Origami & Folding
what one sheet can do
Concept

The axioms — where it appears

The seven single-fold operations that generate everything a folder can construct. The list is complete because a fold line has two degrees of freedom and there are only so many ways to spend them.

Named by 15 essays across one field — each of them below, with the objects they name alongside it.

pp − 1, factoredcompassfoldingthe regular p-gon32both tools52 · 2both tools72 · 3folding only112 · 5neither132 · 2 · 3folding only172 · 2 · 2 · 2both tools192 · 3 · 3folding only232 · 11neither292 · 2 · 7neither312 · 3 · 5neither372 · 2 · 3 · 3folding only11 is the first prime out of a fold's reach — 11 − 1 = 2 · 5the factor of five is the obstruction, and no arrangement of folds produces onea compass needs a power of two; a fold needs nothing above three

The eleven-sided one nobody can fold

Folding reaches the heptagon, which a compass cannot. It does not reach the hendecagon, and the obstruction is a single prime factor: ten has a five in it, a fold solves cubics, and no arrangement of cubics produces a five.

construction · Constructible polygons 1/25 folds to reach 1/5, and every mark on the way is exactthe solid line is the one fold used at every step; the dashed lines are the stepsnothing here converges — each crossing lands on its fraction and stops

One crossing, and then another

Folding a strip into thirds by Fujimoto's method halves the error at every fold and never reaches a third. There is a construction that arrives instead: cross the square's diagonal with a line through the mark you already have, and the crossing lands on the next fraction exactly — one fold per step, all the way down.

construction · Exact division the bare sheet4 references · 4 linesnothing has been foldedafter 1 fold9 references · 12 lineshalves, and nothing elseafter 2 folds565 references · 92 lineshalves, thirds, fifths — and worsea fold is an alignment, and an alignment needs something already on the paper to align565 references after 2 folds, and the count is finite however many folds are allowed

Cheap where it reaches

Two folds from a bare square put marks at a half, a third, a quarter, a fifth, a sixth, an eighth and a twelfth — and at no seventh, ninth or eleventh at all. A rule that reaches every fraction takes n folds to reach one nth. The systematic route and the short one disagree everywhere, and neither of them knows about the other.

construction · Reference points distinct fold lines this axiom specifies and no other doesaxiomafter 4 pointsafter 9 pointsafter 565 pointsA1 — through two points08121054A2 — one point onto another08142649A3 — one line onto another4564994A4 — perpendicular through a point001661distinct lines in all1292274300the four operations name 38 folds at the first round and draw 12 lines with them

What each axiom is worth

The list of seven folds is complete, and the proof of that says nothing at all about whether its members are independent or equal. Measured on a bare square, one of the four elementary axioms supplies every fold the others cannot and the other three supply nothing. Two rounds later the ranking has inverted, and the one that carried the first round is the least productive of the four.

construction · The axioms the degree of the equation, and what it is made ofnumberdegreemade ofwhere it comes from½11a fold in half√222^1the diagonal of the squareφ22^1the silver rectangle's cousin∛233^1doubling the cube2 cos(2π/7)33^1the regular heptagon∜242^2a square root of a square root∛2 · √262^1 · 3^1a product of two of them2^(1/5)5not twos and threesa fifth root2 cos(2π/11)5not twos and threesthe regular hendecagonchecked by exhaustion: no number here satisfies a rational equation of lower degree with coefficients up to 6

The numbers a fold reaches

Folding solves cubics, which is one fact about one fold. The reason the subject has a theory rather than a bag of tricks is a second fact about all of them: the lengths a folder can mark are closed under addition, subtraction, multiplication, division, square roots and cube roots. Constructions can therefore be built out of constructions — and no tower of them ever arrives at a fifth root.

construction · Origami numbers the bar is how many folds the alignment names, averaged over the trialsa1 — the fold through two points1.000.0% none · 0.0% twoa2 — one point onto another1.000.0% none · 0.0% twoa3 — one line onto another2.000.0% none · 100.0% twoa4 — a perpendicular through a point1.000.0% none · 0.0% twoa5 — a point onto a line, through a point1.5223.8% none · 76.2% twoan alignment with no fold is not a failed construction; it is an alignment the paper cannot make

An axiom may name no fold

The seven operations are stated about points and lines in a plane, and a plane has no edges. On a square, three of them always name exactly one fold and always land it on the paper; placing a line on a line names two, and 13.3% of the folds it specifies are creases the sheet never reaches; and placing a point on a line through a second point names two folds, one, or — 23.8% of the time — none at all.

