Origami & Folding
what one sheet can do
Concept

Fixed point — where it appears

A value an iteration leaves unchanged. An iteration with a contracting one forgets its errors, which is why a convergent folding construction can be more accurate than an exact one.

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

guessoff by 0.16667fold 1off by 0.08333fold 2off by 0.04167fold 3off by 0.02083fold 4off by 0.01042fold 5off by 0.00521fold 6off by 0.00260solid mark — the fold is aiming at 1/3; dashed — at where 1/3 has gonehalving word R L — period 2, from 2^2 − 1 = 3 × 1every fold halves the error exactly, so 6 folds divide it by 64

Folding a strip into thirds

A third cannot be constructed by the axioms, so it is not constructed. It is guessed, and then halved into place — an algorithm rather than a construction, with an error that falls by exactly half at every fold.

construction · Exact division proportion 1.3two shapes, in turn1.3001.5381.3001.5381.300proportion √2 = 1.4142one shape, throughout1.4141.4141.4141.4141.414halving turns a proportion of r into one of 2/r, and those are the same number only at √2the 1.3 sheet is a different shape after every fold; the √2 sheet is the same shape after all of thema square metre at √2 is 840.9 × 1189.2 mm, which is the 841 × 1189 printed on a sheet of A0

The rectangle that keeps its shape

Halving a rectangle across its long side turns a proportion of r into one of 2/r, so almost every sheet comes out of the fold a different shape from the one that went in. Exactly one does not, and it is not a shape anybody chose.

construction · Paper proportion

Named alongside it

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

Rational divisionA-seriesBinary expansionConvergenceError propagationFujimoto's methodHalvingPaper proportionReference point

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