Origami & Folding
what one sheet can do
Concept

Kinematics — where it appears

The study of motion without asking what causes it. Rigid origami is kinematics: the fold angles move together according to closure conditions and no force appears anywhere in the account.

Named by 7 essays across 2 fields — each of them below, with the objects they name alongside it.

nearly flatwidth ×0.91 length ×0.98ν = -0.22half closedwidth ×0.66 length ×0.88ν = -0.52nearly packedwidth ×0.45 length ×0.55ν = -2.93both dimensions shrink together — pulling it open in one direction opens it in the other

A sheet with one freedom

A Miura-folded sheet can move in exactly one way. Pull it open in one direction and it opens in the other — a negative Poisson's ratio, arriving entirely from the crease pattern and not at all from the paper.

tessellation · Miura the pattern as it isevery edge keeps its length6.7e-16the same pattern, moved by 0.01and one of them cannot1.7e-21e-181e-161e-141e-121e-101e-81e-61e-41e-21largest change in any edge length, in panel widthswhat an isometry has to do, and what it manages5 × 4 panels, at 50% folded, every edge of both comparedthe moved pattern is fitted the best single panel its own edge lengths allow before being folded at allso the gap is not a bad choice of panel — it is what is left when the best choice has been made

Panels instead of paper

Flat-foldability asks whether a pattern can reach a flat state. Rigid-foldability asks whether it can get there without any face bending on the way. The second is much stronger, and everything that gets manufactured lives inside it.

rigid · Rigid folding -0.02-0.02-0.03-0.04-0.06-0.10-0.19-0.49-0.05-0.06-0.07-0.10-0.15-0.23-0.43-1.01-0.10-0.13-0.16-0.21-0.30-0.45-0.81-1.79-0.20-0.24-0.30-0.39-0.53-0.79-1.36-2.93-0.34-0.41-0.50-0.64-0.85-1.24-2.10-4.41-0.54-0.65-0.79-0.99-1.31-1.88-3.13-6.48-0.80-0.94-1.14-1.42-1.87-2.64-4.36-8.950.120.200.300.420.550.700.850.080.200.320.440.560.680.800.90panel slanthow far the sheet is closedwhat the map saysevery cell negativefrom -0.02 to -8.95a factor of 535.6a material has one value;this has a working point,and it is chosendarker means more negative

A property you can dial

Steel has one Poisson's ratio. A Miura-folded sheet has a surface of them, and where on that surface it sits is set by the panel shape and by how far it happens to be folded — which is why it is a mechanism rather than a material.

tessellation · Mechanical metamaterials grown symmetricallyoffset to one sidepanels overlap over 4% of their areawhich is the jam every thick-panel design meetspanels do not overlap at alland the hinge axes have not movedwhich is not free: fold it tighter and this offset runs out toosheet 0.16 panel-lengths thick, folded to 100° — the overlap is measured from the geometry

Panels with somewhere to go

Every way of giving a folded panel real thickness costs something. Tachi's offset-panel technique costs the least interesting thing there is — it stops the panels being a surface, and leaves the hinges exactly where the zero-thickness pattern put them.

rigid · Thickness 1.7e-6 at 1e-61e-61e-51e-41e-31e-21e-61e-51e-41e-31e-2how far each vertex of the flat pattern was moved, in panel widthslargest edge-length errorfitted slope0.9999over four decadesat a displacement ofexactly zero the erroris 6.7e-16, which iswhere the arithmeticstops and not wherethe geometry does5 × 4 panels at 50% folded, with one set of displacement directions scaled across the decadesa slope of one is the claim: the failure is first order, so no displacement is small enough to be free

The only pattern that moves

A rigid motion is not a generic property of a folded pattern. Move one vertex of a Miura by a thousandth of a panel and the sheet has no isometric folded position of that kind at all — and the amount by which it fails is first order in the displacement, so no move is small enough to be free.

rigid · Rigid folding the opposite crease, ×1the odd crease, ×0.26794910how far the vertex has foldedsectors 60° · 90° · 120° · 90°constant to 8.2e-13 over 199 points of the motionand equal to cos((α+β)/2) ÷ cos((α−β)/2), which the solver never forms

The vertex is geared

A rigid four-crease vertex has one degree of freedom, which says that one number decides everything and not how. The how is a fixed ratio: the tangents of the half fold angles at two creases stay in constant proportion for the whole of the motion, and the proportion is a function of the sector angles and nothing else.

rigid · Rigid folding the period cell of the Miuraone period, with its neighbours round it2 interior vertices in the cell7 crease pieces drawnperiod 1.000 × 2.000one column wide and two rows high, because the zigzag returns after twothe cell is a rectangle of ordinary paper until somebody says its edges are one edge

A mechanism that closes on itself

A rigid-foldable pattern is a mechanism: panels as rigid plates, creases as hinges, and a motion counted by degrees of freedom at each vertex. Close the sheet into a tube and the mechanism has to come back to itself after a circuit — a constraint that is not at any vertex and that the degree-of-freedom count does not see.

rigid · Rigid folding

Named alongside it

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

Degrees of freedomMechanismRigid-foldabilityMetamaterialMiura-oriPoisson's ratioAuxeticConstraintDesign spaceFold angleGluingHinge axis

All concepts