Origami & Folding
what one sheet can do
Series

Shrinkage — the series

6 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. patternhow much smaller it foldscreasingper sheet-widthpreliminary8.0 layers, 8 at the deepest8.0×4.81.65×footprint × depth = 1.004 of the sheetmiura8.1 layers, 16 at the deepest8.1×6.21.31×footprint × depth = 1.001 of the sheetyoshimura32.0 layers, 36 at the deepest32.0×11.82.71×footprint × depth = 1.000 of the sheetwaterbomb31.6 layers, 32 at the deepest31.8×14.32.22×footprint × depth = 0.992 of the sheettwist3.0 layers, 9 at the deepest3.0×4.70.64×footprint × depth = 0.995 of the sheetthe shrinkage is the pattern's, not the paper's — nothing here knows what the sheet is made of

    What a corrugation costs

    Every tessellation this repository can fold, measured the same way: how much smaller it gets, how deep the stack becomes, and how much creasing was needed to buy it. The last column is the one nobody quotes and the one a folder feels.

    part 1 · tessellation
  2. 24682468times shorter across the sheettimes shorter down itaccordionMiuraleaf corrugationYoshimurasquare twistwhat each one doesaccordion8.00× across, 1.00× downMiura2.73× across, 1.75× downleaf corrugation3.57× across, 1.22× downYoshimura4.00× across, 4.00× downsquare twist1.52× across, 1.52× downthe dashed line is a directionthe fold never touchesone direction untouched: 1 of 5 · equal both ways: 2 · neither: 2the single number a corrugation is usually quoted by is these two multiplied together

    A shrink is two numbers

    How much smaller a folded sheet gets is quoted as a single factor, and that factor is a product. Measured along each axis separately, an accordion turns out to leave one direction of the paper exactly alone, a twist draws in equally both ways, and the Miura does neither — which is the whole of what makes it a Miura.

    part 2 · tessellation
  3. one similarity, five tilingslong: a cell of the flat sheet · short: where it lands foldedthe square grid ×0.41037344the triangular grid ×0.41037344the honeycomb ×0.41037344the elongated triangular tiling ×0.41037344the rhombille tiling ×0.41037344turned 36.62°, the same on every onethe scale is a property of the pleat, and the tiling does not enter it

    Folding it flat is one similarity

    Where a cell of paper goes when a twist tessellation collapses is a scale and a turn: multiply the plane's lattice by 0.410373441 and rotate it by 36.62°. That is the answer on the square grid, the triangular grid, the honeycomb, the elongated triangular tiling and the rhombille alike, agreeing to eight figures — while the collection's other answer to how much smaller it gets gives those five tilings five different numbers.

    part 3 · tessellation
  4. the folded period of the Yoshimuradrawn periods across the top123456the turnsame way upslides240°yesno120°yesno0°yesyes240°yesno120°yesno0°yesyesa turn of 240° comes back to nothing after three of them, and that is the folded periodthe drawing repeats every one, which is what makes it a tessellation

    The turn a column costs

    The Yoshimura's drawing repeats every column. Folded flat, it does not: the fold carries one column onto the next by a turn of two hundred and forty degrees, so the folded state repeats every third column and not before. A pattern has two periods and only one of them has ever been written down.

    part 4 · tessellation
  5. each family's dial, run, in the plane of the two directional factorsmeasured on the folded state of every pattern, not predicted from its rule1234561234along the patternacross itthe accordionthe square twistthe Miura foldthe Yoshimuraa point on the lower edge leaves one direction alone; a point on the dashed diagonal draws in equally both ways

    The plane the five points were in

    Five corrugations measured at one setting each gave five points, and the space between them was left as an open question: forbidden, or merely unvisited. Every one of those patterns has a dial nobody turned. Turned, they trace curves — the accordion's is a line with integers on it, the square twist's is the diagonal and nothing else, and the Miura's turns round on itself, so a steeper slant stops buying a smaller sheet.

    part 5 · tessellation
  6. the cross factor of a leaf corrugation against its zigzag anglemeasured off the folded state's own extent, at every angle drawnno change0.600.680.760.840.921.00zigzag angle, radians0.981.001.02below one between 0.7 and 0.88 radians, at worst 0.98555 — and the areal factor never falls below 1.360

    The direction that gets longer

    A shrink factor below one is a direction in which the folded sheet is bigger than the flat one, and the leaf corrugation has one. The cause is not the taper and not the angle: a corrugation's folded extent across its own creases is a constant of the cell, the same number at two rows and at ten, so the cross factor is the sheet's height divided by a fixed length — a straight line through the origin that crosses one at a definite row count.

    part 6 · tessellation

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