construction · The axioms the bar is the share of each round's new references that lie on the sheet's own edgeafter one fold80%4 on the edge · 1 inside · 12 fold linesafter 2 folds9%48 on the edge · 508 inside · 92 fold linesan edge is a line a fold can cross twice, so it yields marks with the folds and not with their pairs

The edge was there first

A folder's first fold has nothing to align to but the sheet's own outline, and it shows in where the marks land: four of the five references the first fold adds are on the paper's edge. By the second fold the edge holds forty-eight of five hundred and fifty-six new ones. The rim is where references are cheap and it fills up, because an edge is a line a fold can cross twice while two folds inside the paper cross once each.

construction · Reference points the bar is the largest gap between anywhere on the sheet and a referencea third fold specifies more folds than can be listed, so a sample of them is taken insteadnone of them0.089565 references, from two folds50 of them0.0763,498 references · 0.02% of the round100 of them0.0508,056 references · 0.04% of the round200 of them0.04723,480 references · 0.07% of the round400 of them0.02274,694 references · 0.15% of the round800 of them0.014270,882 references · 0.29% of the roundevery row is a lower bound on what the whole round would buy, because leaving folds out can only make the gap larger

The third fold cannot be listed

Two folds from a bare square reach five hundred and sixty-five reference points. The third round specifies three hundred and seventy-eight thousand folds, of which two hundred and seventy-four thousand are distinct — and the crossings of those with each other run to the tens of billions. The closure stops being computable at exactly the depth a folder starts working at, and what can be said instead is a bound rather than a list.

construction · Reference points axiom 1through two pointslinearaxiom 2point onto pointlinearaxiom 3line onto linelinearaxiom 4through a point, square to a linelinearaxiom 5point onto a line, through a pointquadraticaxiom 6two points onto two linescubicaxiom 7point onto a line, square to a linelinearthe degree each axiom can solve — one of them is why paper beats the compass

Which of the seven survive

The seven axioms are the complete list of ways one fold can be specified by aligning marked things. Every one of them names points and lines on a sheet, three of them quietly assume that a line has two sides, and on a closed sheet a line need not — so the list is complete for a disc and shorter for anything else.

construction · The axioms degree is how far out the number is; height is what the construction costsnumberdegreesteps in the towerwhat the steps are½10 stepsa fold in half√221 stepthe diagonal of the squareφ21 stepthe silver rectangle's cousin∛231 stepdoubling the cube2 cos(2π/7)31 stepthe regular heptagon∜242 stepsa square root of a square root∛2 · √262 stepsa product of two of them2^(1/5)5no tower reaches ita fifth root2 cos(2π/11)5no tower reaches itthe regular hendecagon2^(1/8)83 stepsthree square roots2^(1/9)92 stepstwo cube roots2^(1/12)123 stepstwo squares and a cube2^(1/16)164 stepsfour square roots1 pairs invert: 2^(1/9) is of degree 9 and 2 steps, 2^(1/8) of degree 8 and 3a degree 2^a · 3^b is a steps of square root and b of cube root, so the height is a + b and nothing else

Reachable is not cheap

The closure is what makes folding a theory rather than a bag of tricks: constructions can be built out of constructions. What that also means is that constructions have lengths and the lengths compose, so every reachable number has a height as well as a degree — the number of extension steps the shortest tower to it must take. The two orderings disagree, and a ninth root is a shorter tower than an eighth.

construction · Origami numbers what the plane specifies, and what the paper carries2 rounds from a bare sheet, every proportion at the same rulesheeton the paperlost to the edgesquare1 × 1.000565 references72%1,440 off the paperA-series1 × 1.414114,927 references47%102,162 off the paper3 : 21 × 1.500135,281 references48%123,872 off the paperdouble square1 × 2.00044,673 references59%65,294 off the paper2 rounds of a1, a2, a3, a4 · a crossing is a reference only where the paper is · square loses the largest share, at 72%

The field has no edge

Origami numbers are a field on the unbounded plane and a folder has a piece of paper. A fold line runs forever; a crossing of two of them is a number in the field wherever it lands, and it is a reference somebody can put a finger on only where there is paper under it. Counted rather than assumed, two rounds of the four linear axioms on a square put seventy-two per cent of their crossings off the sheet.

