Box pleating — where it appears
Named by 22 essays across 6 fields — each of them below, with the objects they name alongside it.
Designing on a grid
Box pleating gives up the efficiency of a free packing and buys creases that land where they are supposed to. For a design with hundreds of folds that is not a compromise — it is the only thing that makes it foldable.
The crease has a radius
A fold does not go through a line. It goes round a small arc, and the arc uses more paper than the stack advances by — a fraction of a millimetre per crease, and several millimetres across a grid, which is why an ambitious tessellation comes out short.
Publishing the pattern instead of the sequence
A diagram sequence is one picture per step and a crease pattern is one picture. When designers began releasing patterns rather than diagrams, the cost of publishing a model fell by two orders of magnitude and the difficulty moved onto the reader — which is what made the complex era possible and what made most of it unfoldable.
What the grid settles
Box pleating is usually defended as a trade: give up efficiency, buy creases that land where they should. There is a second thing it buys and nobody quotes it — on a lattice the best possible packing is a finite question with an answer, while off the lattice nobody knows the best packing of six circles in a square and probably never will.
Spelling a tree on a grid
Box pleating asks every limb of a design to be a whole number of grid squares, which sounds like rounding and is not. Rounding each limb to its own nearest whole number is one way to choose the numbers, and at most resolutions it is not the best way — the best whole-number version of a subject is often a coarser one, with fewer squares and a shape twice as close.
The whole alphabet of a grid
Box pleating is defended as a trade — give up packing efficiency, buy creases that land where they should. There is a third thing it buys and it is much stronger than either: on a forty-five degree grid there are exactly six kinds of interior vertex a flat-foldable design can contain, ever. On a thirty degree grid there are thirty.
The second term
A feature grafted into a finished design costs exactly the paper slid in for it, and the bill has one term. Add a second feature across the first and it has two: the rectangle where the strips cross is paper both features are charged for and neither uses. At a strip a fifth of the sheet wide it is nine per cent of the bill; at four fifths it is nearly a third.
The sheet remembers
Perfect memory is the fourth idealisation, and the least examined of the four. It is usually read as a complaint that paper will not lie flat again; the large half is the opposite. A sheet folded once is no longer blank, so folding a second model into it is folding the union of two patterns — and a union folds flat only where every new crease meets every old one at a right angle.
A tie is not a decision
The crimp reduction decides a vertex by folding its smallest sector away, and where two sectors tie for smallest it has no forced move and must try each of them. That search is not rare — on the vertex at the centre of the first base anybody folds it happens for fourteen of the sixteen letterings — and it has never once changed the answer.
Which vertices are the random ones
Every measurement on this site that begins 'over 373 random degree-four vertices' is a statement about a population nobody declared. There is no canonical way to pick a crease pattern at random, four defensible ways of doing it disagree about the same three questions by factors rather than by margins, and the disagreement reaches a sentence this site has published as though it were general.
The other grid
Box pleating is drawn at forty-five degrees, and the twenty-two-and-a-half-degree grid is usually described as the same thing done finer. It is not a refinement, it is a different alphabet: five kinds of vertex become fifty-six, and the share of letterings whose decision needs a search falls from 60 per cent to 22. A finer grid is a larger vocabulary and a less ambiguous one.
What a grid costs in circuits
Box-pleating puts every crease on a square grid, and a square grid is the shape with the most short circuits per panel that this collection draws. On the sixteen-by-sixteen grid a designer actually works on, one mountain-valley labelling in a hundred agrees with itself. A search still finds one in two hundred and sixty-one steps.
A corrugation never backtracks
As a box-pleating grid goes from two divisions to sixteen, the share of random letterings that agree with themselves falls from a hundred in a hundred to one. The cost of finding one that does stays at exactly one step per panel — four, nine, sixteen, twenty-five, and two hundred and fifty-six — with not a single wrong guess anywhere in the family.
Ninety-nine in a hundred pass
A designer checks a box-pleated pattern the way every text teaches: vertex by vertex, counting mountains and valleys, watching the smallest sector. At sixteen divisions that check passes a hundred letterings in a hundred, and one of them folds. The check that separates them costs a single sweep over the crease list and is in no recipe anywhere.
The designer's grid is the dearest thing here
Two hundred and fifty-six panels of box-pleating grid take two hundred and fifty-six search steps to letter — exactly one per panel, at every size from two divisions to sixteen, with not one decision withdrawn. That is the most any pattern in this collection costs per panel of paper. A twist tessellation costs half of it, and a tilted corrugation a quarter.
A grid glued
Box pleating is the designer's grid: every crease on a line, every angle a right angle or forty-five degrees, and a whole design method built on the convenience of it. Roll the grid into a tube and half the column counts stop folding, on a pattern whose whole selling point is that it always works.
Which ceiling is binding
Two constraints hold a design's layer count down and both are ceilings on the same number. The stack gets better as the paper thins; the grid gets better as the sheet grows, because piling layers needs divisions and a division cannot be finer than a folder can place it. They cross at a sheet size that rises as the paper thins — so on the papers a classical folder had, the substrate really is the limit, and only at tissue weights does the hand take over.
A graft needs a square line
A strip can be slid into a finished crease pattern only along a line every crossed crease meets square, because only such a crease continues across the strip as itself. Tried on every line between the vertex columns and rows of eight printed patterns, four take no strip in either direction and three take one in a single direction. The eighth, the fold-and-cut triangle, appears to take strips both ways, and every line it admits runs through blank margin. No printed pattern takes a strip across its creases in both directions — and one diagonal crease in a plain grid removes exactly the row and the column it sits in.
Six rectangles and one term
Two grafted strips crossing leave one rectangle both features are charged for and neither uses. Three strips crossing two leave six, and the obvious budget adds them up. It does not have to: the six rectangles sum to the product of the two families' total widths, exactly, so a design with any number of features is priced by two numbers rather than by a double sum — and a family of strips in one direction alone carries no crossing term at all, however many of them there are.
A design that keeps its lines clear
A strip can be slid in only along a line every crossed crease meets square, so a diagonal crease spends the lines it crosses — and the census of eight printed patterns found four taking no strip in either direction. What decides how fast a design spends them is not how many diagonals it has but which rows and columns they sit in: six diagonals on a six-by-six grid leave twelve clear lines when they share a row and none at all when no two do, from the same six creases and the same amount of paper.
The width is charged in grid
A grafted strip may be any width at all and the bill in paper is exactly width times length, with no second term — which is what the first of these essays established, and is a statement about area. The grid is a different property of the same pattern and it is not conserved: a strip whose width is not a whole number of spacings puts every vertex past the cut onto a finer grid, and a width four per cent away from a spacing divides the grid by twenty-five where a width half a spacing away divides it by two. A width nearly right costs far more than one plainly wrong.
Rounding in the cheap direction
A tree spelled on a grid has every limb rounded to a whole number of units, and the rounding is chosen to keep the subject's proportions. Each rounding is also a small move of length between edges, and the edges have prices. Rounding the bird's dearer edges down and its cheaper ones up gives the largest model of every rounding tried, on grids of four, six and eight units — 2 to 6 per cent larger than rounding to the nearest unit, and larger than the unrounded bird itself on all three. The proportions pay for it, by five points of error on eight units and by thirty-four on six, which is the trade the grid had been making silently in whichever direction the arithmetic happened to fall.
Named alongside it
The objects these essays reach for when they reach for this one.
GridCrease patternDesign techniqueAssignmentGraftingSector anglesCrimpingDesign costIdealisationSearch costAreaThe big-little-big lemma