Origami & Folding
what one sheet can do
Concept

Design technique — where it appears

A move a designer makes to get a property a base does not have by itself — a colour change, a graft, a stretched flap. Each of them is paid for in paper, and the price is usually computable.

Named by 16 essays across 3 fields — each of them below, with the objects they name alongside it.

00.10.20.30.40.500.20.40.60.81flap width, as a fraction of the sheetarea showingthe two are equal at a thirdfront showingreverse showingtotal facemeasured on the folded state at 4 flap widths, and the marks are those measurements

Bringing the other side to the front

Paper has two sides and most models show one. A colour change shows the other, and it is not a crease problem — which panels can show the reverse is settled by the pattern's two-colouring, and what it costs is twice what it shows.

design · Colour change 2345678910-0.15-0.1-0.0500.050.10.15discsshortfall of the symmetric search14.6%1.7%0.0%-0.0%5.6%0.5%10.8%10.4%4.7%symmetry: mirror · both searches at 90 restartsneither number is a proved optimum — this compares two searches

When symmetry costs

Design software and designers both reach for symmetry, and for a good reason: it makes the search enormously easier. It is a heuristic and not a theorem, and how much it gives away can be measured — including the case where the optimum is symmetric about an axis nobody imposed.

design · Symmetry 123456760%65%70%75%80%sheet, longer side to shorterfraction of the sheet claimedbest at 7.0 to 17 flaps · every sheet the same areathe sheet's shape is a design variable that origami paper hides by being sold square

The square is a choice

Every packing on this site has been into a square, because origami paper is sold square. Hold the area fixed and vary the shape instead and the efficiency turns out to be spiky rather than smooth — with the same peak value at every proportion that is a ratio of two factors of the flap count, and nowhere else.

design · Sheet shape the design as it stood20.00 of paper · 5 creases meet the cutthe same design, one strip wider21.80 of paper · the strip is 0.45 acrossthe band is 0.45 × 4 = 1.8000, and that is the entire difference between the two patternsall 44 creases away from the cut keep their length; the 5 that cross it are longer by 0.45 and by nothing elsea design grows by accretion because the arithmetic of growing it is this short

Paying in paper

A feature added to a finished design costs exactly the paper inserted for it. Cut the crease pattern along a line, slide in a strip, and every existing crease continues across it unchanged — so the bill is the strip's width times the length of the cut, and there is no second term.

design · Grafting 1½¼what one flap costsin the middle · a whole disc0.2463 of the sheeton an edge · half of one0.1232 of the sheetin a corner · a quarter0.0616 of the sheeteach one integrated over the sheetrather than taken from the fractionat 0.28 sheet-widths a flap costs 0.2463 of paper in the middle, 0.1232 on an edge and 0.0616 in a cornerso an efficient design fills the boundary first, and the edge of the sheet is the cheapest paper on it

The corner is worth four times the middle

A flap consumes every point of paper within its own length of its tip — but only the paper that is actually there. On an edge of the sheet that is half a disc, and in a corner a quarter, so the same flap costs four different prices depending on where it stands and the boundary is the cheapest paper on the sheet.

design · Sheet shape 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 the bar is the share of the footprint showing the side that started face upthe rest of it shows the other side, and neither is chosen by anybodyThe preliminary base0.3%2 of 8 panels in viewThe square twist47.4%7 of 9 panels in viewThe hexagon twist90.8%10 of 13 panels in viewFold and cut — the triangle100.0%1 of 7 panels in view · 2 statesa panel out of view from above is not hidden — it is under the pile, and turning the sheet over shows a different set

Which side arrives

A colour change is described as a choice: bring the reverse of the sheet to the front where the design wants it. On a pattern whose panels can be ordered, nobody chooses. The preliminary base shows the side that started face up over one part in a thousand of its own footprint, and two of its eight panels are the only ones in view at all.

design · Colour change the outline, shrunk — and the one corner that moves outwardwhat the shrink finds5 edges, 1 of them meeting at a corner that turns back3 skeleton nodes7 arcs traced by the cornersthe last of them forms at 0.181 of a sheet

The corner that splits the shrink

The universal molecule fills a convex polygon by shrinking it, and at a corner that turns back the shrink does something no convex polygon does: the region breaks in two. That event can now be computed — the skeleton of a non-convex outline is available here for the first time, and it is what lets one straight cut reach a star. It does not give the molecule back, because a molecule needs the shrinking region to stay one piece and a split is exactly the moment it stops.

design · Molecules the vertex nobody listedthe two lines meet at 22.9°sectors 157.1° 22.9° 157.1° 22.9°alternating sums 314.2° and 45.8°Kawasaki fails — it holds only at a right angle2 mountain and 2 valleyMaekawa fails — a crossing can only be 4–0, 2–2 or 0–4mountainvalleyraw edge

The reader decides the junction

Five of the eight patterns printed here have places where one crease ends on another — four on the preliminary base, eight on the square twist, twelve on the hexagon twist, nineteen on the Yoshimura, three on the fold-and-cut triangle. At each of them a reader has to decide whether two lines meet or pass through one another, and no notation, caption or teaching text in the subject mentions that the decision is being made.

history · Pedagogy 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 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 what each sheet's shape costs in conditionsa square0 loopsχ = 1 · no loop that cannot be shrunka slit from the rim0 loopsχ = 1 · the same paper, topologicallyone hole1 loopχ = 0 · one parity conditiona cylinder1 loopχ = 0 · the same sheet as one holetwo holes2 loopsχ = -1 · two independent conditionsa torus2 loopsχ = 0 · two conditions, no rim at alla slit inward from the rim changes nothing, and a closed cut changes everything

The cut that changes nothing

A slit goes right through the material and leaves the sheet exactly the object it was. A closed cut removes almost no paper and produces a different sheet with a condition it did not have. Kirigami is made almost entirely of the first kind, which is why every result about it survives the distinction untouched.

material · Kirigami 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 the bar is the footprint, split by how deep the other side liesone layer downtwo layers downnot under this point at allThe preliminary base8 panels · 1 stateThe square twist9 panels · 1 stateThe hexagon twist13 panels · 1 stateFold and cut — the triangle7 panels · 2 statesa 2 × 2 Miura patch4 panels · 1 statea 3 × 2 Miura patch6 panels · 3 statesa 3 × 3 Miura patch9 panels · 6 statesa 4 × 3 Miura patch12 panels · 11 statesthe letter fold3 panels · 2 statescounted over every folded state and from both faces, so nothing here is one lucky pile

One sheet down

A colour change has been priced by how deep the pile is — eight layers over every point of the preliminary base, six on a small Miura. Ordered, the piles say something else: wherever the other side of the paper lies under a point, it is the next sheet down on every pattern with one folded state, and never more than two down on any. And relettering the same creases cannot reach it. Of 112 letterings of the preliminary base that fold, every one shows either the printed face or the whole face turned the other colour; the square twist's eight only turn its face round.

design · Colour change

Named alongside it

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

Box pleatingGridCircle packingTwo-colouringColour changeCrease patternGraftingLayer orderAssignmentBoundaryConservationEfficiency

All concepts