Origami & Folding
what one sheet can do
Concept

Symmetry breaking — where it appears

A structure with symmetric alternatives ending up in one of them. It is what a self-folding sheet must achieve and what a pattern repeating a single vertex makes hard, because the symmetry is exactly what stops a neighbour telling the two branches apart.

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

-10010000.20.40.60.81how far the vertex is drivenstored energybranch oneMVMMbranch twoMVVVboth run downhillfrom the flat state,and end at zeroso the energy does notprefer either branch —the noise decides

Paper that folds itself

A self-folding sheet has to supply the fold and then choose what to fold into. The second half is where these things fail, and no amount of torque helps, because the two outcomes are equally downhill.

rigid · Self-folding -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 how many folded states each crease of a four-by-four Miura leavesdriven to 0.6 radians, with every consistent assignment enumerated rather than the first eight1 state4an actuator belongs on one of these2 states82 states, so the sheet has a choice4 states44 states, so the sheet has a choice8 states88 states, so the sheet has a choicethe four that leave one are c:3:2, c:3:3, r:3:2, r:3:3 — all of them at the same corner of the sheet

Only four creases decide a Miura

Driving one crease of a rigid quadrilateral mesh settles every other one — except that on the pattern everybody builds it often does not. Enumerated properly, four of a four-by-four Miura's twenty-four creases leave exactly one folded state and the other twenty leave two, four or eight. A mesh whose vertices all differ leaves one from every crease. The ambiguity is not a property of quadrilateral meshes; it belongs to the symmetry.

rigid · Self-folding 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 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

Named alongside it

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

Energy minimisationConvergenceCorrugationCrease radiusMembrane hingeActuationBranch selectionRigid foldingSelf-foldingBifurcationBucklingCylinder

All concepts