Origami & Folding
what one sheet can do
Concept

Footprint — where it appears

The region of the plane a flat folded state covers. It is what a silhouette records, and its area times the average layer count returns the sheet, which is the conservation law behind every shrink factor here.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

the patternthe panels, foldedsheet 12.000footprint 1.966 · 6.11 layers on average · 12 at the deepest1.966 × 6.11 = 12.007, which is the sheet

The paper is all still there

A folded sheet is smaller than it was and none of it has gone anywhere. How much smaller it is and how many layers deep it is are not two properties of a pattern — they are one number, and their product is the sheet.

flat-folding · Conservation 00.10.20.30.40.500.20.40.60.81flap width, as a fraction of the sheetarea showingthe two are equal at a thirdfront showingreverse showingtotal facemeasured on the folded state at 4 flap widths, and the marks are those measurements

Bringing the other side to the front

Paper has two sides and most models show one. A colour change shows the other, and it is not a crease problem — which panels can show the reverse is settled by the pattern's two-colouring, and what it costs is twice what it shows.

design · Colour change 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.

tessellation · Shrinkage 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.

tessellation · Shrinkage 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.

tessellation · Shrinkage 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.

tessellation · Shrinkage

Named alongside it

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

ConservationFolded statePacking ratioCorrugationDeploymentLayer countMeasurementShrinkageTessellationAnisotropyBounding boxColour change

All concepts