The family the Miura belongs to

Move one vertex of a Miura by a thousandth of a panel and the sheet has no isometric folded position of that kind at all, and the amount by which it fails is first order in the displacement, so no move is small enough to be free. That is a strong result and it leaves an obvious hole: it says the Miura is special without saying what it is special within.

The family exists, it is large, and it is picked out by a condition nobody would guess from looking at a crease pattern.

One rule, and Kawasaki comes free #

Draw a set of straight lines — call them the columns. Draw a polyline across them — a row — that reflects in every column line it crosses, the way a billiard ball reflects off a cushion. Keep drawing rows.

That is the whole construction, and it produces a flat-foldable pattern without anything being solved for. At each vertex the column crease runs straight through, and the two row creases sit at equal and opposite angles to it. So the four sectors come out as β, 180° − β, 180° − β, β, and the alternating sums are two pairs of supplementary angles, which are equal because supplementary angles always are.

Measured over these meshes the worst departure from Kawasaki is 5 × 10⁻¹⁴ degrees, which is what “identically” looks like when a computer is asked. The rule does not merely tend to produce flat-foldable patterns; it cannot produce anything else.

That is a lot of patterns. Every row is free to start where it likes on the first column and to lean however it likes, and every one of those choices gives a developable, flat-foldable quadrilateral mesh. Two numbers per row, and the fan’s angle, and the whole family is written down.

It is worth noticing what has been bought and what has not. Flat-foldability at a vertex is two conditions on the angles, and a construction that satisfies them by symmetry rather than by solving is not cheating — it is the same trick the twist tessellations use, where the polygon’s corner is the supplement of the tiling’s own angle beneath it and Kawasaki comes out as 1 − 1 rather than as a computation. What has not been bought is anything at all about whether the sheet moves.

What folding rigidly asks for on top #

A flat-foldable pattern has a folded state. Whether it can be reached — by a motion in which every panel stays rigid and every crease stays a crease — is a different question with a different answer, and here the difference has a shape that can be written down.

Start with what one vertex does. A degree-four vertex is geared: fixing one fold angle fixes the rest, by a ratio that depends on the sector angles and not on how far the vertex has folded. At the vertices this construction makes, the gearing has a particularly clean form —

tan(ρ_column ⁄ 2) = cos β · tan(ρ_row ⁄ 2)

— where β is the angle between the row crease and the column running through it. The cosine of one angle, and nothing else. No length, no position, no reference to any neighbour.

That expression is a closed form and it is not how it is checked. The check composes four rotation matrices round the vertex and asks whether the product is the identity; the two agree to 7 × 10⁻¹⁶ radians, which is two computations meeting rather than one restating itself.

The condition, which is about a table #

Now put the vertices together, and the arithmetic writes itself.

A column crease runs straight down a whole column, so it has one fold angle along its length. A row crease runs along a whole row, so it has one too. Call them S for column i and Z for row j. At the vertex where they cross, the gearing says

tan(S_i ⁄ 2) = cos β_ij · tan(Z_j ⁄ 2)

for every i and every j at once. Rearranged: the table of numbers cos β_ij has to be a column of numbers times a row of numbers. It has to be rank one.

That is the condition, and it is not a condition at a vertex. It is a condition between vertices, and it is the first one in this subject that cannot be checked by looking at any bounded piece of the sheet: rank is a property of the whole table.

On the meshes built by the fan rule with every row leaning the same way, it holds exactly. The measured departure from rank one is between zero and 9 × 10⁻¹⁶, and the sheet then folds with every panel where its neighbours expect it — two panels’ opinions of where a shared corner went agreeing to about 10⁻¹⁵ of a sheet-width.

There is a way of stating the rank condition that makes it feel less abstract. It says: the ratio of any two columns’ fold angles is the same in every row. If column three is always folded twice as far as column one on the top row, it has to be twice as far on every row, at every moment of the motion. That is a demand about coordination across the sheet, and a crease pattern that fails it is a sheet whose two halves want to fold at different rates.

