Miura — where it appears
Named by 27 essays across 5 fields — each of them below, with the objects they name alongside it.
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.
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.
Which crease to push
Deciding one fold angle settles every other one on a quadrilateral mesh, which is what makes a self-folding sheet buildable with a single actuator. It leaves a question that sounds like an afterthought: which crease. Driving each of a mesh's twenty-four in turn gives twenty-four different answers to how far an error in it travels — and on the sheet that repeats one vertex, it gives several answers to what shape the sheet takes.
The Miura folds two ways
One vertex repeated is what makes the Miura buildable: identical panels, identical creases, one degree of freedom. It is also what makes it ambiguous. At one fold angle on one crease the sheet has two folded states, differing in three letters and in half its width, and both of them close exactly — while a mesh with no two vertices alike has one.
Nothing to average over
A folded corrugation is reported with a Poisson's ratio, and both of this site's measurements of one were made on a sheet that repeats a single cell. On such a sheet every cell behaves the same way and the cell's number is the sheet's number. On a sheet with no repeating cell the cells run from −3.5 to +0.4 — some widening while others narrow — and the sheet's own figure describes none of them.
Sixty-four rules, sixteen fold
The Miura fold's letters are usually given as a recipe: rows one way, columns changing at every row. Write down every rule of that shape — the letter on a crease depending only on which row and which column it is in — and there are sixty-four. Sixteen fold flat. They are exactly the ones whose columns change at every row, the row letters do not matter at all, and every one of the forty-eight refusals is the counting theorem's alone.
The loop is in the rule
Of the forty-eight repeating rules that do not fold a grid corrugation, thirty-eight send four panels round in a circle and ten merely fail the count. Which is which can be read off three of the rule's six bits, without building the pattern, folding it or walking a single arrow — and the closed form agrees with the arrows on all sixty-four rules of both grid families.
Half the recipe is decoration
Every account of the Miura fold gives its letters as two instructions: the rows go one way, and the columns change letter every time they cross a row. Enumerate all sixty-four repeating rules and the second instruction is the whole of the condition — all four ways of writing the rows appear among the sixteen that fold, in every combination. The first instruction has never constrained anything.
The leaf's rules are the Miura's
A plicate leaf packs into a corrugation that is broad in the middle and narrow at both ends. Enumerate every repeating mountain-valley rule it admits and the table is the Miura fold's table, rule for rule, number for number — and it stays that table when the leaf is redrawn with even columns, a violent taper, taller rows or a steeper zigzag. The plant's geometry cannot reach its own letters.
The corrugation that closes on itself
A Miura cell crosses one crease per period in one direction and four in the other. So one of its two directions has a parity condition that half the sizes fail, and the other direction cannot fail at any size — the same sheet, the same drawing, and two gluings that behave completely differently.
One node per panel, with the rim gone
A rectangle of repeating pattern cut out of the plane costs exactly one node of search per panel, on every family and at every size. Take the rim away and the total falls and the cost per panel rises, because the letters that were removed were the ones that could not be wrong.
A metamaterial with no edge
A folded metamaterial's properties are quoted per unit cell, because a material is supposed to be the same everywhere and a cell is supposed to stand for the whole of it. Every cell this collection has measured has been cut out of a patch, with a rim round it — and a rim is the one place a repeating material is not like itself.
A mechanism that closes on itself
A rigid-foldable pattern is a mechanism: panels as rigid plates, creases as hinges, and a motion counted by degrees of freedom at each vertex. Close the sheet into a tube and the mechanism has to come back to itself after a circuit — a constraint that is not at any vertex and that the degree-of-freedom count does not see.
The tube that gets built
Every folded structure that leaves a laboratory is a sheet joined to itself — a boom, a stent, a bellows, an airbag, a packed antenna. The mathematics has been done on flat rectangles for the whole history of the subject, and the object is a cylinder, which is a different sheet with different counts and a condition the rectangle does not have.
A paper limits spacing, not density
The density ceiling put a sheet's limit at parallel creases a crease-width apart, 1⁄w of crease a square metre, and claimed no arrangement carries more. Crossing families do: they meet at vertices, which is shared ground, and never come closer than w anywhere else. Measured over every pair of creases that do not share a vertex, density times closest spacing settles at about 1.9 for both the Miura and the waterbomb, so each can be folded nearly twice as fine as the density bound said — 262 cells a side on copier paper for the Miura rather than 139, and 141 for the waterbomb rather than 74.
The pattern cheapest to trust
Demonstrating that a one-shot deployment will open takes a number of successful tests proportional to its hinge count, so the pattern that needs fewest tests for what it delivers is the one with fewest hinges per layer of compaction. That criterion is a count nobody computes, and computed on the printed shelf it ranks the patterns differently from crease length per layer: the preliminary base is first, at exactly one hinge per layer, and the square twist rises from seventh to fourth. As patterns are refined the difference sharpens. The waterbomb settles at 2.47 hinges a layer and the Yoshimura at 1.47, but the Miura climbs without levelling — 1.32 at two cells a side, 5.69 at eight — so every finer Miura costs more tests for each layer it adds, and the pattern that gets built is the only one of the three that gets dearer to trust as it gets finer.
