Origami & Folding
what one sheet can do
Series

Thickness — the series

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

    part 1 · rigid
  2. 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

    Getting thickness round a corner

    There are half a dozen ways to build a fold in a panel that has depth, and the useful way to arrange them is not by what the cross-section looks like. It is by what each one gives away.

    part 2 · rigid
  3. 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.

    part 3 · rigid
  4. 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.

    part 4 · rigid
  5. 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.

    part 5 · rigid
  6. 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.

    part 6 · rigid
  7. the bar is the share of the footprint at least half as deep as its deepest pilethe note is how deep that pile is and how much of the footprint it coversThe preliminary base100.0%uniform · deepest 8 layers over 99.7%The Yoshimura pattern100.0%uniform · deepest 60 layers over 100.0%The waterbomb tessellation96.6%uniform · deepest 32 layers over 96.6%The tapered corrugation87.9%graded · deepest 16 layers over 0.9%The Miura fold75.5%graded · deepest 16 layers over 11.9%The hexagon twist24.5%island · deepest 7 layers over 24.5%The square twist17.5%island · deepest 9 layers over 17.4%Fold and cut — the triangle3.0%island · deepest 7 layers over 2.9%uniform: the pile is the pattern · island: the deep region is a patch · graded: deepest on a sliver, half as deep nearly everywhere

    Three kinds of pile

    A thick-panel technique is priced at the deepest pile a pattern has, and the depth of that pile says nothing about where it is. Mapped over the folded footprint, the printed patterns fall into three kinds. On a uniform pile the deepest count is the whole footprint — sixty layers everywhere on the Yoshimura. On an island it is a patch and the rest is shallow. On a graded pile it is a sliver — under one per cent of the tapered corrugation — while nearly nine tenths is at least half as deep, and the Miura, the pattern that gets built, is graded.

    part 7 · rigid
  8. 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.

    part 8 · rigid
  9. a Miura 16 columns wide and one row deep, folded flat and laid along its length: layers at each pointeach shade is one depth; the key gives the share of the footprint at eachslant 20.05°: the columns overlap 3.10 deep on average1 deep 11.1%2 deep 26.1%3 deep 41.8%4 deep 21.1%slant 15.00°: the columns overlap 4.00 deep on average1 deep 10.5%2 deep 10.5%3 deep 10.5%4 deep 68.4%

    The slant that stacks in one depth

    A folded Miura's rows lie exactly on one another, and its columns each lie one step along the folded strip from the last, so the pile at any point is the rows times the number of column images over it — on average two over the sine of twice the slant. When that number is an even whole number the pile is one depth everywhere but the ends: two columns deep at 45 degrees, four at exactly 15. When it is odd the pile stays mixed, because neighbouring columns lean opposite ways. The printed Miura at 20.05 degrees sits beside the odd count three, and its graded pile is that and nothing more; at 15 degrees the same single-freedom mechanism stacks every panel away from its ends over one depth.

    part 9 · rigid
  10. three ways to fold a corrugation of 24 columns, laid along its length and shaded by depth a roweach shade one depth a row; the two rows fold exactly onto each other1 deep2 deep3 deep4 deep5 deep6 deep7 deepequal columns: 78% at 4 deepthe taper at its best single slant: 55% at 4 deepthe taper, each zigzag leaning its own amount: 62% at 4 deep

    The slant belongs to the line

    A corrugation's columns pile 2h ⁄ (w sin 2a) deep a row — twice the row height over the column's width times the sine of twice the slant — so a tapered corrugation at one slant piles its narrow columns deeper than its broad ones. The slant looked like a property of the whole pattern. It is a property of each zigzag line: at a corrugation's vertex the row creases are one straight line and the zigzag leans the same amount either side of upright, so the angles pass Kawasaki's condition whatever each zigzag's own lean. Leaning every zigzag for the columns beside it takes a 24-column leaf taper from 55 per cent of its footprint four deep a row, at the best single slant, to 62, and cuts the part piled deeper than four from 22 per cent to 3.

    part 10 · rigid
  11. straight creases at 60°: the zigzags between 78° and 132°the middle has turned 23° out of the plane of the first columnevery straight crease folds at one angle; each zigzag at its own, set by its lean

    Parallel creases forgive every lean

    A corrugation graded so its tapered columns pile evenly leans every zigzag line its own amount, and the question left was whether it still folds rigidly on the Miura's one freedom. It does, and it could not have failed to. With straight creases that are parallel, a zigzag reflecting in them keeps one angle to them all the way along, so the table of cosines that decides rigid folding has a constant in every row whatever the leans. The published rule that such rows must be copies of one another holds only when the straight creases fan. What grading costs is the shape of the motion: the zigzags close at their own rates, the strip bends up to 24 degrees out of its plane early in the fold, and the leaf opens from its ends.

    part 11 · rigid

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