The condition is local after all, on four vertices #

The rank statement is described above as the first condition in this subject that cannot be checked on any bounded piece of the sheet, and that is not right. Rank is a global property of a table and it has a completely local certificate, and the certificate is worth having because it turns the condition into something a designer can apply with a protractor.

A table with no zero entries has rank one exactly when every two-by-two minor vanishes, and it is enough to check the minors formed by adjacent rows and columns. If

$$\cos\beta_{ij}\,\cos\beta_{i+1,j+1} = \cos\beta_{i,j+1}\,\cos\beta_{i+1,j}$$

holds for every neighbouring pair of columns and every neighbouring pair of rows, then the ratio between any two columns’ entries is the same in every row, the rows are proportional, and the table is a column times a row. Nothing more is needed.

Four vertices at a time, then — the four at the corners of one cell of the mesh. That is a bounded piece of the sheet, it is the same bounded piece the loop conditions on a general quadrilateral mesh live on, and there is one equation per cell rather than one condition on a whole table.

Which makes it a test rather than a diagnosis #

The practical consequence is that the condition can be applied to a drawing before anything is folded, by measuring angles.

At each vertex, β is the angle between the row crease and the column crease running through it — a quantity a protractor reads off the flat pattern. Take the cosines. For each cell of the mesh, multiply the two diagonally opposite ones together and compare the two products. Equal, and the cell is fine; unequal, and the mesh does not fold rigidly, and the cell where they differ is where to look.

That is a considerably better instrument than the residual the figures report. A gap between two panels is measured on a folded state and tells a designer only that something is wrong somewhere; the cell test is measured on the drawing, costs four multiplications, and names the cell. The map figure above makes exactly that point from the other side — the largest disagreements are not at the row that moved — and the reason is that a residual is a symptom while the vanishing minor is the fault.

It also explains why moving one row breaks the mesh everywhere rather than locally. Shifting a row changes β at every vertex along it, so every cell that row bounds has a minor that no longer vanishes — a whole band of cells rather than one — and each of them contributes its own disagreement to a folded state that has to accommodate all of them at once.

So the condition is neither a vertex condition nor a property of the whole sheet. It sits between them, on cells, which is exactly where this subject’s genuinely global difficulties have turned out to live every other time somebody has looked.

The refusal, and it is the whole point #

Take one of those meshes and move a single row sideways — change where it crosses the second column, and let the reflection rule propagate the change along the rest of that row.

Every vertex still reflects. Kawasaki still holds to 5 × 10⁻¹⁴ degrees, which is the same number to the same decimal place as before the row moved. The pattern is exactly as flat-foldable as it was, and every checker in this subject says so.

The table is no longer rank one, by 0.005 at a two per cent displacement and 0.04 at twelve per cent. And the sheet no longer folds: the panels leave a shared corner 6.6 × 10⁻⁴ of a sheet-width apart at two per cent and 7.8 × 10⁻³ at twelve. On a sheet the size of a book, the second is a gap of a couple of millimetres that has to be closed with force.

What the family looks like #

The parallel member is the Miura, and its neighbours are corrugations that curve.

The curving is worth measuring on the right object. A corrugation zigzags, so neighbouring panels are at a large angle to one another whether or not the sheet is curved, and comparing panel orientations reports the zigzag rather than the shape. What does report the shape is the straight creases: a Miura’s stay parallel through the whole motion, and a fan’s do not.

At a fan of 5.2° between neighbouring columns, the outermost two straight creases are 30.9° apart on the flat sheet and 25.9° apart in the folded one. At 9.2° they are 55.0° and 47.3°; at 13.8°, 82.5° and 73.3°. So folding does not merely preserve the fan — it closes it, by about a sixth in each case, and by an amount that depends on how far the sheet has been taken.

There is a reading of that worth keeping. A flat Miura is a flat sheet and a folded Miura is a flat-ish slab; a flat fan member is a flat sheet and a folded one wraps part of a cone. The pattern decides which, and it decides it through a single number — how far the columns fan — that is invisible in any single vertex of the drawing. One vertex repeated is what makes a tessellation a material; this family is what happens when the repetition is not quite a repetition.

