How much line is on the paper

Every description of a crease pattern in this subject counts creases. The vertex conditions count them — Maekawa is a difference of two counts — the enumerations count them, and the phrase “a pattern with thirty-eight creases” is how a pattern’s size gets stated.

A folder does not spend creases. They spend length: a crease has to be laid in along its whole extent, and a pattern with a few long creases is more work than a pattern with many short ones.

The two quantities are both properties of the same drawing, both are computed the same way from its coordinates, and they do not agree about which patterns are big.

The two orders #

The Yoshimura has the most folding on the shelf and it also has the most creases, so at the top of the table the two agree. Lower down they do not: the preliminary base has eight creases and more folding than the square twist’s twelve, and the Miura has thirty-eight creases against the tapered corrugation’s forty-five with eight millimetres less folding between them.

Two point four metres of crease on a sheet a hundred and seventy millimetres across. That is the number worth sitting with. A folder working a Yoshimura is putting in nearly two and a half metres of line, by hand, on a piece of paper the size of a postcard, and every millimetre of it has to land where the pattern says.

Where the count and the length part company #

The clearest disagreement on the shelf is the Miura against the square twist. Thirty-eight creases against twelve — a factor of three — and 1,049 millimetres against 704, a factor of one and a half. Counting creases says the Miura is three times the job; measuring them says it is half again.

The reason is what the creases are. A Miura of six columns and four rows is a grid of parallelograms whose creases are unbroken lines from one edge of the paper to the other — four full-width zigzags and six full-height ones — so it has few creases and each is a substantial fraction of the sheet. The square twist has short creases around a central polygon, so it has fewer of them and they are shorter still. The Yoshimura’s creases are courses and diagonals between junctions — laid down in segments rather than as one edge from side to side — so it has many and each is a fraction of the width, and it ends up at the top of both columns for two unrelated reasons.

This measurement was wrong for a long time and is corrected here, by a factor of six on the Miura: the sum of a pattern’s crease lengths is in the pattern’s own coordinates, and turning it into millimetres needs the pattern’s own width as well as the sheet’s printed size. A length needs a scale is the whole account of it.

That is why the count and the length disagree, and it is a real distinction rather than a bookkeeping one. Counting creases counts decisions: how many places the pattern says something about. Measuring them counts work.

How the length is measured #

The number comes off the drawing and nowhere else, and each step in getting it is a place a different number could have been produced.

The pattern’s own coordinates. Every crease is a segment between two vertices in the sheet’s units, and its length is the distance between them. A pattern is a unit square by convention here, so the sum is in sheet-widths; multiplying by the size the sheet is printed at gives millimetres.

The boundary is not crease. The four edges of the paper are marked B and are not folded, so they are excluded — which matters more than it sounds, since on a small pattern the outline is longer than everything inside it. The fold-and-cut triangle is 1.72 sheet-widths of crease inside an outline of four.

And a crease drawn as several segments counts once. The Yoshimura’s courses are laid down between successive junctions rather than as one edge from side to side, for reasons that have nothing to do with length — a course drawn as a single edge passes through every junction without being seen there, and every one of those vertices would be analysed as carrying four creases instead of six. Summing the segments gives the same total either way, which is the one thing about this measurement that is insensitive to how the pattern was built.

What subdivision costs #

Draw one tessellation on the same sheet at finer and finer subdivision and the total climbs in the way the cell count does.

Per cell, the length is nearly constant — the cell is the same shape at every subdivision, just smaller, and its perimeter shrinks in proportion while there are more of them. So the total is essentially the cell count, and a pattern with four times as many cells is four times the folding.

That is the sentence a designer needs and it is the opposite of what a picture suggests. A finely subdivided tessellation looks like a refinement, a detail added to something already there. It is not: it is the same pattern more times, and the hand pays proportionally.

The count and the length answer different questions #

It is worth being precise about what each of the two numbers is for, because both are correct and neither substitutes for the other.

