Origami & Folding
what one sheet can do
Series

Genericity — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. -3-2.5-2-1.5-1-8-6-4-20tolerance (log₁₀ radians)fraction inside it (log₁₀)1 vertex · slope 1.002 vertices · slope 2.013 vertices · slope 3.0140,000 random vertices, none of them constructed to fold and none of them folding

    Almost every pattern fails

    Kawasaki's condition is one equation for each interior vertex, and a drawing satisfies an equation with probability zero. Every pattern on this site folds because it was constructed to, and the fraction that would fold by accident can be measured.

    part 1 · flat-folding
  2. 0.5°1%1°2%2°3%5°8%10°16%20°33%how far each sector would have to moveshare that would fold40,000 random four-crease verticesthe median vertex is 31.31° per sector from folding, the mean 33.91°none of them folds, and almost none of them nearly does either

    A near miss is nearly as rare

    Flat-foldability is a coincidence of measure zero, which is usually where the argument stops. Measure how far a random vertex is from folding rather than whether it does, and the answer is thirty-one degrees a sector — so the tolerance real paper has does not buy back anything at all, and a pattern that nearly folds had to start near one that did.

    part 2 · flat-folding
  3. the bar is the second-smallest sector at a typical vertexthe triangular patch at seven turns, with the same panels and the same creases at all of themturn 0.261.92°smallest sector 60.00° · next 61.92° · no lettering exists, proved by exhaustionturn 0.2160.71°smallest sector 60.00° · next 60.71° · no lettering exists, proved by exhaustionturn 0.215560.06°smallest sector 60.00° · next 60.06° · no lettering exists, proved by exhaustionturn 0.21660.00°smallest sector 60.00° · next 60.00° · a lettering existsturn 0.2260.00°smallest sector 59.52° · next 60.00° · a lettering existsturn 0.2560.00°smallest sector 56.10° · next 60.00° · a lettering existsturn 0.3560.00°smallest sector 46.15° · next 60.00° · a lettering existsthe verdict changes exactly where that sector passes sixty degrees and stops being the second smallest

    Where a sector crosses sixty

    Turn the twist polygons of a tessellation patch a hundredth of a radian further and the pattern goes from having no mountain-valley labelling at all to having one immediately. Nothing about its graph changes across the transition — the same eighty-three panels, the same hundred and forty-two creases, the same four labellings at every one of its sixty vertices. What changes is which sector at a vertex is the smallest one.

    part 3 · flat-folding
  4. each cell is one patch, searched to a verdictgreen: a lettering exists · magenta: none exists, by exhaustion0.150.250.350.50.70.91.11.3turn angle, in radianssquare2626262626262626elongated1515323231313232hexagonal1515394545464545triangular1515393939373737the number in a cell is the nodes the search visited; 6 of 32 patches have no lettering at all

    A region with no lettering

    One turn angle at which a tessellation patch has no consistent lettering was found by sweeping a dial. Sweeping two dials finds nine patches with none, across three tilings, filling a corner of the parameter space — and never touching the square tiling, whose sectors have no sixty degrees to cross.

    part 4 · flat-folding
  5. the angles that admit a thirdφ₁ − φ₂ + φ₃ a multiple of a straight angle30°30°60°60°90°90°120°120°150°150°60°, 120°the first crease's angle, against the secondevery other pair of angles folds nothing,at any length and any positions

    How rare a band that folds is

    Almost every crease pattern fails to fold flat, and the usual way of saying so is a count over discrete choices. A glued band fails for a reason that no count can reach: its crease angles have to satisfy an equation, and a set defined by an equation has no volume in the space it sits in.

    part 5 · flat-folding

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