How the signs were found #

There is a piece of the calculation that had to be measured rather than assumed, and it is instructive about what a crease pattern does and does not carry.

The gearing gives the size of every fold angle and not its sign, and the signs are the mountains and valleys. Swapping them along a row describes a different fold of the same lines. The candidates form a small family — each of the two crease families’ signs may alternate with the row index, the column index, both or neither, with either polarity — and exactly one of them puts every panel where its neighbours expect it. The next best leaves between a fifth and two fifths of a sheet-width open, which is not a near miss.

So the letters on these figures are read off the motion rather than drawn, and then put past Maekawa and the big-little-big lemma — which are theorems about letters and know nothing about a motion. They pass. That is a check rather than a restatement, because the two computations share no code and there was no reason in advance for the fold the sheet makes to be one the letter theorems permit.

There is a smaller finding inside that. A vertex here admits two labellings satisfying Maekawa, and they are the two branches it can fold along. Which one the sheet takes is not the vertex’s decision: it is fixed by the requirement that the whole mesh close, and the other branch — perfectly legal at every vertex on its own — is a fold no sheet of this family makes.

What it costs to leave the family #

The refusal above is stated as a gap, which is the honest measurement, and it is worth translating into what a maker would notice.

A quadrilateral mesh outside the family can still be made. Panels can be cut, hinges fitted, and the thing assembled flat; it is only when it is asked to move that the arithmetic arrives. What happens then is not that it jams: it is that closing one part of it opens another, and the assembly accommodates by bending panels that were meant to be rigid. The hinge is where the error has to go when the panels will not take it, and the gap measured above is how much error there is to place.

The practical rule that falls out is unexpectedly simple: in a mesh of this kind, the rows have to be copies of one another. Scaled about the fan’s centre if the columns meet, translated if they are parallel — but copies. A designer who adjusts one row for local reasons has left the family, and no adjustment to that row alone will bring it back.

Where the model stops #

Nothing here counts freedoms. The words degree of freedom do not appear in any measurement above, and that is deliberate: counting a mechanism’s mobility is the business of kinematics, and this site’s licence covers rigid origami’s own patterns rather than the counting machinery. What is measured is a residual — an angle that fails to close, and a gap that fails to shut.

The columns are straight, and that is a restriction. Quadrilateral meshes in which both families of creases zigzag are a larger class, and the rank-one condition above was derived under the assumption that one family runs straight through each vertex. Whether it generalises is not settled here.

The rank-one condition is necessary, and sufficiency is measured rather than proved. Every mesh whose table is rank one folds, on every member tried; the argument that it must is a paragraph of algebra that has not been written out, and the figures report a measurement.

Rigid folding is not flat folding. A member of this family reaching a flat state is a further question, and the motions above stop well short of one.

No material anywhere. The panels are rigid because they are declared rigid. What a real panel does — how much it bends, what the hinge costs, where the error goes when a crease is misplaced — is a question about stiffness and belongs to structures rather than to geometry.

Who found it, and when #

The rigid foldability of quadrilateral meshes was worked out in the folding-mathematics literature over the two decades to about 2010, principally by Tomohiro Tachi, whose generalised quadrilateral meshes are the family this construction lands in. The reflection rule is a way of drawing them rather than a new object, and the rank condition is a way of stating the compatibility requirement rather than a new theorem.

What is this site’s own is the measurement: that flat-foldability does not move at all under a displacement that destroys rigid foldability, quantified on the same sheet, with a gap a maker could put a ruler against.

Where the ladder goes next #

The obvious continuation is the mesh whose columns are not straight, which is the general quadrilateral case and where the table’s entries stop being a single cosine.

The other direction is the one this family makes newly askable. Every member folds with one angle, and every member curves by a different amount — so a designer wanting a particular curved surface has a family to search in, and the search would be over the fan angles rather than over the crease pattern. What surfaces are reachable, and how nearly, is a design question with the machinery now in place to ask it.