Origami & Folding
what one sheet can do
Concept

Fujimoto's method — where it appears

An iterative way of dividing a strip into odd fractions by repeated halving. It never lands on the fraction and its error contracts at every fold, which makes it self-correcting where an exact method is not.

Named by 3 essays across 2 fields — 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 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

The kindergarten was a geometry class

Froebel put paper folding into mass education in the 1830s, and did it as mathematics rather than as craft. His three categories — the folds of life, of beauty, and of knowledge — are the first systematic treatment of folding anybody wrote down, and the third one is a geometry syllabus.

history · Pedagogy 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

Dividing a loop into n

Fujimoto's method divides a strip into any number of equal parts by folding badly and then folding the error in half, over and over. It converges because each step halves what is left over. On a closed loop there is no edge for the leftover to sit against, and what replaces the edge is the loop's own closure.

construction · Exact division

Named alongside it

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

ConvergenceError propagationExact divisionRational divisionBinary expansionBoundaryConstructive proofFixed pointFroebelGluingPedagogyReference point

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