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 3 essays across 2 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

Named alongside it

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

CorrugationKawasaki's theoremLeaf foldingAssignmentThe big-little-big lemmaClosureConvergenceCylinderDeployable boomMaekawa's theoremMiuraPacking ratio

All concepts