Origami & Folding
what one sheet can do
Rigid folding

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.

Assumes Solved is not built and Solving every face at once.

20 min read 9 figures From craft to hardwarePaper is not ideal

Cut a solved mesh’s every dimension a fifth of a millimetre wrong on a sheet 150 millimetres across and the closure is gone. Cut a Miura’s every dimension five per cent wrong and it still folds exactly, because what makes a Miura fold is a property of its structure and what makes a solved mesh fold is an equation.

Both halves of that are right — the Miura’s exactness comes from one vertex repeated and the mesh’s from a system of equations — and the first one has a hidden quantifier in it. A fifth of a millimetre is a size, and the errors it was measured over all pointed the same way. The solutions are not a point, and once that is taken seriously the number stops being a tolerance and becomes a tolerance in one direction.

The same error, three directionsHow far a solved mesh is from closing after every one of its lengths is cut wrong by the same amount, against the size of that error on a sheet 150 millimetres across, in three directions. Across the surface of solutions the closure is lost in a fifth of a millimetre. Along it, the same error a hundred times larger costs less.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
Fig. 1 How far a solved mesh is from closing after every one of its lengths is cut wrong by the same amount, along three directions in the space of lengths. The vertical scale is logarithmic and the three lines are not variations on one another: at the same error, the top and the bottom differ by a factor of five thousand.

Four equations in twenty lengths

A four-by-four quadrilateral mesh has twenty free lengths and four interior faces. Each face contributes one closure equation — walk the fold angles round that face’s loop and come back to where the walk started — and solving all four together is what makes the mesh fold.

Four equations in twenty unknowns do not have a solution; they have a solution set, and near any solution that set is a surface of dimension sixteen. The four is not an assumption: it is the rank of the closure’s own derivative, computed at the solution and reported with the singular values that produced it.

How many directions there are of each kindThe lengths a quadrilateral mesh has, sorted by whether the closure conditions see a change in them. Four of the directions are constrained and the rest are not, so the surface of solutions is wide and an error is much more likely to point somewhere harmless than somewhere fatal.20 lengths · 4 constrained directions · 16 free ones4 directions the closure sees — a step here is paid for in proportion16 it does not — a step here changes nothing to first orderwhich is why one figure of a fifth of a millimetre describes neither
Fig. 2 The twenty lengths of the mesh, sorted by whether the closure conditions see a change in them. Four directions are constrained. The other sixteen are the surface, and a step along any of them leaves the mesh folding.

So a cutting error is a vector in a twenty-dimensional space, and the space splits into two parts that behave completely differently.

Across the surface — along one of the four directions the closure’s derivative points in — the residual grows in proportion to the step, which is the behaviour an error has when it has nowhere to go. That is the direction in which a fifth of a millimetre matters, and it is where the earlier number came from.

Along the surface — along any of the sixteen the equations do not see — the residual does not change to first order at all. What is left is second order, which is to say small and getting smaller as the step shrinks.

How many directions there are of each kindThe lengths a quadrilateral mesh has, sorted by whether the closure conditions see a change in them. Four of the directions are constrained and the rest are not, so the surface of solutions is wide and an error is much more likely to point somewhere harmless than somewhere fatal.20 lengths · 4 constrained directions · 16 free ones4 directions the closure sees — a step here is paid for in proportion16 it does not — a step here changes nothing to first orderwhich is why one figure of a fifth of a millimetre describes neither
Fig. 3 The same surface reached from a different mesh. Four equations in twenty lengths leaves sixteen directions to move in without leaving the solution set, and which sixteen they are depends on where on the surface you are standing — which is why a tolerance quoted as one number is quoted in the wrong units.

What the difference costs, in millimetres

Numbers, on a sheet 150 millimetres across, with the mesh’s own lengths at about one panel each.

A step of a hundredth of a millimetre across the surface leaves a closure mismatch of 8×10⁻⁴ radians. The same step along it leaves 1.3×10⁻⁸ — sixty thousand times smaller. At half a millimetre the two are 5×10⁻² and 5×10⁻⁵, a factor of about a thousand; and at the largest step measured the ratio is five thousand.

The ratio moves with the step because the two curves have different shapes: one is a straight line through the origin and the other is a parabola. On a logarithmic plot they have different slopes, which is the visible signature of first order against second, and it is the reason the phrase a tolerance of 0.2 mm cannot be repaired by choosing a better number.

