Footprint — where it appears
Named by 13 essays across 5 fields — each of them below, with the objects they name alongside it.
The paper is all still there
A folded sheet is smaller than it was and none of it has gone anywhere. How much smaller it is and how many layers deep it is are not two properties of a pattern — they are one number, and their product is the sheet.
Bringing the other side to the front
Paper has two sides and most models show one. A colour change shows the other, and it is not a crease problem — which panels can show the reverse is settled by the pattern's two-colouring, and what it costs is twice what it shows.
What a corrugation costs
Every tessellation this repository can fold, measured the same way: how much smaller it gets, how deep the stack becomes, and how much creasing was needed to buy it. The last column is the one nobody quotes and the one a folder feels.
A shrink is two numbers
How much smaller a folded sheet gets is quoted as a single factor, and that factor is a product. Measured along each axis separately, an accordion turns out to leave one direction of the paper exactly alone, a twist draws in equally both ways, and the Miura does neither — which is the whole of what makes it a Miura.
The plane the five points were in
Five corrugations measured at one setting each gave five points, and the space between them was left as an open question: forbidden, or merely unvisited. Every one of those patterns has a dial nobody turned. Turned, they trace curves — the accordion's is a line with integers on it, the square twist's is the diagonal and nothing else, and the Miura's turns round on itself, so a steeper slant stops buying a smaller sheet.
The direction that gets longer
A shrink factor below one is a direction in which the folded sheet is bigger than the flat one, and the leaf corrugation has one. The cause is not the taper and not the angle: a corrugation's folded extent across its own creases is a constant of the cell, the same number at two rows and at ten, so the cross factor is the sheet's height divided by a fixed length — a straight line through the origin that crosses one at a definite row count.
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.
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.
A sheet that turns with the twist
Holding a different panel of a folded twist can give a different pair of shrink factors: on a square sheet the three- and six-sided twists gave two pairs each and the five-sided twist five. Cut each twist from a regular sheet of its own number of sides and every panel gives the same pair. The reason is a count of rotations. A K-sided twist's panels hold its folded state in frames that are multiples of 180⁄K or 360⁄K degrees; a sheet that shares g of the twist's rotations makes the widths agree in every frame those rotations reach, with the half-turn a width cannot see thrown in. On twenty-two pairs of twist and regular sheet, that count gives the exact number of answers every time — a hexagon twist on a triangle gives one, a pentagon twist on a square five.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
Folded stateLayer countMiuraShrinkageDeploymentMeasurementConservationPacking ratioAnisotropyCorrugationHingeTessellation