Origami & Folding
what one sheet can do
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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.
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.

 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.

 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.

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