembled from the middle outward, which is exactly how a traditional base is taught and why the teaching survives without the measurement.

Two patterns with the same total #

The clearest demonstration is a pair. The Yoshimura and the waterbomb tessellation have total folding lengths within two per cent of each other at the sizes they are printed. Their profiles run 0.79/1.01/1.09/1.22/1.78 and 0.89/0.93/1.26/0.95/1.35 — similar, which is the point: two tessellations with the same total and the same distribution really are the same kind of job.

Now put the preliminary base beside them. Its total is a fraction of theirs — twelve creases against ninety — and its central band carries 4.95 against their 1.78 and 1.35. A pattern with a twentieth of the folding demands three times the density where it matters.

That is the sense in which a total is misleading. It is a good measure of how long a pattern takes and a poor one of how hard it is, and the second is what a reader asks about a pattern they have not folded before.

What the densest band costs in paper #

There is a physical limit under all of this and the profile is the right instrument for finding it.

A crease has a radius: the paper turns through a region a few tenths of a millimetre wide rather than at a line, and the region takes up paper. So a band carrying five times its share of the folding is a band where five times as much of the paper has been spent on turning, and there is a density at which a band has no flat paper left in it at all.

For the preliminary base at 150 mm, the innermost band is a square about 15 mm across carrying about 950 mm of crease — which at a crease radius of a tenth of a millimetre is around 190 mm² of turned paper in a band of 225 mm². That is most of it, and it is why the centre of a collapsed base is a knot rather than a stack.

The same arithmetic on the waterbomb tessellation’s innermost band gives a fraction under a fifth. The tessellation could be drawn several times finer before its densest band ran out of paper; the base could not be drawn finer at all.

The two orders, again #

Counting a pattern’s creases and measuring them rank the printed shelf differently, which was the finding this ladder started from. Adding the profile gives a third ordering, and it agrees with neither.

By count, the shelf runs from the preliminary base’s twelve creases up to the waterbomb’s ninety-two. By length, from the fold-and-cut triangle’s 258 mm to the Yoshimura’s 2,380. By peak density — the highest ratio any of its bands reaches — it runs from the waterbomb’s 1.35 up to the fold-and-cut triangle’s 6.80, which puts the shortest pattern on the shelf at the top.

Three measures, three orders, one shelf. That is not a defect of any of them; it is what happens when a single word — how much folding — is doing the work of three different questions. How long it takes is length. How many decisions it contains is count. How hard the worst part is, is peak density.

A reader choosing a pattern to fold on a wet afternoon wants the third, and it is the one no notation records.

Where the measurement is thin #

Bands are a crude shape. Distance from the edge of a square is a reasonable coordinate for these patterns and a poor one for a pattern with structure that is not concentric — a Miura’s profile is mostly an artefact of which rows fell in which band, and a finer division would move the numbers around without changing the shape.

The rim band is the largest and the least interesting. Thirty-six per cent of the paper is a lot to average over, and a pattern with all its rim creasing along one edge reads the same as one with it spread round all four.

The sampling is a hundred points per unit length, which is fine for a crease running across a band and crude for one lying along a band boundary — such a crease’s length is split between two bands by where its samples happen to fall. No crease in these patterns runs along a boundary, because the boundaries are set by the sheet and the creases are not, but a pattern designed on concentric squares would need a different division.

And it says nothing about direction. Two creases crossing a band at right angles to each other are the same length as two parallel ones, and a folder’s hands know the difference. Which way the creases run is a separate quantity and one this collection has not measured.