Origami & Folding
what one sheet can do
Concept

Hinge — where it appears

The joint a fold becomes when a sheet is replaced by panels with thickness. Where its axis is placed decides how far the fold can travel and whether the original kinematics survive.

Named by 8 essays across 2 fields — each of them below, with the objects they name alongside it.

010203040-3-2-10how far open — degrees for the cut sheet, the same fraction of the motion for the foldPoisson's ratiocut into squares−1 everywhere, exactlya fold, slant 0.35-0.12 at the startand without limit at the endthe two cross onceand agree nowhere elsethe flat line is a finite difference of two measured widths, taken the same way as the curve beside ita material made of matter cannot change its Poisson's ratio as it deforms; a material made of geometry can

Bought with holes

A Miura-folded sheet gets wider as it is pulled, and by how much depends on its panels and on how far it happens to be folded. A sheet cut into squares joined at their corners does the same thing and holds the value at exactly minus one, everywhere in its motion — the same property, bought with different geometry, and paid for in holes.

material · Kirigami cuts at 85% of the pitch0.15 of material between one cut and the nextcut length, as a fraction of the pitchligament6 piecesone piece at every length up to 97%and the count is worked out from the cuts, not measured off a picture

One cut short of falling apart

Everything a cut sheet can do is bought out of the material between the end of one cut and the start of the next. That material shrinks to nothing in a straight line as the cuts grow, and the sheet stays in one piece the whole way down — until the instant it does not, and then it is in six.

material · Kirigami how far the fold closes, against where the hinge sitslower faceupper facemid-surface180°0°closing upwardclosing downwardtapering the panels buys most of it back: 166° of the 180°, less twice the taper

Thickness has a sign

Swap every mountain for a valley and back again. Kawasaki does not notice, Maekawa gets the same condition the other way round, the lemma still asks the two creases to differ, and the layers come out mirrored. Every theorem on this site is blind to which side of the paper it is looking at — and a hinge in a panel with depth is not. The fold closes one way and jams at nothing at all the other.

rigid · Thickness 0°61°108°travel before contact110.0°measured by contact testpanel thickness22% of the panel lengthwhat it gives upmaterial at the crease, sothe panel is thinnest whereit is worked hardesta zero-thickness pattern says the panels meet along a line; nothing that is built does

Thickness round a closed loop

Real panels have thickness, and every technique for accommodating it works by shifting a hinge off the ideal crease by a small amount. On a flat sheet the shifts accumulate outward and end at the edge. On a closed sheet they accumulate round a loop and have to come back to where they started, which is a condition none of the techniques was designed to satisfy.

rigid · Thickness four Miuras at a slant of 0.35, folded flat and drawn at one scalethe first two differ only in rows, the second two only in columns4 columns by 2 rows0.939 by 2.4364.07 layers deep4 columns by 6 rows0.939 by 2.43612.21 layers deep2 columns by 4 rows0.939 by 1.7506.04 layers deep6 columns by 4 rows0.939 by 3.1229.23 layers deep

The rows are free

A Miura's hinges per layer of compaction were found to climb without a ceiling as the pattern is refined, and the footprint was blamed for growing with the sheet. Measured one direction at a time it grows in only one. Every band between two straight creases folds exactly onto the band before it, as an accordion's pleats do, and the folded box does not change at all as they are added. Every band between two zigzags walks the stack a step along itself, the sine of the slant each. So the hinges a layer costs settle as rows are added and climb as columns are, and the same sixty-four cells cost 1.94 hinges a layer laid out long in their rows and 16.14 laid out long in their columns.

rigid · Deployables a 16 by 4 Miura cut along its zigzags into modules, folded and stackedslant 0.35; each module folds to its own box and the stacked modules share itmodulescolumns eachbox lengthlayers, stackedhingeshinges a layeractuators1166.55111.01089.791283.80819.71045.292442.43632.6962.954821.75048.3801.6681611.40764.0480.7516a cut removes one zigzag's hinges and adds one module to drive; the box and the hinges a layer both fall

A cut along the zigzag needs a tether

A Miura's folded stack walks along itself by the sine of its slant for every column, so a sheet cut along its zigzags into modules folds into boxes a fraction as long and stacks far deeper for the hinges. But the pieces cannot stay joined by a crease: a zigzag crossed by row creases is a line of four-crease vertices, and joined there the modules are the one Miura again. The join has to be a tether and every module its own actuator — and a cut, which removes a zigzag's worth of hinges, pays for itself in complete success only when an actuator fails less often than those hinges together.

rigid · Deployables 00.511.520.60.70.80.91failures crowded into one deployment in 10^xchance10 modules, share expected10 modules, at least 90%one freedom10 modules, all of it300 hinges at 0.999 each on average, actuators 0.99 · to the right, the same failures crowd into fewer deployments

Splitting insures only the failures that come alone

Three hundred hinges split among ten separately driven modules were found to open 96 per cent of their area on average against 73 for the same hinges as one mechanism. That gain rests on every hinge failing on its own. Let the same average failure rate arrive in a few bad deployments instead — a cold soak, a crushed stowage — and the one mechanism gains far more than its modules do, because a shared cause makes three hundred hinges behave like one. At one bad deployment in a hundred the lead is gone. A tether that passes a jam to its neighbour erodes it from the other side, and takes the chance of nearly all the area long before it takes the average. Neither touches the chance that everything opens.

rigid · Deployables a 10 by 10 Miura at a slant of 20.1°, with two of its places markedzigzag line 3: 10 hinges, one a rowthey all fold onto one segmentcolumn 3, odd lines: 5 straight hingesthey fold onto one segment toocut between columns into 5 modules

A stack is a shared fate

Hinges that fold onto one segment of a folded stack share whatever happens to that segment. On a Miura every zigzag line puts one hinge from every row in the same place, so a sheet cut between its rows into five modules leaves every place in every module, and one jam anywhere stops all five. Cut between its columns, each place stays inside one module — except at the slants where two over the sine of twice the slant is an even whole number, the very slants that pile the stack evenly, where columns that far apart share places across the cut.

rigid · Deployables

Named alongside it

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

DeploymentMiuraActuationDegrees of freedomFolded stateFootprintIndependenceKirigamiManufacturingThe offset-panel techniqueReliabilityThickness

All concepts