Origami & Folding
what one sheet can do
Theme

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.
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.

 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.

 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.

 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.

 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.

 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.

 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.

 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.

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