Origami & Folding
what one sheet can do
Concept

Poisson's ratio — where it appears

The ratio between how much something narrows one way and stretches another. For a folded corrugation it is a property of the configuration rather than of a substance, and on a sheet with no repeating cell it is not even one number.

Named by 5 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 00.20.40.60.80123how far the sheet is closed−ν, so every curve shown is a negative ratioMiura, slant 0.25Miura, slant 0.42Miura, slant 0.6accordionexactly zerothe same paper,three behaviours,chosen by the creasepattern alone

A material made of creases

A folded sheet has a Poisson's ratio, a stiffness and a packing behaviour that the flat sheet did not. None of them belong to the paper — they belong to the pattern, and changing the pattern changes them without changing the material at all.

tessellation · Mechanical metamaterials -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 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 the bar is the average number of layers over the folded footprint0.5 of the sheet3.891 whole of 9 · 89% cut by the rim0.42 of the sheet3.641 whole of 9 · 89% cut by the rim0.34 of the sheet4.049 whole of 9 · 0% cut by the rim0.28 of the sheet4.309 whole of 21 · 57% cut by the rim0.22 of the sheet4.349 whole of 25 · 64% cut by the rim0.18 of the sheet4.7925 whole of 45 · 44% cut by the rima bulk property arrives as the boundary leaves, and neither a page nor a sheet of paper reaches the end of it

The property a patch does not have

A folded corrugation is described as a material — a packing ratio, a stiffness, a Poisson's ratio — and every one of those is a statement about an unbounded medium. Fold the same tiling at six sizes on the same square and the compaction climbs from 3.89 layers to 4.79 as the share of units the rim cuts falls from nine tenths to four, and it has not settled at the fine end. The number a patch gives is the material's number minus its own boundary.

tessellation · Mechanical metamaterials

Named alongside it

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

AuxeticMetamaterialKinematicsPacking ratioTunabilityBoundary vertexDegrees of freedomDesign spaceFolded stateGeometric propertyHingeKirigami

All concepts