Origami & Folding
what one sheet can do
Series

Tolerance — the series

9 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 05101520253000.050.10.150.20.250.3creasesdrift, in panel widthsthe same way each timeat randomeach crease 0.5° out · 40 strips averaged for the random casethe drift is a composition of reflections, and would be the same on paper

    Error is folded too

    A folded position is a composition of reflections, and a reflection in a line that is slightly off turns everything beyond it by twice as much. So an error does not stay where it was made — and whether it grows with the crease count or with its square root depends on whether it is the same error every time.

    part 1 · rigid
  2. 0°40°71°travel before contact72.1°measured by contact testpanel thickness22% of the panel lengthwhat it gives upa gap, and with itstiffness and a pattern nolonger quite the one onpapera zero-thickness pattern says the panels meet along a line; nothing that is built does

    Nowhere to put the error

    Paper takes a misplaced crease and spreads it along its whole length as a curvature nobody notices. A panel is flat by definition and cannot, so the error arrives at the hinge — and the room to receive it is a length that has to be drawn, is paid for in fold angle, and has to grow with the crease count.

    part 2 · rigid
  3. the ratio of each column's cosine to the first column'scol 1col 2col 3col 4col 5the sheet that folds1.0000-1.08221.0000-1.08221.00001.0000-1.08221.0000-1.08221.00001.0000-1.08221.0000-1.08221.0000row 1row 2row 3one row moved by 14 per cent1.0000-1.08221.0000-1.08221.00001.0000-1.13971.0000-1.13971.00001.0000-1.08221.0000-1.08221.0000row 1row 2row 3worst disagreement between rows: 4e-16 before, 0.0575 after

    Where an error goes

    A misplaced crease in a folded sheet has to be paid for somewhere, and this subject has two answers already — the error is folded too, and the hinge is where it ends up. There is a third. In a quadrilateral mesh a mistake in one row has no consequence in that row at all: it is felt by the columns, which is to say by every other row on the sheet.

    part 3 · rigid
  4. the solved mesh, cut wrong byworst mismatch left, radians0.017 mm0.00210.051 mm0.00580.169 mm0.01990.508 mmno closure at alla Miura, cut wrong by0 per cent2e-141 per cent6e-155 per cent4e-15

    Solved is not built

    A mesh that folds because an equation holds and a mesh that folds because one crease family runs straight through every vertex are not two examples of the same thing. Cut a Miura's every dimension five per cent wrong and it still folds exactly. Cut a solved general mesh a fifth of a millimetre wrong on a 150 mm sheet and the closure is gone.

    part 4 · rigid
  5. 00.20.40.60.81-7-6-5-4-3-2-1error in every length, millimetres on a 150 mm sheetclosure mismatch (powers of ten)across the surface of solutionsthe direction the closure's own derivative points ina direction chosen without regard to the surfacewhich has a component along bothalong the surface of solutionsone of the twelve directions the equations do not seethe same step costs 5,130 times as much one way as the other

    A tolerance is a direction

    Cut a solved mesh a fifth of a millimetre wrong and its closure is gone. That is true of the errors it was tried with and false of errors in general: the solutions form a surface sixteen directions wide, an error along it costs five thousand times less than the same error across it, and the fifth of a millimetre is the allowance in one direction out of twenty.

    part 5 · rigid
  6. the height is the allowance in millimetres on a 150 mm sheetthe horizontal axis is the fold angle of the driven crease, in radians0.425 mm0.033 mm0.32.4fold angle of the driven creasea budget of 0.02 radians on a 150 mm sheet12.9 times less allowance at the closed endthe mesh is most forgiving where it is doing least

    The allowance is spent at the end

    A tolerance on a solved mesh was priced at one fold angle, because that is where a tolerance is priced. The surface of solutions turns out not to move as the sheet folds — the free directions at a third of a radian are the free directions at two and a half, to twelve figures — and the price of leaving it rises by a factor of thirteen along the way.

    part 6 · rigid
  7. the bar is the vertices the drawing has and the list does notThe preliminary base09 listed · panels closeThe Miura fold035 listed · panels closeThe square twist016 listed · panels closeThe hexagon twist022 listed · panels closeThe Yoshimura pattern045 listed · panels closeFold and cut — the triangle011 listed · panels closeThe tapered corrugation040 listed · panels closeThe waterbomb tessellation041 listed · panels closethe square grid, assembled064 listed · panels closethe triangular grid, assembled1282 listed · panels 1.73 apartthe honeycomb, assembled1884 listed · panels 2.00 apartthe rhombille tiling, assembled12138 listed · panels 1.86 apartthe elongated triangular tiling, assembled576 listed · panels 1.73 apartevery pattern with a bar has panels that cannot be placed, and every pattern without one places exactly

    How deep is a crossing

    A crossing is a verdict with no middle: two creases either pass through one another or they do not, and the first makes a pattern unfoldable while the second leaves it untouched. Measured on the patches where they occur, the shallowest crossing runs 0.16 mm past the end of the crease it meets, on a sheet 150 mm across. Five of the forty-seven are under half a millimetre, which is thinner than the line a pencil draws.

    part 7 · rigid
  8. the other fault a drawing can have, counted and measureda stub is a crease with a free end; the distance is how far from the rim it stoppedpatcheswith a crossingwith a stubstubsdistinct depthsshallowestdeepest120761566270.27 mm35.1 mm0.27 mm35.09 mm66 stubs27 depthsdistances at the 150 mm these patterns print at; the scale is logarithmic because the range is a factor of 128

    A stub is never alone

    A crossing is a crease running past another and it has a depth. A stub is a crease that simply stops, and it has one too — how far from the rim it stopped, which is also how much shorter than a crease it is. Measured across a hundred and twenty drawings: sixty-six stubs, from 0.27 mm to 35 mm at printed size, every one of them paired with another at exactly the same distance, and not one on a drawing that did not already have a crossing.

    part 8 · rigid
  9. drawings carrying each fault, of those drawna junction is split in two: the fault, and the rim ending the reading also calls a junctionextended, of 120clipped, of 120a crossing760two creases pass through each othera stub150a crease stops in the middle of the papera crease ending on a crease00the junction as a faulta crease ending on the rim120120the junction as the reading counts ita fragment25a crease too short to seeevery stub is on a drawing with a crossing; every clipped fragment is on a drawing with nothing else wrong

    Two faults, not four

    A drawing departs from its crease list in four named ways — a crossing, a stub, a junction and a fragment — and a checker tests for all four. Counted side by side on two hundred and forty drawings, two of the tests find nothing the others do not. No crease anywhere ends on another crease: every junction the reading finds is a crease meeting the rim, which is where creases are meant to end. Every stub is on a drawing that already has a crossing. What is left is two independent faults, and the clipped construction, which makes no crossings, still makes the second — creases a thousandth of a millimetre long, in pairs.

    part 9 · rigid

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