Tolerance — where it appears
Named by 20 essays across 5 fields — each of them below, with the objects they name alongside it.
Almost every pattern fails
Kawasaki's condition is one equation for each interior vertex, and a drawing satisfies an equation with probability zero. Every pattern on this site folds because it was constructed to, and the fraction that would fold by accident can be measured.
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.
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.
A near miss is nearly as rare
Flat-foldability is a coincidence of measure zero, which is usually where the argument stops. Measure how far a random vertex is from folding rather than whether it does, and the answer is thirty-one degrees a sector — so the tolerance real paper has does not buy back anything at all, and a pattern that nearly folds had to start near one that did.
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.
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.
Exact is not accurate
There are two ways of dividing a strip into equal parts: a ladder that lands on the fraction as a rational number, and Fujimoto's method, which never arrives. Read as mathematics that settles it. Read as instructions for somebody with a sheet of paper it settles nothing, and past four parts the method that never arrives is the one whose crease lands nearer the mark.
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.
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.
What buys the reach costs the accuracy
A reference is a crossing of two creases, and a crossing transmits a folding error multiplied by one over the sine of the angle the creases make. Measured across the whole closure on a square, the four linear axioms never produce a crossing shallower than thirty-seven degrees — and the conic axiom, the one that sends a point onto a line and reaches the heptagon, produces crossings under seven.
The helix chooses the lattice
A strand can cross to the next helix only where its backbone faces that helix, and a double helix turns about 34.3° a base. Three neighbours a third of a turn apart are faced exactly every seven bases. Four neighbours a quarter of a turn apart are never faced exactly by any whole number of bases — the nearest miss by 4.3°, and the misses do not average out, they add: 17° over thirty-two bases, 137° over two hundred and fifty-six. The lattice a design is drawn on is decided by the molecule before any shape is chosen.
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.
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.
A proud hundredth buys almost nothing
A folded design's sheet is set by its mean layer count and its paper by its deepest point, and the ratio of the two moved the size at which a design uses both equally. That ratio is two numbers of a distribution. Read the whole distribution of every printed pattern's folded footprint and the shelf splits in two: six patterns reach their deepest depth over at least a tenth of the footprint, and two reach it only on a sliver — the fold-and-cut triangle on 2.9 per cent, the tapered corrugation on 0.78. A bound that lets part of the footprint stand proud of the stack's ceiling does nothing at all until the proud share passes the deepest depth's share, so a hundredth moves one pattern of eight, and a twentieth moves two.
The rule has a width
On an equilateral triangle of paper the regular polygons that beat the largest circle are exactly the multiples of three. Tilt the apex a degree and every one of them loses share in proportion, while the circle loses almost nothing — a multiple's advantage is a corner and the circle's is a hilltop. So the rule holds only in a window, 44° to 79° of apex for the hexagon and 56° to 65° for the 24-gon, and past it the second place goes to polygons like the hendecagon that no single fold can build.
The worst angle sets the price
On a triangle of paper that is nearly equilateral, the largest regular hexagon loses 0.448 percentage points of the sheet for every degree the triangle is out of true, the nonagon 0.267 and the dodecagon 0.193. Those rates were measured. They have a closed form, 2n tan²(π/n) / 9√3 a radian, and it depends only on the worst of the triangle's three angles, whatever its shape. Rate times sides falls to 2π² / 9√3, and the window in which each polygon beats the circle shrinks as ninety degrees over its number of sides.
The window has an exact edge
How far an isosceles triangle may stray from equilateral before its largest regular hexagon falls behind its inscribed circle was measured — 15.6 degrees closing the apex, 19.1 opening it — and estimated to first order as the polygon's lead over its rate. Neither number needs measuring. The fitting problem has three sides and three unknowns, so its bound is an equality, the polygon keeps a side on the base, and its share is one line. That line carries the same factor as the circle's, so the circle's own loss cancels out of the comparison entirely, and each edge of the window is an angle in closed form. The opening half is wider because opening the apex lengthens the perimeter. And the window does not shrink as ninety degrees over the sides, as the first-order estimate says, but as (2 − √2)·180° over them — about 105 — because the corner's reach bends back as fast as it rises.
Two sides at sixty are enough
On a nearly equilateral triangle every multiple of three beats the inscribed circle, and to first order how long it keeps doing so depends only on the worst of the triangle's angle errors, which makes the window a hexagon in the plane of two errors. Fitted exactly, the window is a scalloped star. It reaches furthest — 21.8 degrees for the hexagon, against 15.6 and 19.1 on isosceles triangles — where one angle opens and another closes with the third left at sixty, because two sides still meet at sixty and the polygon can lie flush against both. Far outside it, wherever all three sides lie along the polygon's own directions, the polygon wins again.
No placement cleans every drawing
Five of a hundred and twenty clipped twist patches carry fragments — creases a fraction of a millimetre long, left where the sheet's edge cut through the pattern — and the proposed repair was to nudge the sheet. A fragment turns out to be a corner of a twist polygon lying within a seventh of a millimetre of the rim, and moving the pattern 0.127 mm clears any one drawing. No single movement clears them all. Laid at random, a sheet puts a fragment on one drawing in nine; of twenty-four random placements the cleanest still leaves three drawings with one, and the centred placement every drawing was made at leaves five. A fragment is not a fault of the construction. It is the threshold meeting a placement, and the only clean set of drawings is one placed drawing by drawing.
A fragment counts creases, not length
A fragment was a piece of crease under a thousandth of the sheet, a threshold chosen for being smaller than a pencil line. Set it instead at the width a crease occupies in real paper, which is the first threshold a folder would recognise, and the rate at which a sheet carries one can be derived from the drawing without placing it: the crease width, times about 2.5, times the number of creases on the sheet. Measured over 1,024 placements on each of 120 drawings it comes out at 1.00 against 0.99 at the width of a crease in kami, with no drawing off by more than a quarter. It reads how many creases there are, and hardly how long they are. It also says what it cannot, which is the clean share: 63% of placements at the width of copier paper, against the 30% independent pieces would leave, because a rim that catches one corner catches its whole row.
Named alongside it
The objects these essays reach for when they reach for this one.
IdealisationError propagationCrease patternMeasurementPatchFold angleKawasaki's theoremManufacturingOptimisationQuadrilateral meshRegular polygonRigid folding