Origami & Folding
what one sheet can do
Concept

The Huzita–Hatori axioms — where it appears

The name usually given to the seven fold axioms, after the two people who published them in 1991 and 2001. Beloch had the crucial sixth in 1936, which is why the naming is a case study in attribution.

Named by 14 essays across 2 fields — each of them below, with the objects they name alongside it.

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

One fold at a time, and there are exactly seven of them

A fold is specified by bringing points and lines into coincidence. There are seven ways to do that, the list is provably complete, and one of the seven does something no compass can.

construction · The axioms the cubic8x³ + 4x² − 4x − 1its roots-0.900969-0.2225210.6234903 real common tangentsone fold for each rootand the fold gives cos 2π/7each curve is the set of folds that puts one point on its line; a line touching both does the two at oncea compass intersects circles and gets two answers; a fold touches parabolas and gets up to three

Fifty years in the wrong language

Margherita Beloch showed in 1936 that one fold solves a general cubic. The result was correct, published, and in a mathematics journal — and the subject that needed it did not find it until 1991. The cost of a paper nobody reads is measurable, and it is most of a century.

history · Attribution nφ(n)its prime factorscompassone foldtwo at once322422542 · 2622762 · 3842 · 2962 · 31042 · 211102 · 51242 · 213122 · 2 · 31462 · 31582 · 2 · 21682 · 2 · 217162 · 2 · 2 · 21862 · 319182 · 3 · 32082 · 2 · 221122 · 2 · 322102 · 523222 · 112482 · 2 · 2the 11-gon is the first a single fold misses, and two simultaneous folds reach itthe 23-gon is the first that needs more than two, because 22 has an 11 in it

Two creases at once

The seven axioms describe what one fold can do, and the restriction to one fold is a rule somebody imposed rather than a property of paper. Allow two creases to be made simultaneously and the reachable degree rises — and the hendecagon nobody could fold becomes foldable.

construction · Multifold 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

A fold needs something to align

Every axiom names things that must already be on the paper — a point to fold onto a point, a line to bring to a line. So what a folder can build is bounded by what they can refer to, and that set is finite at every depth: nine references after one fold, several hundred after two, and every one of them computable in advance.

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 2 folds from a bare sheet, every proportion at the same areamarks separated by at least 0.3 mm on a 150 mm sheetthe squarewhat origami paper is sold as565 marks · 92 distinct foldsthe A serieshalves into itself45,705 marks · 752 distinct foldstwo squaresa square cut the long way26,155 marks · 540 distinct foldsthe 1 : √3 rectanglethirds into itself42,746 marks · 732 distinct foldsthe golden rectanglenot in the halving family43,233 marks · 732 distinct folds

The sheet decides which points exist

Every measurement of what folding can locate has been made on a square, because origami paper is sold square. Hold the area fixed and change the proportion: one fold reaches nine marks on a square and twenty-nine on the A-series rectangle, and two folds reach 565 against 45,705. The square is the worst of five proportions at both depths, and the reason is its own symmetry.

construction · Reference points one round of folds through two points and folds placing one point on anothera crossing that lands inside the hole is not a reference and is not drawnwith a hole: 212 referencessolid: 9from 8 corners and 8 edgesfrom 4 corners and 4 edges

A hole is an edge

A folder's first fold has to be specified by aligning things that are already there, and what is already there is the sheet's outline. Cut a square hole in the middle and the outline doubles: one round of alignments reaches nine references on a plain square and two hundred and twelve on a holed one — more than the plain square reaches in two rounds.

construction · Reference points 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 axiom is satisfied by both, and says nothing about whichthe two are square to one another, alwaysthe thin lines are the two the axiom is given; the two heavy ones are its answers

The axiom that names two folds

Bring one line onto another and the operation is satisfied by either of two folds, always exactly perpendicular to one another, creasing the paper in completely different places. Over four thousand random line pairs on a square, both folds land on the sheet two thousand nine hundred and sixty-five times, and the statement of the axiom does not say which one is meant. The fifth axiom is worse: its two answers are at any angle at all, from half a degree apart to square.

construction · The axioms 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 the count that was made, and the count that was notboth are floors: neither enumeration tracks which fold an alignment attaches tofreedomspaper onlywith simultaneous creasesneeding oneone fold277—two at once4228664three at once650296246four at once895791696five at once1016117921631at two folds the omission is 64 operations of 86 — 74% of them, and none can be described without naming the other creasea pair of hands cannot make a condition between two creases it is making; a jig holding two lines can

Twenty-two is a floor

The enumeration that gives seven single-fold axioms spends each fold line's two degrees of freedom on alignments to points and lines already on the paper, and its own account says what it leaves out — an alignment may refer to a crease being made in the same instant. Adding those back leaves the single-fold count at seven and takes the two-fold count from twenty-two to eighty-six, of which sixty-four cannot be stated in terms of the paper at all.

construction · Multifold three enumerations of the same thinga fold line has two freedoms, so m of them have 2m — the question is whether the budget is one pool or m pursesfolds at oncepaper onlypooled, with crossingseach alignment attachedthe ratio1777× 1.022286105× 1.23502963042× 10.3495791145,211× 183.6516117929,782,771× 5459.1the last column is what tracking which fold an alignment names is worth, and it grows because the naming itself grows

Each fold needs its own two

The enumeration that gives seven axioms spends a fold line's two degrees of freedom on alignments; run for m folds it spends 2m from one pool, and a pool can be spent three on one line and one on the other, which determines neither. Attaching every alignment to the fold it constrains repairs that, and two other things — and the two-fold count goes from twenty-two to a hundred and five, of which only twenty-eight have to be made at one instant.

construction · Multifold 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 pointConstructionThe axiomsConstructibilityAxiomEnumerationMultifoldSheet shapeAxiomsCompletenessConstructible numberCrossing

All concepts