construction · Origami numbers how squarely the folds that fix a reference crossand what a crossing at that angle does to an error in the foldingthe four linear axioms565 references · worst 36.9°with the conic axiom16,890 references · worst 6.3°60° to 90°error × 1.264.6%54.4%45° to 60°error × 1.418.2%21.8%30° to 45°error × 2.017.2%17.1%15° to 30°error × 3.90.0%5.2%8° to 15°error × 7.20.0%1.2%under 8°error × 14.30.0%0.3%the linear axioms bottom out at 36.9° — a multiplier of 1.67 — and the conic axiom reaches 6.3°, a multiplier of 9.12 rounds on a square · a reference priced at 1 ⁄ sin of the widest angle its own folds make · shares, so the two sets are comparable

What buys the reach costs the accuracy

A reference is a crossing of two creases, and a crossing transmits a folding error multiplied by one over the sine of the angle the creases make. Measured across the whole closure on a square, the four linear axioms never produce a crossing shallower than thirty-seven degrees — and the conic axiom, the one that sends a point onto a line and reaches the heptagon, produces crossings under seven.

construction · Constructible polygons what each axiom set specifies, and what of it the sheet carrieson a unit square, from its four corners and four edgesaxiomshow manyroundsfold lineson the paperoff itlosta line through two points116500.0%a line through two points126500.0%and a point onto a point218900.0%and a point onto a point22321336432.5%and the other two linear ones411291257.1%and the other two linear ones42925651,44071.8%and a point onto a line through a point51283310475.9%and a point onto a line through a point52———past the capthe conic axiom loses three quarters of its crossings at one round, and cannot be run a second

The axiom that reaches furthest wastes most

The field of origami numbers is defined on an unbounded plane and a folder has a square. Counted axiom set by axiom set on the same sheet, the share of crossings that land off the paper rises with every axiom added: nothing at all from the first two, fifty-seven per cent from the four linear ones at a single round, and seventy-six per cent from the conic axiom at a single round — more, in one round, than the linear four lose in two. The instrument that reaches furthest into the field delivers the smallest share of what it specifies.

construction · Origami numbers after one round from a bare square, the 80 new folds a second round offers, by what each adds alonenine references and twelve lines on the sheet; a fold adds a reference wherever it crosses a line already there, on the paper, somewhere newadds 61620 per cent of the menuadds 545 per cent of the menuadds 4810 per cent of the menuadds 3810 per cent of the menuadds 21215 per cent of the menuadds 145 per cent of the menuadds 02835 per cent of the menuthe average fold on the menu adds 2.50; 16 folds tie for the most, 6; of the 28 that add nothing, 28 never cross the paper at all

A fold is worth what follows it

After one round of folds from a bare square the sheet carries nine references and twelve lines, and a second round of the four linear axioms offers eighty new folds. Made alone, the best of them adds six references and the average two and a half. Made all together they add 556 — nearly seven a fold — because a fold's yield is mostly its crossings with the folds that come after it. So the value of choosing has to be measured along an order: always making the most productive fold next reaches half the round in 28 folds where a random order takes 51, and all of it in 52, which is the fewest folds that can. The conic axiom's 305 extra folds leave the first eight choices exactly as they were.

construction · Origami numbers references on the paper, fold by fold, choosing from a growing menu and from the round's ownthe bound is what k folds can make at most, whatever they are; the growing menu's curve lies on it020040060002040folds madereferences on the paperthe round, 565growing menuthe boundthe round's menuthe round's 565 references take its own menu 52 folds; a growing menu passes 565 at fold 29, and no 28 folds of any kind can

Each new fold can cross every old one

A folder choosing each fold from the second round's fixed menu needs fifty-two folds for the round's 565 references, and no fewer. Let the menu grow — after every fold, every fold the references now on the paper specify — and the count obeys a bound that has nothing to do with axioms: a new line can cross each line already through the paper once, and the rim twice. After k folds that is at most 9 + k(k + 11)/2 references. The greedy folder meets the bound exactly at every one of thirty folds, passes 565 at the twenty-ninth, and no twenty-eight folds of any kind can. But of its 589 references only fifteen are the round's, and they are nearly useless: the median lies a seventieth of a millimetre from its nearest neighbour.

construction · Origami numbers

Named alongside it

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

Reference pointReachable setConstructible numberThe Huzita–Hatori axiomsClosureConstructionOrigami numberSheet shapeField extensionRational divisionAxiomCompleteness

All concepts