The crease count is the size of the combinatorial object. It is the exponent in 2ⁿ when the letterings are enumerated, it is what the vertex conditions are stated over, and it is what decides whether a pattern can be searched at all — the enumerator refuses above twenty-two, and twenty-two is a count and not a length.

The length is the size of the physical job. It is what a hand travels, what a scoring tool traces, what a laser perforates. It has units, it scales with the sheet, and it is meaningless without the size the pattern is printed at.

So a pattern can be large in one sense and small in the other, and both of this shelf’s extremes are of that kind. The Yoshimura is the largest combinatorial object here — eighty-six creases, 2⁸⁶ letterings — and a middling job. The Miura is a modest combinatorial object at thirty-eight and the largest job by a factor of nearly three.

The one place the two must be compared is when a claim moves between them. “This pattern is twice the work” needs the length; “this pattern has twice the freedom” needs the count; and a sentence that starts with one and finishes with the other is the kind of slide this measurement exists to make visible.

What the length buys #

Length is a cost, so the interesting quantity is the exchange rate: how much folding is spent per unit of what folding is for.

The natural unit of what it is for is compaction — the sheet’s area divided by the footprint it packs into, which is the average number of layers over that footprint and is measured off the folded state rather than designed. Divide one by the other and the eight patterns come out in an order nothing like the first.

The Yoshimura converts crease into layers nearly three times better than the Miura does. Fourteen sheet-widths of folding for sixty layers, against six for nine.

That is not a criticism of the Miura, and the reason is worth stating because it is the whole design argument. A Miura is not trying to compact; it is trying to fold into a flat rectangle with one degree of freedom, and it spends its crease on being deployable rather than on being small. The Yoshimura folds into a line and has no motion worth the name. Compaction per crease and usefulness are different axes, and the exchange rate above measures one of them.

The ratio of the two numbers #

The two columns of the first table are a count and a length, and dividing the one into the other gives a third quantity that is neither: the average length of one crease. It is the exact factor by which the two orderings differ, and on this shelf it separates the patterns more cleanly than either column does on its own.

In sheet-widths, the preliminary base averages 0.60 of a sheet-width per crease, the square twist 0.39, the hexagon twist 0.34, the fold-and-cut triangle 0.29, the waterbomb 0.19, the Yoshimura 0.16, the Miura 0.16 and the tapered corrugation 0.15.

Read the top of that list against the sheet. The preliminary base’s average crease is more than half the width of the paper, which is what a pattern of four full diagonals and midlines has to look like, and it is why eight creases are more folding than the square twist’s twelve.

The shelf falls into two groups on this number. The classical base and the twists average between a third and just over a half of a sheet-width. The corrugations and the tessellations — including the Miura — average between a seventh and a fifth. Nothing sits between the groups, and the gap between them is a factor of about two.

That is worth having because it states the essay’s disagreement as one number per pattern instead of as a discrepancy between two tables. A pattern whose mean crease is long is a pattern whose count understates it; a pattern whose mean crease is short is one whose count overstates it. And the mean is only the first moment — a pattern with a few very long creases and a crowd of short ones has the same mean as one of uniform middling creases, and those are different evenings at the table. Which of the two a pattern is remains unmeasured here.

The millimetres are not scale-free and the ratios are #

One caution goes with quoting any of these figures, and it is the reason the table carries a paper size at all.

Crease length is a length. Print the same pattern at twice the width and every crease doubles, so the total doubles with it: the Yoshimura’s 2.38 metres at a hundred and seventy millimetres becomes 4.76 metres at three hundred and forty, and 1.19 metres at eighty-five. Nothing about the pattern has changed. The folding has.

The density and the mean crease length, quoted in sheet-widths instead of millimetres, do not move at all. They are ratios between two lengths on the same drawing, and a drawing has no size. So the sheet-width figures are properties of the pattern and the millimetre figures are properties of a decision about paper — and the exchange rate above is of the first kind, which is why it can be quoted without naming a sheet while the first table cannot.