What a tolerance is, per directionHow far a solved mesh may be cut wrong before its closure passes a stated budget, in three directions. Across the surface of solutions it is a fifth of a millimetre. In the other two the budget is never spent at all within the range measured, so there is no number to report and the bar is not drawn.budget: a closure mismatch of 0.02 radians, on a sheet 150 mm acrossacross the surface of solutionsthe direction the closure's own derivative points in0.211 mma direction chosen without regard to the surfacewhich has a component along bothno allowance to report — the budget is never spentalong the surface of solutionsone of the twelve directions the equations do not seeno allowance to report — the budget is never spent
Fig. 4 The same measurement as the number a manufacturer wants: how far the mesh may be cut wrong before its closure passes a stated budget. Across the surface, 0.211 millimetres. Along it there is no such distance within the range measured, so no bar is drawn — an absence rather than a large number.

The 0.211 millimetres is worth pausing on. It is, to three figures, the fifth of a millimetre the earlier rung reported — so nothing there was wrong. What was missing is that the number describes one direction out of twenty, and the other sixteen do not have one.

A random error is mostly harmless

The third line on the first figure is the one a manufacturer actually cares about, because a real error does not point along an axis anybody chose.

A direction taken without regard to the surface has a component along each part, and in twenty dimensions with only four of them constrained, most of its length lies in the sixteen. Measured: a random direction costs about six times less than the worst one, and within the whole range swept it never spends the budget at all.

That is not luck and it is not an argument for carelessness. Where an error goes in a quadrilateral mesh has an answer of its own — a mistake in one row has no consequence in that row — and this is the same fact counted in directions rather than in rows. It is a statement about proportions: the constrained subspace is a fifth of the whole, so a random unit step has about a fifth of its length in it. What it means practically is that the same tolerance specification produces wildly different failure rates depending on what the errors are correlated with — a systematic error, such as every panel cut from a template that is slightly wrong, points in one direction and can point in the worst one; independent random errors spread themselves over the space and mostly do not.

The same error, three directionsHow far a solved mesh is from closing after every one of its lengths is cut wrong by the same amount, against the size of that error on a sheet 150 millimetres across, in three directions. Across the surface of solutions the closure is lost in a fifth of a millimetre. Along it, the same error a hundred times larger costs less.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 3,504 times as much one way as the other
Fig. 5 What an error costs, by direction rather than by size. A displacement along the surface costs nothing at any magnitude; a displacement off it costs immediately and in proportion. The other way this site prices an error — how far it has travelled from where it was made — is a different measurement and answers a different question.

The measurement a workshop would recognise

Put the same numbers the way a drawing office would.

Take the mesh, cut every one of its twenty lengths wrong by the same amount, and ask how far the closure has drifted. That is one experiment with one answer, and it is the experiment behind every tolerance ever quoted. What it hides is that cut every length wrong by the same amount names a direction — the one where all twenty errors are equal and positive — and there was never a reason to think that direction was representative.

Three experiments instead of one:

The last column is the allowance for a budget of 0.02 radians:

the errors point 0.03 mm 0.27 mm allowance
across the surface 3.2×10⁻³ 2.6×10⁻² 0.211 mm
without regard to it 5.5×10⁻⁴ 4.4×10⁻³ never reached
along the surface 2.0×10⁻⁷ 1.3×10⁻⁵ never reached

The first row is the specification. The third row is the same twenty numbers rearranged. A workshop that hit the first tolerance would be paying for a precision that sixteen of its twenty dimensions did not need, and a workshop that missed it in a free direction would ship a mechanism that folds.

What a tolerance is, per directionHow far a solved mesh may be cut wrong before its closure passes a stated budget, in three directions. Across the surface of solutions it is a fifth of a millimetre. In the other two the budget is never spent at all within the range measured, so there is no number to report and the bar is not drawn.budget: a closure mismatch of 0.02 radians, on a sheet 150 mm acrossacross the surface of solutionsthe direction the closure's own derivative points in0.175 mma direction chosen without regard to the surfacewhich has a component along both1.021 mmalong the surface of solutionsone of the twelve directions the equations do not seeno allowance to report — the budget is never spent
Fig. 6 The measurement a workshop would recognise, on a second mesh. Each column is how far one length may be cut wrong on its own before the residual leaves the budget — and the columns differ by more than an order of magnitude, which is the whole content of saying that a tolerance is a direction.
The same error, three directionsHow far a solved mesh is from closing after every one of its lengths is cut wrong by the same amount, against the size of that error on a sheet 150 millimetres across, in three directions. Across the surface of solutions the closure is lost in a fifth of a millimetre. Along it, the same error a hundred times larger costs less.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
Fig. 7 The same pricing against a stated budget. Where the cost curve crosses the budget is the allowance in that direction, and the crossing points are spread across two decades — a single figure quoted for the whole mesh would have to be the smallest of them, and would refuse most of the room that is actually there.

The two laws are in the table

The claim that one curve is first order and the other second is the whole of the argument, and the table above settles it without any appeal to the plot’s slopes.

