Origami & Folding
what one sheet can do
Concept

Grid — where it appears

A lattice a design's creases are restricted to lie on. It costs efficiency and buys accuracy, and it restricts the vertices a pattern can contain to a catalogue short enough to write out in full.

Named by 21 essays across 7 fields — each of them below, with the objects they name alongside it.

3 equal partsestimated by eyethe left-hand divisions are exact — a consequence of the fold, not of carethe right-hand ones are a guess, and the error compoundsvalleymountain

Dividing without measuring

A square can be divided into any whole number of equal parts by folding alone — exactly, with no ruler, and with no error to accumulate. The construction is one fold and a theorem nobody expected.

construction · Exact division 32 × 32 gridevery crease on a grid line, or at 45°which is why a 64-grid design can be folded at allmountainvalley

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.

design · Box pleating ρ = 0.12 mmarc 0.377 mm, stack advances 0.240 mmlost per crease (π − 2)ρ = 0.1370 mmone crease, at its real radiuson a 150 mm sheet8 × 81.0 mm — 0.6%16 × 162.1 mm — 1.4%24 × 243.2 mm — 2.1%32 × 324.2 mm — 2.8%48 × 486.4 mm — 4.3%lost along every line of the grid, in both directionswhich is why an ambitious grid is folded from thin paper

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.

material · Idealisation flapsfree search4×4 grid8×8 grid30.25430.2500 −1.7%0.2500 −1.7%40.25000.2500 −0.0%0.2500 −0.0%50.20710.1768 −14.6%0.1768 −14.6%60.18760.1250 −33.4%0.1398 −25.5%the 6-flap case on the finest lattice here is one of 3.25e+8 arrangements, and the bound settles all of themthe free optimum is unknown for most of these counts and the lattice optimum is known for all of thema finer lattice costs less and asks for more creases, which is the trade a designer actually makes

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.

design · Box pleating 42:3:4:363:4:6:583:4:6:5125:7:11:8165:7:11:8248:11:17:13each limb roundedthe best whole numbershow wrong the worst limb isgrid units across the longest limbthe numbers under the axis are the best whole-number limbs at that resolution

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.

design · Box pleating every flat-foldable vertex whose sectors are multiples of 45°6 of them, to degree 8passfoldone objectthe most45·45·135·135866145·90·135·90444190·90·90·90888145·45·45·45·90·90302012245·45·90·45·45·90301812645·45·45·45·45·45·45·451121121643 of the 6 carry markings the conditions accept and the paper refuses

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.

design · Box pleating sectorsletterings that branchdecided by the choice22.5° 22.5° 157.5° 157.5°4/16022.5° 45° 157.5° 135°0/16022.5° 67.5° 157.5° 112.5°0/16022.5° 90° 157.5° 90°0/16045° 45° 135° 135°4/16045° 67.5° 135° 112.5°0/16045° 90° 135° 90°0/16067.5° 67.5° 112.5° 112.5°4/16067.5° 90° 112.5° 90°0/16090° 90° 90° 90°14/160

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.

design · Box pleating the four cheap tests are polynomial in the drawing; the fifth is notreading across a row is one pattern put to all fivecrease pairsverticespanelscreasessearch nodesthe square twist6649127,565the Miura fold703152438refusedthe waterbomb sheet2,850255276refusedthe Yoshimura3,655226586refuseda square patch3,486364984refuseda rhombille patch39,621126157282refuseda refused search is a pattern about which the expensive test says nothing at all, at full price

The cost is in the coincidences

How big an instance is, is what a hardness statement is about, and it is the weaker predictor of what deciding one costs. Hold the degree fixed and vary only how many of a vertex's sectors are equal: the work of deciding it rises by a factor of nearly three, against a factor of two for doubling the number of creases. The expensive instances are the ones a designer draws on a grid.

complexity · Hardness of folding the bar is how many of a hundred random letterings agree with themselveson the orthogonal grid a box-pleated design is drawn on, at five sizes4 by 4949 interior vertices · 16 panels · found in 16 nodes6 by 67325 interior vertices · 36 panels · found in 37 nodes8 by 85849 interior vertices · 64 panels · found in 65 nodes10 by 103681 interior vertices · 100 panels · found in 100 nodes12 by 1215121 interior vertices · 144 panels · found in 145 nodes16 by 161225 interior vertices · 256 panels · found in 261 nodesevery interior vertex is a four-panel circuit, so the number of places a contradiction could sit is the number of vertices

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.

design · Box pleating each point is one pattern: panels across, nodes up00100100200200one node per panelnodes visitedpanels2 by 2 to 16 by 16, and not one backtrack anywhere in the family

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.

tessellation · Miura 16 × 16 gridevery crease on a grid line, or at 45°which is why a 64-grid design can be folded at allmountainvalley

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.

design · Box pleating labellings a vertex keeps, against nodes a panel costs0.000.250.500.751.00481530the box-pleating gridthe tapered leafa crumple, deepeningthe waterbombthe Yoshimura, as drawnthe Yoshimura, tiltedthe twist patcheslabellings the conditions leave at a vertexthe dashed line is one node a panel, which four of these families sit on exactly

