Origami & Folding
what one sheet can do

What one sheet can do.

A square of paper, uncut and unstretched, is a surprisingly strict set of rules — and nearly every theorem in this subject is a consequence of refusing to remove material. These are essays about what follows: one idea at a time, illustrated until the argument is visible, and with every crease pattern checked against the theorems rather than drawn to look right.

Trisecting 63° with one foldAbe's construction. Two horizontal creases give a reference; then a single fold carries the corner onto the lower one at the same moment as it carries the point above onto the ray. The two creases that result divide the angle into exact thirds — a construction provably out of reach of straightedge and compass.hh/263°42.0°21.0°a third of 63° is 21.00° — the fold found it, nothing was drawn at a thirdthe corner reaches the lower crease and the marked point reaches the ray at the same instantmountainvalley
Fig. 1 Trisecting an angle with one fold. Two horizontal creases give a reference, and then a single fold carries the corner onto the lower one at the same instant as it carries the point above onto the ray. The angles marked are computed from the folded positions, not drawn at a third — and this construction is provably beyond straightedge and compass, because it needs a cubic.

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19 essays

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 Axioms and construction

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.

6 figures hh/263°42.0°21.0°a third of 63° is 21.00° — the fold found it, nothing was drawn at a thirdthe corner reaches the lower crease and the marked point reaches the ray at the same instantmountainvalley Axioms and construction

Folding beats the compass, by exactly one degree

Straightedge and compass solve quadratics. A single fold solves cubics. That one-step difference settles two problems Greek geometry could not, and leaves a third exactly as impossible as it was.

6 figures 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 Axioms and construction

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.

6 figures VMMM60°90°120°90°Kawasaki60° + 120° = 180°90° + 90° = 180°both 180° — satisfiedMaekawa3 mountains, 1 valleysdifference 2exactly 2 — satisfiedangles sum to 360°which is what a flat sheet requiresmountainvalley Flat-folding

Two conditions at a point

Whether a single vertex folds flat is decided completely by two tests — one on the angles, one on the assignment. They are independent, they are easy to check, and together they settle the case entirely.

6 figures MVMwalk the folded edge and count the turns:each mountain turns +180°, each valley −180°the walk closes, so the total is ±360° — which forces |M − V| = 2the sheet must come back to where it started Flat-folding

Why the difference is two

Maekawa's theorem says mountains and valleys differ by exactly two at every flat-foldable vertex. The constant is not empirical — it is a full turn, and the theorem is about winding rather than about paper.

6 figures MVMM40°foldsopposite across the small sectorMMVM40°does not foldthe same on both sidesboth satisfy Kawasaki and Maekawa — the angles and the counts are identical Flat-folding

The smallest sector decides

Two assignments can satisfy both flat-folding theorems and only one of them folds. What separates them is a condition about the smallest angle, and it is the first rule in the subject that is not about counting.

6 figures 6 interior vertices, every one satisfying both theoremswhat the local tests seeangles at each vertexassignment at each vertexwhat they cannot seewhether layer 3 passes through layer 7whether a flap has room to existwhether the order is consistent everywhereBern and Hayes, 1996: NP-hardso this pattern is checked, not proved Flat-folding

Local is not global

Every vertex can satisfy every condition and the sheet still not fold. Deciding whether a whole crease pattern folds flat is NP-hard, which means no figure will settle it and no algorithm will scale.

7 figures Levery point within L is spentthe flapLthe circle is not a metaphor — it is the paper the flap consumesso designing a base is packing circles Designing a base

A flap costs a circle

A flap of a given length uses up every point of the sheet within that distance of it. Two flaps whose circles overlap are asking for the same paper twice — and that one observation turned origami design from an art into an algorithm.

6 figures leglegarmarmheadthe checkclosest approach 0.0000no overlap — the packing is validcircles use 71% of the sheetthe rest becomes the bodyefficiency is how much of thesquare the circles can claim,and it is an open problemthe dashed skeleton is the subject; the circles are what it costs Designing a base

Packing is the hard part

Once a subject is a set of circles, designing the model is fitting them into a square. That step has no general algorithm, no known optimum, and it is where every remaining difficulty in origami design now sits.

6 figures 16 × 16 gridevery crease on a grid line, or at 45°which is why a 64-grid design can be folded at allmountainvalley Designing a base

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.

6 figures at every vertexthree of one, one of the other15 interior vertices, all identicalwhat the sheet gainsone degree of freedom, not manyit opens and closes in both directions at oncea negative Poisson's ratio22 mountain and 16 valley creases · 39.3 sheet-widths of foldingmountainvalleyraw edge Tessellations

One vertex, repeated

Take a single flat-foldable vertex and tile the plane with it. The sheet stops being a sheet and becomes a material — with a stiffness, a packing behaviour and a Poisson's ratio that the paper never had.