That distinction is the practical half of the whole measurement. A folder comparing two patterns wants the scale-free numbers, because they say which pattern is more work per sheet whatever sheet is used. A folder deciding whether to attempt one wants the millimetres, and has to be told the size before the number means anything.

Where the density is highest #

Total length is one number for a whole sheet; the density is the same number per unit of paper, and since every pattern here is drawn on a unit square the two are the same figure read differently.

The printed shelf runs from 1.72 sheet-widths of crease per unit area to 14.3 — a factor of eight between the emptiest pattern and the busiest. For comparison, a plain sheet folded in half is 1.0, and the fold-and-cut triangle at 1.72 is not much busier than that.

That range is worth holding next to what the same patterns do to a hand at the table. A pattern at density 2 is a few creases and an afternoon. A pattern at density 14 is a grid that has to be scored before it can be collapsed at all, because the creases interfere: laying the seventy-sixth in by hand, on paper that already carries seventy-five, is not the same operation as laying the first.

Nothing here measures that interference, and it is the obvious place this anchor goes next. What the density gives is the number the interference would be a function of.

Three things this does not measure #

It is not a time. How long a pattern takes to fold depends on precreasing, on order, on collapse, and on whether the folder has done it before. Length is the extent of line that must be laid in and nothing more.

It is not a difficulty. The square twist has twelve creases and 704 mm, and a beginner will find its collapse harder than a Miura’s — which has ten times the folding. Difficulty lives in the collapse and in how many creases must move at once, and neither appears here.

And it is not a property of paper. No thickness, no fibre, no crease radius. Where a crease has a radius the length becomes an area and the number changes; that is a different measurement and this one is deliberately upstream of it.

The traditional patterns are the short ones #

One more reading of the first table, and it is the one that explains why the subject counts creases rather than measuring them.

The three shortest jobs on the shelf are the fold-and-cut triangle at 258 mm, the square twist at 704 and the preliminary base at 724. Two of those are traditional — the preliminary base is under the crane, the lily and half the classical repertoire — and the third is a construction. The three longest are all tessellations or corrugations, patterns that were designed after 1970 and mostly for engineering.

Crease length is the quantity that separates the two traditions, and it separates them by an order of magnitude. A classical base is under a metre of folding at the sizes people use. A corrugation is several. That is not a difference of ambition; it is a difference of what the pattern is trying to do, and it arrived in the subject with the machines and the mathematics rather than with the paper.

It also explains the notation. A dashed line and a solid line record which creases are mountains and which are valleys, and a folding sequence records the order they are made in — neither records how long any of them is, because on a crane it does not matter. On a pattern with seven metres of crease it is the first thing a folder wants to know, and no notation in the field reports it.

The anchor this opens #

Crease length is a new thing to measure on this site and it is worth saying what the ladder above it is for, since a first rung ought to be able to name the next few.

Where the length is, not only how much. Every number here is a total. A pattern can carry its length in a few long creases or in many short ones, and those are different jobs — the first wants a straightedge, the second wants a grid. That is a distribution rather than a sum, and nothing above computes it.

What the length costs in accuracy. A crease laid in by hand has an error, and a long crease accumulates it along its length. Exact is not accurate made that argument about a construction; the same argument about a pattern would be priced in crease length.

And whether the density has a ceiling. A sheet has a thickness, a crease has a radius, and creases closer together than a few radii are not separate creases. That gives a maximum density for a given paper, which would be the first result in this anchor that is about paper rather than about drawing.

None of the three is here. What is here is the measurement they would all be stated against, and the finding that it does not agree with the count the subject has always used instead.

What a folder should take from it #

Ask for the length, not the count. A pattern’s crease count is what its notation records and its length is what an evening costs. They differ by an order of magnitude in how they rank the same eight patterns.

Subdividing a tessellation is buying more of it. The per-cell length is fixed, so doubling the subdivision in each direction is four times the work, and the sheet has not changed size.

And the exchange rate is worth knowing before starting. A Yoshimura gives sixty layers for fourteen sheet-widths of crease. A Miura gives nine for six, and gives something else instead.