Origami & Folding
what one sheet can do
Concept

Completeness — where it appears

The property of a list having nothing left out of it. The seven fold axioms are complete for single folds and the four local conditions are complete only at degree four, which is the difference between a finished list and a merely useful one.

Named by 8 essays across 4 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 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

Why the list stops at seven

The seven axioms are not seven useful folds somebody collected. They are every fold there is, and the proof is an exercise in counting degrees of freedom that takes about a minute.

construction · The axioms evenly spaced — 4 creasesany flat folding16 of 16some-layers16 of 16all-layers16 of 16one-layer2 of 16crimping only6 of 16one short segment — 4 creasesany flat folding4 of 16some-layers4 of 16all-layers0 of 16one-layer2 of 16crimping only4 of 16uneven — 4 creasesany flat folding8 of 16some-layers8 of 16all-layers0 of 16one-layer2 of 16crimping only0 of 16a machine that takes fewer layers is weaker, not more patientthe paper is joined, so what it declines to hold it also cannot move

The machine that may choose

Three restricted machines lose patterns that fold perfectly well. Give one of them a choice — any block of layers, top or bottom — and the loss vanishes: over a hundred and seventeen spacings, every flat folding of every strip became reachable. Being forced was the whole problem.

complexity · Simple foldability 4681012010203040sides of the polygoncreasescreases in the patternperpendicularsskeleton arcsa 12-sided outline needs 36 creases and one skeleton nodea convex outline is the cheap casea reflex corner splits the shrinking front, and this solver refuses those rather than guessing

What universality costs

The fold-and-cut theorem says any straight-line drawing can be flattened onto a single line. It says nothing about how much crease pattern that takes, and the amount is a measurable quantity — computed here by running the construction rather than by estimating it.

complexity · Flat-foldability 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 polygonsectors at the twist vertexKawasakitiles the plane3-gon60.0 · 60.0 · 120.0 · 120.0180.0° = 180.0°yes — 6 round a point4-gon90.0 · 90.0 · 90.0 · 90.0180.0° = 180.0°yes — 4 round a point5-gon108.0 · 108.0 · 72.0 · 72.0180.0° = 180.0°no6-gon120.0 · 120.0 · 60.0 · 60.0180.0° = 180.0°yes — 3 round a point7-gon128.6 · 128.6 · 51.4 · 51.4180.0° = 180.0°noevery one of these twists satisfies the local theorems and folds flat on its ownthe interior angle has to divide 360° for the twists to meet, which only 3, 4 and 6 dobeyond 7 sides the assignment search runs out — 21 free creases, and the enumerator refuses above 22

Which polygons twist

Twist tessellations come in three kinds — triangle, square, hexagon — and it is natural to read that as a fact about twists. It is not. A twist can be built around any regular polygon and every one of them folds; what stops at three is the tiling, and the tiling is a fact about the plane.

tessellation · Twists quadrilateralconvexmolecule builtpentagonconvexmolecule builtL, one reflex cornerone reflex cornerconstruction refusedthe polygon admits a shrinkingdart, one reflex cornerone reflex cornerconstruction refusedthe polygon admits a shrinkinga convex polygon shrinks inward and stays a polygon; a reflex corner is a wall the shrink runs intoso a non-convex region is split into convex pieces first, and choosing the split is a searchwhich is where a construction that always works hands the difficulty to whatever comes before it

The molecule that does not exist

The universal molecule fills any convex polygon, always, which is what makes it the part of the tree method with no special cases. Hand it a reflex corner and it does not produce a worse pattern — it produces nothing, and the difficulty moves backwards to whoever chose the polygons.

design · Molecules 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

Named alongside it

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

Constructible numberThe Huzita–Hatori axiomsStraight skeletonThe axiomsAlignmentAssignmentAxial polygonAxiomThe big-little-big lemmaConservationConstructionDegrees of freedom

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