6 figures nearly flatwidth ×0.87 height ×0.50half closedwidth ×0.58 height ×0.82nearly packedwidth ×0.11 height ×0.99both dimensions shrink together — pulling it open in one direction opens it in the other Tessellations

A sheet with one freedom

A Miura-folded sheet can move in exactly one way. Pull it open in one direction and it opens in the other — a negative Poisson's ratio, arriving entirely from the crease pattern and not at all from the paper.

7 figures found in crushed drink cans, tree bark and deployable boomsthe pattern is a consequence of thin-wall buckling, not of a designmountainvalley Tessellations

Patterns nobody designed

Crush a thin cylinder and it folds into a diamond lattice. Nobody chose the pattern — it is the buckling mode with the lowest energy, and it satisfies the flat-folding theorems because it just folded.

6 figures folds flatfolds rigidlyas panelsin the outer setthe bird basemost traditional modelsanything with a squash foldin the inner setthe Miura foldthe Yoshimura patterneverything ever manufacturedpaper cheats by bending very slightly; sheet metal does not Rigid folding

Panels instead of paper

Flat-foldability asks whether a pattern can reach a flat state. Rigid-foldability asks whether it can get there without any face bending on the way. The second is much stronger, and everything that gets manufactured lives inside it.

6 figures zero thicknesspanels meet exactlyfour layers of real materialeach fold has to clear the ones belowhinge offset to the surfacethe panel rotates about the right linea crease pattern describes a surface with no thicknessand everything anybody builds has some Rigid folding

The sheet has a thickness

Every crease pattern describes a surface with no thickness. Everything anybody builds has some, and getting it around a corner is the central problem of turning origami into hardware.

6 figures Miura solar array17×Space Flyer Unit, 1995airbag folding25×stored for years, opens in 30 msheart stent6×threaded through an arterystarshade11×26 m disc, 2.5 m launch tubemap fold9×the original problempackeddeployedthe ratio is what is bought; one degree of freedom is what makes it reliable Rigid folding

Folding that gets built

Solar arrays, airbags, stents and starshades. The requirement is always the same — large in use, small in transit, along a path nobody has to trust to chance — and folding is what answers it.

6 figures the patternconcentric arcs, alternatingwhat the sheet doesa shape with no flat state at allthe curve is in the crease; the saddle is the paper refusing to stretch Curves and material

A crease that curves

Bend a crease and the paper either side is forced into a shape nobody creased. The flat-folding theorems say nothing about it, because they are statements about straight creases meeting at a point.

6 figures cylinderreachablecurved one way onlyconereachablecurved one way, from a pointsphereunreachablecurved two ways — impossiblesaddleunreachablecurved two ways — impossible Curves and material

What a flat sheet can become

A sheet that cannot stretch can only take shapes that are flat in one direction at every point. Cylinders and cones are reachable; a sphere is not, and no amount of folding will get one.

6 figures no thicknesslayers add up; a 64-grid model is millimetres thick at the coreno stretchpaper stretches a little, which is why wet-folding works at allcreases are linesa crease has a radius; sharp folds tear and soft ones springperfect memorypaper relaxes, so a model opens slightly the moment it is put downthe theorems are exact statements about a sheet nobody has ever folded Curves and material

Four things that are not true

Zero thickness, no stretch, creases that are lines, and perfect memory. Every theorem on this site rests on all four, every one of them is false, and the interesting engineering is exactly where each fails.

6 figures

Threads running through

themes, not chapters

One sheet, no cuts

A single square, uncut, unstretched. It is an arbitrary rule that turns out to be a generative one — nearly every theorem in the subject is a consequence of refusing to remove material.

8 essays

The pattern is the object

A crease pattern is not a picture of a model. It is the model, written down — complete, checkable, and foldable by anyone who has the paper.

5 essays

Local rules, global behaviour

Two conditions at a single vertex decide whether it folds flat. Whether a whole sheet does is a different question, and a much harder one.

7 essays

Folding beats the compass

Straightedge and compass solve quadratics. A fold solves cubics, which is why paper trisects an angle and Euclid's tools cannot.

3 essays

Flat is rare

Almost no crease pattern folds flat. The ones that do are a vanishingly small, highly structured set, and that scarcity is what makes them worth studying.

3 essays

Paper is not ideal

Zero thickness, no stretch, infinitely sharp creases, perfect memory. Every one of those is false, and the interesting engineering lives in exactly where each fails.

6 essays

From craft to hardware

The same mathematics that folds a paper crane deploys a solar array, packs an airbag and threads a stent through an artery. The scale changes; the constraints do not.

6 essays