Origami & Folding
what one sheet can do
Concept

The offset-panel technique — where it appears

A thickness technique that keeps every hinge axis on its ideal crease line by moving the panels instead. It preserves the zero-thickness kinematics exactly and pays for it in the space the offsets take.

Named by 6 essays across one field — each of them below, with the objects they name alongside it.

zero thicknesspanels meet exactly4 layers of real materialeach fold has to clear the ones belowhinge offset to the surfacethe panel rotates about the right linea crease pattern describes a surface with no thicknessand everything anybody builds has some

The sheet has a thickness

Every crease pattern describes a surface with no thickness. Everything anybody builds has some, and getting it around a corner is the central problem of turning origami into hardware.

rigid · Thickness 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 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 deepest pile on the shelf: 60 layersin office copier paper, that is 18.5 mm of paper to find at one creaseThe Yoshimura pattern60 layers · 18.5 mm · 59× the two-layer allowanceThe waterbomb tessellation32 layers · 9.7 mm · 31× the two-layer allowanceThe Miura fold16 layers · 4.7 mm · 15× the two-layer allowanceThe tapered corrugation16 layers · 4.7 mm · 15× the two-layer allowanceThe square twist9 layers · 2.5 mm · 8× the two-layer allowanceThe preliminary base8 layers · 2.2 mm · 7× the two-layer allowanceThe hexagon twist7 layers · 1.9 mm · 6× the two-layer allowanceFold and cut — the triangle7 layers · 1.9 mm · 6× the two-layer allowancetwo layers

The pile, not the panel

Every technique for building a fold out of panels with depth is drawn, described and priced at one crease between two panels. A folded model has two layers nowhere except at its last fold: the printed patterns here reach eight, sixteen, thirty-two and sixty, and the length a thick panel has to find at those creases is not the published allowance but fifty-nine times it.

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 the bar is the share of the paper in panels that lie over more than one deptheach panel's folded image is laid on the depth map and the depths under it are countedThe preliminary base0.0%uniform · 8 panels · 1 depth under eachThe Yoshimura pattern0.0%uniform · 65 panels · 1 depth under eachThe waterbomb tessellation0.0%uniform · 52 panels · 1 depth under eachFold and cut — the triangle90.0%island · 7 panels · 1 to 3 depths under eachThe hexagon twist92.5%island · 13 panels · 1 to 4 depths under eachThe square twist94.2%island · 9 panels · 1 to 4 depths under eachThe Miura fold100.0%graded · 24 panels · 3 to 4 depths under eachThe tapered corrugation100.0%graded · 28 panels · 4 depths under eacha panel over one depth can be given one thickness; a panel over several cannot

A panel is not the unit of depth

A thick-panel design gives each panel a thickness, an offset or a taper, so a graded pile could be met panel by panel only if every panel's folded image lay over one depth. On the Miura none does. Every one of its twenty-four panels lies over three or four of the four depths its pile takes, and on the tapered corrugation every panel lies over all four. The uniform piles are the opposite — every panel of the Yoshimura, the waterbomb and the preliminary base lies over exactly one depth — which is why panel-by-panel techniques look adequate on the patterns they are drawn for. On the Miura the steps between depths cross the middle of panels, and they run parallel to the panels' own sides.

rigid · Thickness

Named alongside it

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

ThicknessThickness accommodationHingeLayer countManufacturingMembrane hingeTapered panelBoundaryConservationGluingHinge axisKinematics

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