Origami & Folding
what one sheet can do
Concept

Taper — where it appears

Cutting a panel's edge back at an angle so that a thick fold can close further than its own depth would allow. It buys travel in proportion to the depth removed and costs the paper that is no longer there, which is the trade every thickness accommodation makes in some form.

Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.

18 interior vertices26 mountains · 19 valleyscolumns taper 2.44 : 1packs to 11.2% of flatmountainvalleyraw edgethe taper is in the columns, because Kawasaki does not mention their widthtapering the rows instead puts the alternating sums at 186.4° and 173.6°

A leaf packs by corrugating

A corrugation is the cheapest fold there is — parallel creases, no interior vertex to think about — and a leaf that uses one has to taper it, because a leaf is broad in the middle. Which direction the taper is allowed to run turns out not to be a matter of taste.

biology · Leaf folding the flat pattern6 columns · 5 rowsthe course, closedradius 0.16676 sides of 0.1667 close into a ring of radius 0.1667, in the sheet's own unitsthe circle through the corners is 4.72% longer than the ring, which is what a 6-sided polygonowes its circle

The cylinder the pattern chooses

A Yoshimura pattern folds into a tube, and the tube's diameter is not a property of the paper. The course of diamonds has to go round exactly once, so the sheet's width is spent on the circumference the moment the columns are drawn — and what a larger sheet buys is a longer tube, never a fatter one.

tessellation · Found patterns the bar is how many repeating rules fold, and it is the same bar six timesthe same corrugation redrawn at six geometries, every one of them swept in fullas printed1648 refused, all by the count · 38 of them close a circle of foureven columns1648 refused, all by the count · 38 of them close a circle of foura one-sided ramp1648 refused, all by the count · 38 of them close a circle of foura violent taper1648 refused, all by the count · 38 of them close a circle of foursix taller rows1648 refused, all by the count · 38 of them close a circle of foura steeper zigzag1648 refused, all by the count · 38 of them close a circle of fourno vertex condition reads a column width, a row height or a row count, so none of them can move the table

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.

biology · Leaf folding three ways to fold a corrugation of 24 columns, laid along its length and shaded by depth a roweach shade one depth a row; the two rows fold exactly onto each other1 deep2 deep3 deep4 deep5 deep6 deep7 deepequal columns: 78% at 4 deepthe taper at its best single slant: 55% at 4 deepthe taper, each zigzag leaning its own amount: 62% at 4 deep

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.

rigid · Thickness straight creases at 60°: the zigzags between 78° and 132°the middle has turned 23° out of the plane of the first columnevery straight crease folds at one angle; each zigzag at its own, set by its lean

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.

rigid · Thickness

Named alongside it

The objects these essays reach for when they reach for this one.

CorrugationKawasaki's theoremMiuraLeaf foldingThicknessAssignmentThe big-little-big lemmaClosureConvergenceCylinderDeployable boomLayer count

All concepts