The folded strip lies at its own slant
The two directional factors read a folded state against the axes the pattern was drawn on, with its first panel held where it was drawn. A flat-folded Miura is not lying along those axes: it is a strip of fixed width laid at exactly its own slant to them, and the leaf corrugation's strip lies at exactly its zigzag. So the Miura's cross factor changes with the number of columns — 1.748 on four, 0.948 on ten, at one slant — and on the leaf, on every Miura with an odd number of columns and on the twists of three and six sides, it changes with which panel is held. Read without axes, both families compact least at 0.80 radians, not at the 1.05 the box on the axes reports.
The rows are free
A Miura's hinges per layer of compaction were found to climb without a ceiling as the pattern is refined, and the footprint was blamed for growing with the sheet. Measured one direction at a time it grows in only one. Every band between two straight creases folds exactly onto the band before it, as an accordion's pleats do, and the folded box does not change at all as they are added. Every band between two zigzags walks the stack a step along itself, the sine of the slant each. So the hinges a layer costs settle as rows are added and climb as columns are, and the same sixty-four cells cost 1.94 hinges a layer laid out long in their rows and 16.14 laid out long in their columns.
Every panel holds a frame
A flat-folded pattern can be put down on any of its panels, and on some patterns the pair of shrink factors changes with the choice. Holding a panel does not move the paper so much as turn the frame it is measured in, by twice the alternating sum of the crease directions crossed to reach that panel — a number Kawasaki's condition makes independent of the route. So the frames are read off the crease pattern before anything is folded. A pattern with one frame gives one answer whatever its folded shape; the five-sided twist has five frames and five answers; and the four-column Miura agrees on every panel although its folded strip is not its own mirror image, because a box cannot tell a shape from its half-turn.
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.
The floor at degree four
A flat-foldable vertex creases some of the paper round it twice, and the ground it doubles had been found by search never to fall below the evenly spread vertex's — one band width squared at degree four. At degree four it can be proved. Kawasaki's condition leaves two free sectors; two identities of the kite turn the closed form into a function of their mean and half their difference; and two derivatives, each one line, show that parting the sectors always costs ground and that, with them equal, opening the mean to a right angle always saves it. So every flat-foldable degree-four vertex creases at least one band width squared twice, and only four right angles crease exactly that. The same pairing that makes the proof work is missing at degree six.
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.
A cut along the zigzag needs a tether
A Miura's folded stack walks along itself by the sine of its slant for every column, so a sheet cut along its zigzags into modules folds into boxes a fraction as long and stacks far deeper for the hinges. But the pieces cannot stay joined by a crease: a zigzag crossed by row creases is a line of four-crease vertices, and joined there the modules are the one Miura again. The join has to be a tether and every module its own actuator — and a cut, which removes a zigzag's worth of hinges, pays for itself in complete success only when an actuator fails less often than those hinges together.
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.
A stack is a shared fate
Hinges that fold onto one segment of a folded stack share whatever happens to that segment. On a Miura every zigzag line puts one hinge from every row in the same place, so a sheet cut between its rows into five modules leaves every place in every module, and one jam anywhere stops all five. Cut between its columns, each place stays inside one module — except at the slants where two over the sine of twice the slant is an even whole number, the very slants that pile the stack evenly, where columns that far apart share places across the cut.
Two stacks shuffled in glued pairs
A Miura patch three columns wide and r rows high has a central binomial number of folded states — 2, 3, 6, 10, 20 and on to 252 at nine rows — and the count was measured without a reason. The reason is a shuffle. In every state the middle column lies at the bottom and the two end columns lie above it, interleaved like two halves of a deck. Some pairs of an end column's panels can never be parted, because the row crease joining them lies inside the other column's folded footprint, and the two columns take those welds in turn, one at every row crease. That leaves r + 1 unbreakable blocks, and the number of ways to interleave them is the binomial.
An array is priced by its vertices
One slit along a Miura's zigzag crease multiplies the share of letterings that pass every local test by four, and one along a straight crease by three. An array of slits is not the product of those factors. Two slits that meet at a vertex release it once, not twice, and on a small Miura the slits' own factors price correctly exactly the 22 of its 128 arrays in which no two slits touch. What prices every array is the product of each vertex's own share — the ring conditions where nothing was cut, the strip rule wedge by wedge where something was. So the cheapest way to release every vertex is a matching: zigzag slits laid in pairs that never meet, half as many slits as vertices. And the count can be made before anything is cut only on a pattern whose vertices decide independently, which the Yoshimura's do not.
Named alongside it
The objects these essays reach for when they reach for this one.
CorrugationFolded stateRigid foldingDeploymentFootprintGluingUnit cellKawasaki's theoremLayer countMaekawa's theoremParityPeriodicity