One step per panel is a table size

Four families of crease pattern search at exactly one step per panel — a grid at nine sizes, a leaf, a Miura, six crumples — and it was read as a law about patterns that fill their own sheet. It is a number: the conditions at each of their vertices admit eight labellings. Where the conditions admit four, the cost is half. Where they admit thirty, it moves again, and the same pattern at two proportions demonstrates it with everything else held still.

flat-folding · Sector angles the box-pleating grid: nodes against panels0100200one a panel0 panels256every vertex of this family keeps 8 labellings

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.

design · Box pleating the period cell of the gridone period, with its neighbours round it1 interior vertices in the cell4 crease pieces drawnperiod 1.000 × 1.000one square, because a grid repeats at every linethe cell is a rectangle of ordinary paper until somebody says its edges are one edge

A grid that will not close

Take the simplest crease pattern there is — a square grid — and join a cell of it into a torus. With an even number of squares across it folds. With an odd number it has no flat folded state at all, and the obstruction is a parity that has nothing to do with the pattern being difficult, because a grid is not difficult.

flat-folding · Two-colourability the period cell of the gridone period, with its neighbours round it1 interior vertices in the cell4 crease pieces drawnperiod 1.000 × 1.000one square, because a grid repeats at every linethe cell is a rectangle of ordinary paper until somebody says its edges are one edge

A map with no edges

Counting the ways a rectangular map folds is the oldest open problem in the subject, and every version of it assumes the map has an edge. Join the map's opposite edges and the question changes shape: half the sizes have no folded state at all, and the ones that do have no bottom layer to count from.

complexity · Map folding the pieces that are one panelleft and right edges identified — 9 pieces, 6 panels9 pieces on the drawing6 panels on the sheet10 creases, 4 verticeskeeps the sidetwo pieces of one shade are one piece of paper, a cell apart

The tube a map makes

Join one pair of a map's edges and the result is a tube — a real object, foldable in the hand, and neither the strip's problem nor the torus's. It has one loop that cannot be shrunk instead of two, it keeps its bottom layer because it keeps half its rim, and half its sizes are refused by a parity the flat map does not have.

complexity · Map folding the period cell of the gridone period, with its neighbours round it1 interior vertices in the cell4 crease pieces drawnperiod 1.000 × 1.000one square, because a grid repeats at every linethe cell is a rectangle of ordinary paper until somebody says its edges are one edge

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.

design · Box pleating the bases at which a crossover is possiblethe backbone turns 360° over a turn's worth of bases, and a crossover needs it facing the neighbourhoneycomb — three neighbours6 exact, 0 near071421283542square — four neighbours2 exact, 4 near4.3°4.3°4.3°4.3°07142128354210.5 bases to a turn · a mark is a base whose backbone faces a neighbour to within 5°, and the number under it is the miss

The helix chooses the lattice

A strand can cross to the next helix only where its backbone faces that helix, and a double helix turns about 34.3° a base. Three neighbours a third of a turn apart are faced exactly every seven bases. Four neighbours a quarter of a turn apart are never faced exactly by any whole number of bases — the nearest miss by 4.3°, and the misses do not average out, they add: 17° over thirty-two bases, 137° over two hundred and fifty-six. The lattice a design is drawn on is decided by the molecule before any shape is chosen.

biology · Molecular folding the lines a graft could useevery gap between vertex columns, and every gap between vertex rows, triedpatternvertical lineshorizontal linesgraftsThe preliminary base0 of 20 of 2nowhereThe Miura fold6 of 130 of 4one wayThe square twist0 of 60 of 6nowhereThe hexagon twist0 of 42 of 11one wayThe Yoshimura pattern0 of 120 of 5nowhereFold and cut — the triangle2 of 82 of 8both, in margin onlyThe tapered corrugation7 of 150 of 4one wayThe waterbomb tessellation0 of 80 of 8nowherea 4 by 4 grid4 of 44 of 4both waysthe grid, one square creased diagonally3 of 43 of 4both waysa line is admissible when every crease it crosses is square to it; the entry is admissible of the lines between vertex columns or rows

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.

design · Grafting a 6 by 6 grid, and the lines it still admits a strip onthe same diagonals placed three ways — the count is of lines, not of creases05100123456diagonal creases in the patternlines a strip could be slid into, both directionsno two in a row or a columnall in one rowdropped at randoma diagonal costs the row and the column it sits in, so a design that keeps its diagonals in a few rows keeps its lines clear

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.

design · Grafting a 8 by 8 grid, spacing 0.125 — what each strip width does to itthe bar is how many times finer the grid becomes — one means unchanged0.1251x1 spacings — the grid is unchanged0.251x2 spacings — the grid is unchanged0.3751x3 spacings — the grid is unchanged0.06252xthe grid becomes 0.0625, 2 times finer0.15xthe grid becomes 0.0250, 5 times finer0.1325xthe grid becomes 0.00500, 25 times finer0.25xthe grid becomes 0.0250, 5 times finera width four per cent away from a spacing divides the grid by twenty-five; a width half a spacing away divides it by two

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.

design · Grafting

Named alongside it

The objects these essays reach for when they reach for this one.

Box pleatingAssignmentDesign techniqueSearch costEnumerationError propagationGluingParitySector anglesBoundaryConstraint propagationCorrugation

All concepts