Origami & Folding
what one sheet can do
Concept

Sector angle — where it appears

The angle of paper between two neighbouring creases at a vertex. Kawasaki's condition is a statement about sector angles alone, and a narrow sector also decides how far the bands of its two creases overlap in real paper.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

the pale bar is the share that folds in a ring, the dark one in a line40 vertices from each population at each degreeangles at random, degree 425% in a ring · 70% in a lineangles at random, degree 613% in a ring · 46% in a linemultiples of 45°, degree 432% in a ring · 83% in a linemultiples of 45°, degree 630% in a ring · 87% in a linemultiples of 30°, degree 429% in a ring · 79% in a linemultiples of 30°, degree 623% in a ring · 70% in a line

A ring and a line

A vertex has a certain amount of paper at it, and the paper either closes round or it does not. Holding the sectors fixed and changing only that: the ring has twice as many letterings to choose from and folds in a quarter of them, the line has half as many and folds in seven-tenths, and cutting a ring open has never once cost a lettering.

flat-folding · Boundary the crease each pattern has least room on, once the ground near its ends is creased twicea narrow sector pushes the overlap out along both creases, as one over the sine of the anglepatternnarrowest sectorreach, both endsthat crease, mmroom to shrinkThe tapered corrugation65.9°2.1020.416×The waterbomb tessellation45.0°2.4128.320×The hexagon twist60.0°2.1525.520×The Yoshimura pattern60.0°2.3128.320×The Miura fold69.9°2.0626.722×The square twist90.0°2.0036.130×Fold and cut — the triangle58.2°1.1828.640×The preliminary base45.0°1.4175.088×reach is in band widths; room to shrink is the crease's length over that reach, on copier paper with a band 0.6 mm wide

A vertex creases the paper twice

Two creases that meet share ground near the point, and their bands overlap out along each of them to w⁄sin θ for a sector angle θ below a right angle. Add the overlap at both ends of a crease, and a crease no longer than that is overlap from end to end — a crease only in the drawing. That third bound binds before the spacing does: the finest Miura on copier paper is 135 cells a side by its vertices against 262 by its spacing, and the finest waterbomb 82 against 141. Both land within a tenth of the density bound, which had the wrong argument and nearly the right number.

material · Crease density

Named alongside it

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

The big-little-big lemmaBoundary vertexCrease lengthCrease radiusFlat-foldabilityIdealisationMaekawa's theoremStripThicknessVertex degree

All concepts