Across the surface, the step goes from 0.03 mm to 0.27 mm — a factor of nine — and the residual goes from 3.2×10⁻³ to 2.6×10⁻², a factor of 8.1. A straight line through the origin would give nine, and the shortfall is the curve beginning to bend at the far end, where the mesh is approaching the point at which it stops folding at all.

Along the surface, the same factor of nine in the step takes the residual from 2.0×10⁻⁷ to 1.3×10⁻⁵, a factor of sixty-five. A parabola would give eighty-one. Nothing else in the neighbourhood of those two numbers is a plausible law: a straight line would have given nine and a cubic seven hundred.

The middle row settles the third claim. A direction taken without regard to the surface goes from 5.5×10⁻⁴ to 4.4×10⁻³, a factor of exactly eight — so it is first order too, as it must be, because a direction chosen without regard to a subspace is not orthogonal to it. What it is is first order with a smaller coefficient, and the size of that coefficient is the next question.

Why six, and not some other number

The random direction costs about six times less than the worst one, and that factor is not arbitrary. It is a consequence of the dimension count and can be predicted before the measurement.

A unit vector taken without regard to any particular direction in a space of twenty dimensions has, on average, a component of about one over the square root of twenty along any single chosen direction — which is 0.224, a factor of four and a half. The worst direction is the steepest of the four constrained ones, so four and a half is the factor to expect from the geometry alone.

The measured six is a little larger, and the difference is the spread among the four sensitivities. The constrained directions are not equally expensive: the closure’s derivative has four singular values and they are not equal, so a random vector’s cost is an average weighted across all four while the worst-direction cost uses only the largest. The more unequal those four are, the further the ratio rises above the square root of the dimension.

That gives the finding a shape a specification can use. The protection a random error enjoys is roughly the square root of the number of dimensions divided by the number of constraints — a property of the mesh’s size and shape rather than of the workshop — and it can be estimated from the drawing before anything is cut. A mesh with more free directions protects a random error better; a mesh with more faces protects it less; and neither fact is visible in any tolerance quoted per dimension.

The bar that is not drawn

The tangent allowance is reported as an absence rather than as a number, and it is worth saying what the number would have been and why it is not quoted.

Extrapolating the parabola gives a coefficient of about 1.8×10⁻⁴ radians per square millimetre, so a budget of 0.02 radians is spent at roughly ten millimetres — fifty times the across-surface allowance, and seven per cent of the sheet.

That number must not be used, and the reason is in the section above about where the model stops. Ten millimetres is far outside the neighbourhood the derivative describes; the surface curves, the free directions at the start are not the free directions ten millimetres away, and the mesh stops folding entirely somewhere past half a millimetre in the worst direction. An extrapolation across four orders of magnitude of a locally fitted parabola is arithmetic rather than a measurement. The honest report is that within the range where the measurement means anything, the tangent direction never spends the budget — which is what the empty bar says.

Which theorem was checked, and how

The claim is a claim about a derivative, so the derivative is computed rather than argued about.

The rank is measured. The closure’s Jacobian — one row per equation, one column per length — has its singular values computed by a Jacobi sweep on the small matrix its rows produce, and the number of them above a threshold is the count of constrained directions. It comes out four, at every solution reached from every starting mesh tried, and it must be capable of coming out lower: the whole point of the earlier rung was that a mesh whose crease families run straight is degenerate, and the rank is what detects that.

The tangent direction is constructed rather than found. The four rows are orthonormalised, a fixed pseudo-random vector has the row space projected out of it, and what remains is a direction the closure’s derivative cannot see. Fixed rather than random, so that the same figure is the same figure twice.

The residual is recomputed from scratch at every step. Nothing is linearised: each point of each curve is a mesh with different lengths, put through the same propagation that produced the original solution, and the mismatch reported is the one that propagation leaves.

The refusal is asserted. A step along the surface must have no allowance to report within the range measured, and if one were found the figure would say so rather than drawing an empty bar. That check is what makes the empty bar mean something.

What each mesh's reason for folding survivesEvery length of each mesh made wrong by the stated amount, and what is left of the closure. The solved mesh folds because an equation holds; the Miura folds because one crease family runs straight through every vertex, which a badly cut sheet still does.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
Fig. 8 The measurement this rung refines: how fast the residual leaves zero when the mesh is cut wrong, length by length. Each column there is one axis of the twenty-dimensional space, and the sensitivity it reports is the projection of that axis onto the four constrained directions.

Where the model stops

Past about half a millimetre the mesh stops folding at all. The last point of the worst-direction curve is not a residual: it is the propagation failing to find any configuration, which is a different failure from a large mismatch and is reported as one. The curves are drawn to the last point that is a residual.

