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 5 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 the cotangent of half the anglepatternnarrowest sectorreach, both endsthat crease, mmroom to shrinkThe tapered corrugation65.9°0.7712.226×The Yoshimura pattern60.0°0.8714.227×The waterbomb tessellation45.0°2.4140.028×The hexagon twist60.0°1.3725.531×The Miura fold69.9°1.4328.433×Fold and cut — the triangle58.2°0.9028.653×The square twist90.0°1.0036.160×The preliminary base45.0°1.2175.0104×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⁄2)·cot(θ⁄2) for a sector angle θ below a right angle, and half a band above it. 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 207 cells a side by its vertices against 262 by its spacing, and the finest waterbomb 117 against 141. Both allow half as much again as the density bound, which had the wrong argument and the wrong number.

material · Crease density two vertices, their bands, and the disc every band containsseven band widths across each panel; dots where each crease's band stops overlapping its neighbour'sa Miura vertex, slant 0.35degree 4, narrowest sector 69.9°creased 4 deep: 0.786 w²creased twice or more: 1.06 w²the preliminary base's centredegree 8, narrowest sector 45.0°creased 8 deep: 0.787 w²creased twice or more: 2.83 w²

A disc as deep as the degree

A crease occupies a band a few sheet thicknesses wide, and near a vertex every band contains the point. So the paper there is creased as many times as the vertex has creases — over a disc exactly half a band across, π/4 of a band width squared, at every flat-foldable vertex whatever its angles. Around the disc lies a fringe creased twice, which is what a narrow sector buys: 0.99 band widths squared at a right-angled vertex, 2.86 at the preliminary base's centre. At printed size the doubled ground is a few square millimetres. It is a fixed area per vertex, so it grows as the square of the fineness, and at the finest Miura any paper carries more than half the sheet is creased at least twice.

material · Crease density evenly spread vertices: the doubled ground is a star with straight sidesdashed: each neighbouring pair's kite; the star is their union, and the disc's rim never reaches its edge past degree fourdegree 4, every sector 90°twice or more: 1 = 1.000 w²degree 6, every sector 60°twice or more: √3 = 1.732 w²degree 8, every sector 45°twice or more: 2√2 = 2.828 w²

Where two bands part

Two creases leaving a vertex share a kite of paper beyond the disc every band contains: its sides are the bands' far edges, tangent to the disc, and its tip sits on the bisector. The kite is h²(cot(θ/2) − (π − θ)/2), and a whole vertex is its kites less the overlaps of neighbouring kites — exactly 1, √3 and 2√2 band widths squared for the square grid, the Yoshimura and the preliminary base. The same kite says where the bands part along each crease, (w/2)·cot(θ/2), which is shorter than the w/sin θ these essays had charged: the finest Miura on copier paper is 207 cells a side, not 135, and at that size 57.5 per cent of the sheet is creased twice.

material · Crease density the ground creased twice round a flat-foldable degree-four vertex, over its two free numbersx across: the mean of the two free sectors; y up: half their difference; darkest band nearest one0°30°60°90°30°60°90°x, the mean sectoryone, at four right anglesno vertex past the diagonal: a sector would be nought1 to 1.051.05 to 1.21.2 to 1.51.5 to 22 to 33 to moreband widths squared

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.

material · Crease density

Named alongside it

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

Crease lengthIdealisationThicknessVertex degreeCrease radiusKawasaki's theoremThe big-little-big lemmaBoundary vertexCrease densityDegree-fourFlat-foldabilityMaekawa's theorem

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