Origami & Folding
what one sheet can do
Series

Mechanical metamaterials — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · tessellation
  2. -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.

    part 2 · tessellation
  3. -0.15-0.00-0.12-3.53-0.46-2.69-0.09-0.300.44the sheet as a whole reports -0.175spread 3.97, 23× the sheet's

    Nothing to average over

    A folded corrugation is reported with a Poisson's ratio, and both of this site's measurements of one were made on a sheet that repeats a single cell. On such a sheet every cell behaves the same way and the cell's number is the sheet's number. On a sheet with no repeating cell the cells run from −3.5 to +0.4 — some widening while others narrow — and the sheet's own figure describes none of them.

    part 3 · tessellation
  4. 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.

    part 4 · tessellation
  5. 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 metamaterial with no edge

    A folded metamaterial's properties are quoted per unit cell, because a material is supposed to be the same everywhere and a cell is supposed to stand for the whole of it. Every cell this collection has measured has been cut out of a patch, with a rim round it — and a rim is the one place a repeating material is not like itself.

    part 5 · tessellation

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