Origami & Folding
what one sheet can do
Concept

Energy minimisation — where it appears

Reading a structure's resting state as the lowest point of the energy stored in its hinges. It settles what a spring-loaded pattern does when nothing holds it, and it is what turns a question about forces into a question about the shape of a surface.

Named by 8 essays across 2 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 -3-2-1123050100150turn in each hinge (radians, signed by branch)energy storedrests hereand hereflat: 179the barrier isthe flat state8 folds, hinge 0.05creased to 1.6the two branches are the same sheet folded opposite ways, and the flat sheet is the only state they share

Springs that disagree do not offer a choice

A corrugation of hinges that remember different angles was expected to have more than one position in which nothing pushes. It has exactly one, at the stiffness-weighted mean of what they remember, because a sum of parabolas in one variable is a parabola. The second resting place comes from somewhere else entirely — the mirror pattern — and the flat sheet is the barrier between them, which is also why a creased sheet cannot be pulled flat at all.

biology · Convergence a corrugation remembering 0.8 rad, held at 2.4 and later opened to 0.4torques as shares of the one total both ends of the story share; the barrier as a multiple of its creased valueheld for, τremembered turncontainer suppliesopening needsbarrier0.00.80080%20%1.00×0.51.43049%51%3.19×1.01.81129%71%5.13×2.02.18311%89%7.45×3.02.3204%96%8.41×5.02.3891%99%8.92×the container's share and the opening's share always sum to the whole, whatever the holding time

Holding a fold moves the force

A creased hinge held at an angle slowly comes to remember that angle, so a leaf or a wing packed in a bud for a season is gradually holding itself and the bud has less to do. The force is not used up in the process. The torque the container must supply falls as e^(−T⁄τ), the torque later needed to open the structure rises by exactly the same amount, and the two sum to the same total at every moment of the holding. Held for three relaxation times, a corrugation creased to 0.8 radians and packed to 2.4 needs 4 per cent of the total from its container and 96 per cent from whatever opens it — and the barrier to its mirror image has grown more than eightfold. A packing that lasts buys independence from its container with a harder unfolding.

biology · Convergence 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 four courses holding the word 1010each course clicked shut or left open, read from the bottom upA: made 1.30 H, shut, 1.24 tallB: made 1.45 H, open, 2.16 tallC: made 1.60 H, shut, 1.86 tallD: made 1.75 H, open, 2.61 tall

Four clicks make sixteen lengths

A Kresling tube of four courses has sixteen ways to be at rest, each course shut or open. Pressing and pulling it under one force can reach all sixteen or only five, and the difference is set by how tall the courses were made. Every course shuts at a force that falls as it is made taller, while the force that opens it rises to a peak and then falls. A stack graded below that peak opens shortest first and can only count. Graded above it, it opens tallest first and can be written like a register.

tessellation · Found patterns four courses of alternating hands, ends clamped, holding 0011a right-handed course and a left-handed one shut together, so the ends need not turnright-handed, made 1.30 H: openleft-handed, made 1.45 H: openright-handed, made 1.60 H: shutleft-handed, made 1.75 H: shut

Clamped, a stack counts in pairs

A Kresling tube of four courses, its ends free to turn, can be pressed and pulled into all sixteen of its words. Clamp the ends so they cannot turn and the courses stop being independent switches, because a course that clicks shut turns by a fixed angle — eighty degrees here, whatever height it was made at — and the others must give it back. With every course of one hand the stack does not click at all, at more than twice the force that shuts any course free. With two courses of each hand it clicks in pairs, one of each hand together, so that their turns cancel: every word it reaches has as many right-handed courses shut as left-handed, and it reaches exactly four. The clamp turns a sixteen-word register into a two-switch one whose every state leaves the ends where they were.

tessellation · Found patterns a vertex held at its exact mirror imagesectors 60°, 90°, 120°, 90°, creased to a drive of 1.6 rad, with the rest angles relaxing toward the held angles0510152001234time held, in relaxation timesenergy in the springsthe flat sheetthe creased statethe held state

A crease held the other way

Hold a creased sheet on the far side of flat and its memory crosses zero after τ ln(1 + φ₀⁄φₕ), a time that contains no stiffness and no fold count. Asking what happens at that moment shows that two accounts of a corrugation's energy cannot both stand: a hinge whose memory can cross zero gives one parabola and no barrier at all. The barrier belongs to a degree-four vertex, whose two branches turn out to be at right angles in the space of its fold angles. Held on the other branch its wall dips and never falls, and its two resting states trade depth at nearly the same logarithm, with lengths in place of angles. Only the exact mirror takes every rest angle through zero together, which is the fastest way to erase a crease and the least forgiving.

biology · Convergence 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.

Found patternSymmetry breakingThe Yoshimura patternBucklingConvergenceCorrugationCrease radiusCylinderMembrane hingeCrumplingDeployable boomRigid folding

All concepts