Sixteen directions is a count at one point. The solution set is a surface and its tangent space turns as it is walked, so a direction that is free here is free here; a step large enough to matter has left the neighbourhood the derivative describes, which is exactly why the tangent curve is a parabola rather than a flat line.

Nothing here is a statement about how errors are distributed. There is no estimator, no error bar and no distribution anywhere in the measurement: each number is the residual of a stated perturbation of a stated mesh. The word random above describes one particular direction that was constructed without regard to the surface, and it is one direction rather than a sample of anything.

And nothing here is about a pattern that folds for a structural reason. A Miura folds because a crease family runs straight through every vertex, which is a property no perturbation of its lengths destroys; the surface described here is what a mesh has instead of that property.

And a mesh that closes is still not necessarily an object. Some solved meshes drive a panel through another at every angle of their motion, so a point on this surface can be a mesh that folds and cannot be built — a second condition on the same surface that no equation in it mentions.

What the picture cannot show

A twenty-dimensional space has been drawn as three lines. The three directions are real and the choice of them is not neutral: the worst one is the steepest of the four constrained directions, the tangent one is a particular member of a sixteen-dimensional family, and the random one is a single draw. A figure that showed the distribution of costs over all directions would be a better picture of the situation and would need a way to draw a sphere in twenty dimensions.

What the picture also cannot show is that the surface is curved. Every claim above is about a neighbourhood; the plot’s horizontal axis runs to a millimetre on a 150-millimetre sheet, and whether the sixteen free directions are still free at ten millimetres is a question the derivative cannot answer.

The idealisation, named

The panels are rigid and the hinges are perfect. A real hinge has clearance, and clearance is exactly a small amount of freedom in the fold angle — which changes the problem qualitatively, because a mechanism whose joints have play can absorb a closure mismatch that a mechanism with perfect hinges cannot.

That cuts in the direction that makes the finding matter more rather than less. Play in the joints widens the acceptable region around the surface of solutions; it does not change which directions are cheap and which are expensive, and a specification given as a single number is still describing one direction while being applied to twenty.

The generalisation

A tolerance is a property of a direction, not of a part. The habit of quoting one is not a mistake about arithmetic; it comes from the case everybody learns on, where a dimension is constrained by an equation of its own and the two coincide. Error is folded too makes the same correction to a different intuition: an error does not stay where it was made. A system with more unknowns than equations has a solution set with shape, and on a set with shape the question how far may this be wrong has a different answer at every angle.

The practical form is a rule about specifications. A dimension that is free should be specified loosely and a dimension that is constrained should be specified tightly, and which is which is not visible in the drawing — it is the null space of a derivative, and it takes a linear-algebra step to find. Specifying every dimension to the tightest of them is expensive; specifying every dimension to the loosest is a mechanism that does not close.

And the surprising half, which is the one worth carrying: being close to a solution is not the same as being close to the solution set. A mesh cut half a millimetre wrong in a free direction is nowhere near the mesh that was designed and is exactly on the surface of things that fold. The design was never the point; the surface was.

Outside the family that was already knownHow far the straighter of the two crease families is from running straight through the mesh's vertices. Everything this site could build before this rung sits at the top of the picture.meshworst departure from straight, radiansa Miuraone crease family runs straight through every vertex5e-15the solved meshneither family does, anywhere1.21
Fig. 9 The reason the distinction is easy to miss: some points of the surface are meshes with a straight crease family and some are not, and a solve reaches whichever one it walks to. Two meshes that both fold can be as different as this and still be neighbours on the same surface.

Where the ladder goes next

Two directions lead out of here and only one of them is short.

The short one is the specification: turning the four constrained directions into a set of dimensions a workshop could be given, which means expressing a direction in length space as a statement about particular panels — a change of basis, and a piece of engineering rather than of geometry.

The longer one is the surface itself, and it is the same freedom the family the Miura belongs to describes from the inside. Sixteen dimensions of solutions is a great deal of freedom, and none of it has been used for anything: a designer who needs a mesh that folds and is solid and has panels above some minimum size is choosing a point on that surface subject to conditions nobody has written down. What the surface is for is the question this rung leaves open.

Where this goes next

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. 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. mismatch 0.066 radiansmismatch 8.5e-14 radiansevery vertex of both is developable and Kawasaki-exact to the last bit a double holds Solving every face at once A quadrilateral mesh that folds rigidly has to close round every one of its faces, and the rung that built the general mesh could close one. Four of them at once resisted a descent that drove each free length to its own root, because closing a loop is a condition on several lengths together — and solving them jointly finds a sheet with no two vertices alike that folds, and a surface of them sixteen dimensions wide.

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ClosureFold angleQuadrilateral meshRigid foldingSensitivityTolerance