Origami & Folding
what one sheet can do
Concept

Buckling — where it appears

The departure from flatness a sheet makes when its own metric no longer admits a flat state. It is a geometric consequence rather than a material failure, and the pattern it settles into is often one a folder would recognise.

Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.

found rather than designedcrushed drink cans, tree bark,deployable boomsat every interior vertexsectors 60°, 60°, 60°, 60°, 60°, 60°two courses and four diagonalstwo of one letter, four of the otherwhy the height is not freea steeper diagonal makes the topsector strictly smallest, flankedby two of the same letter20 interior vertices · 23 mountain and 56 valley creasesmountainvalleyraw edge

Patterns nobody designed

Crush a thin cylinder and it folds into a diamond lattice. Nobody chose the pattern — it is the buckling mode with the lowest energy, and it satisfies the flat-folding theorems because it just folded.

tessellation · Found patterns what the shell produced14 interior vertices, all alike17 mountain, 40 valley57 creases carrying a letterand it folds flatchecked, not asserted11.0 sheet-widths of crease, chosen by a buckling loadmountainvalleyraw edge

Found before it was designed

Crush a thin cylinder and it falls into a diamond lattice. That pattern was published in aeronautics in 1951, twenty years before anybody designed with it — and what the buckling load chose was not only the creases but the mountain-and-valley assignment, which is the part a designer gets wrong.

history · Rediscovery 00.20.40.60.8100.511.5radius on the flat sheetgrowth factor Ωhow much each ring grew00.20.40.60.81-2-112radius on the flat sheetcurvature Kclosed formthe curvature that forcesgrown more at the rim · K(0) = −4a = -2.40

A sheet that grows cannot lie flat

A leaf does not decide to buckle. Growth changes the distances between a sheet's own material points, a set of distances determines a curvature, and a curvature that is not zero cannot be laid in a plane by anything — whatever the sheet is made of and however slowly it grew.

biology · Growth as metric one excess of length, 4 ways to spend it2 wavesheight 0.0398∫κ² ds = 173 wavesheight 0.0266∫κ² ds = 395 wavesheight 0.0159∫κ² ds = 1088 wavesheight 0.0100∫κ² ds = 276amplitude × waves is constant, so these are one shape at four scalesexcess 6.0% of the span · every profile below has exactly that excess

The excess does not choose its waves

A rim that has grown longer than its span has to put the extra length somewhere, and two large waves and eight small ones are the same metric — identical arc length, identical excess. Geometry fixes the family and is completely indifferent about the member.

biology · Growth as metric wavesamplitudefits inside2∫κ² = 170.03980.043∫κ² = 390.02660.045∫κ² = 1080.01590.04, 0.028∫κ² = 2760.01000.04, 0.02, 0.0112∫κ² = 6210.00660.04, 0.02, 0.0120∫κ² = 17240.00400.04, 0.02, 0.01depth 0.04 needs 2 waves or more · depth 0.02 needs 4 waves or more · depth 0.01 needs 8 waves or moreexcess 6.0% · amplitude × waves = 0.0797 throughout · floor = that constant ⁄ depth

The container picks the member

Two large waves and eight small ones are the same metric, and geometry has nothing to say about which. Bending has nothing to say either — it rises at every step of the family, so a least-bending rule always answers the fewest waves and never anything else. What decides is the container: amplitude times wave count is constant across the family, so a ceiling on the height is a floor on the number, exactly inversely.

biology · Growth as metric

Named alongside it

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

GrowthMetricArc lengthDifferential growthUnderdeterminationThe Yoshimura patternAssignmentCrumplingEnergy minimisationFound patternFound patternsGaussian curvature

All concepts