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 4 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

Named alongside it

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

KirigamiManufacturingThe offset-panel techniqueThicknessAuxeticBoundaryConnectivityGluingIdealisationMaekawa's theoremMechanismMountain-valley

All concepts