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 7 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 one strip of triangles, two tubes2 courses of a 6-sided Kresling tube, both diagonals at their own length in eachas made: 1.79 a course, turned 80°second state: 0.99 a course, turned 160°long diagonal 2.204, short 1.824long diagonal 2.204, short 1.824

A twisted tube has two heights

Twist a thin cylinder and it falls into a pattern of slanted triangles that clicks between two lengths. The click is usually explained as a fold, but there is no fold. Every crease keeps its own length at exactly two heights, and in between both diagonals have to stretch or shrink. The second height exists only on a course made taller than the one that folds flat, and the Yoshimura's course, the same pattern with no lean, has its second height on the far side of the tube's own axis.

tessellation · Found patterns where the usual Yoshimura lettering puts its panels, foldedevery course one letter, every zigzag the other; 4 by 4; panels drawn at their folded places

The Yoshimura folds flat at a few heights

The Yoshimura's row height has always been treated as a free dial below the equilateral: the sectors close, the counts hold, the lemma has nothing to forbid. Asked for an actual stack of layers, its usual lettering refuses nearly everywhere. Folded, its triangles turn round one centre by twice the base angle at each step, and they close on a ring only when that angle is π over a whole number: a triangle at sixty degrees, a square at forty-five, a pentagon at thirty-six. At any other height the rosette spirals, and seven panels are enough to prove it cannot pass itself. Below π over the number of rows plus two, it does not get far enough round to meet itself, so the small patch hides this too.

tessellation · Found patterns

Named alongside it

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

The Yoshimura patternEnergy minimisationFound patternGrowthMetricArc lengthDifferential growthUnderdeterminationAssignmentCrumplingCylinderFound patterns

All concepts