Origami & Folding
what one sheet can do
Concept

Paper proportion — where it appears

The ratio between a rectangle's two sides, and what folding does to it. Halving an A-series sheet returns a rectangle of the same proportion, and no other proportion has that property, which is the whole reason the series exists.

Named by 12 essays across 4 fields — each of them below, with the objects they name alongside it.

proportion 1.3two shapes, in turn1.3001.5381.3001.5381.300proportion √2 = 1.4142one shape, throughout1.4141.4141.4141.4141.414halving turns a proportion of r into one of 2/r, and those are the same number only at √2the 1.3 sheet is a different shape after every fold; the √2 sheet is the same shape after all of thema square metre at √2 is 840.9 × 1189.2 mm, which is the 841 × 1189 printed on a sheet of A0

The rectangle that keeps its shape

Halving a rectangle across its long side turns a proportion of r into one of 2/r, so almost every sheet comes out of the fold a different shape from the one that went in. Exactly one does not, and it is not a shape anybody chose.

construction · Paper proportion 1 : √3 = 1.7321cut into 3, each part is 1.7321 — the same rectanglethe family1 : √2 = 1.4142 → 2 parts, 1 folds to build1 : √3 = 1.7321 → 3 parts, 2 folds to build1 : √4 = 2.0000 → 4 parts, 3 folds to build1 : √5 = 2.2361 → 5 parts, 4 folds to build1 : √6 = 2.4495 → 6 parts, 5 folds to buildA0 is printed at 1.413793and halves into 1.414634, which is a different rectangleevery ratio here is checked against a square root the construction never takes

One member of a family

A4 halves into A5 and keeps its shape, which is the one thing everybody knows about paper sizes. The property is not about halving and not about two: a rectangle in the ratio √n divides into n copies of itself, for every n, and every one of those rectangles can be folded out of a square one diagonal at a time.

construction · Paper proportion 2 folds from a bare sheet, every proportion at the same areamarks separated by at least 0.3 mm on a 150 mm sheetthe squarewhat origami paper is sold as565 marks · 92 distinct foldsthe A serieshalves into itself45,705 marks · 752 distinct foldstwo squaresa square cut the long way26,155 marks · 540 distinct foldsthe 1 : √3 rectanglethirds into itself42,746 marks · 732 distinct foldsthe golden rectanglenot in the halving family43,233 marks · 732 distinct folds

The sheet decides which points exist

Every measurement of what folding can locate has been made on a square, because origami paper is sold square. Hold the area fixed and change the proportion: one fold reaches nine marks on a square and twenty-nine on the A-series rectangle, and two folds reach 565 against 45,705. The square is the worst of five proportions at both depths, and the reason is its own symmetry.

construction · Reference points the bar is the proportion of the sheet the pattern asks for2 columns, 2 rows1.183(2 + tan 20°) / 2 = 1.1833 columns, 3 rows1.122(3 + tan 20°) / 3 = 1.1224 columns, 3 rows1.455(4 + tan 20°) / 3 = 1.4556 columns, 4 rows1.591(6 + tan 20°) / 4 = 1.5916 columns, 6 rows1.061(6 + tan 20°) / 6 = 1.0618 columns, 6 rows1.394(8 + tan 20°) / 6 = 1.394a square sheet needs a proportion of exactly one, which the counts and the slant have to be chosen for

The paper a pattern asks for

A Miura of c columns and r rows at a slant α wants a sheet whose proportion is (c + tan α) / r — one equation tying the two counts, the angle and the shape of the paper. A square is the case where it comes to one, which needs the tangent of the slant to be a whole number: 45° for a pattern one row taller than it is wide, 63.43° for two, and nothing at all for the slants anybody draws.

tessellation · Miura 3 creases on a Möbius bandthe panels take two coloursseamthe same seam123the right edge onto the left, turned over3 creases, 3 panelsinterior vertices: 0two-coloursthe reflections closeand turn the paper the right waymountainvalleyraw edge

The triangle a strip becomes

A Möbius band of paper folds flat into an equilateral triangle, and the shortest strip that will do it is √3 times its own width. The number is not put in: the crease angles come out of a condition on their alternating sum, the positions come out of two linear equations, and the length is where the drawing stops fitting.

flat-folding · Flat-foldability 1 : √3 = 1.7321cut into 3, each part is 1.7321 — the same rectanglethe family1 : √2 = 1.4142 → 2 parts, 1 folds to build1 : √3 = 1.7321 → 3 parts, 2 folds to build1 : √4 = 2.0000 → 4 parts, 3 folds to build1 : √5 = 2.2361 → 5 parts, 4 folds to buildA0 is printed at 1.413793and halves into 1.414634, which is a different rectangleevery ratio here is checked against a square root the construction never takes

The proportion a band asks for

√2 is a shape: a rectangle either has it or does not, and what it buys is that halving returns the same shape. √3 is what a Möbius band needs, and it is a different kind of number — a minimum rather than a shape, with every longer strip working and no shorter one.

construction · Paper proportion 3 creases on a Möbius bandthe panels take two coloursseamthe same seam123the right edge onto the left, turned over3 creases, 3 panelsinterior vertices: 0two-coloursthe reflections closeand turn the paper the right waymountainvalleyraw edge

Found by people not folding paper

The shortest strip that makes a Möbius band has a literature, and it is in differential geometry rather than in origami. The two subjects have the same number, they reached it by completely different routes, and neither of them cites the other — which is the fourth time this collection has found that shape.

history · Rediscovery what the plane specifies, and what the paper carries2 rounds from a bare sheet, every proportion at the same rulesheeton the paperlost to the edgesquare1 × 1.000565 references72%1,440 off the paperA-series1 × 1.414114,927 references47%102,162 off the paper3 : 21 × 1.500135,281 references48%123,872 off the paperdouble square1 × 2.00044,673 references59%65,294 off the paper2 rounds of a1, a2, a3, a4 · a crossing is a reference only where the paper is · square loses the largest share, at 72%

The field has no edge

Origami numbers are a field on the unbounded plane and a folder has a piece of paper. A fold line runs forever; a crossing of two of them is a number in the field wherever it lands, and it is a reference somebody can put a finger on only where there is paper under it. Counted rather than assumed, two rounds of the four linear axioms on a square put seventy-two per cent of their crossings off the sheet.

construction · Origami numbers the same census, on four sheetsshare of the sheet used, at each polygon's own best rotationsidessquareA series3 : 26 : 5346.4%40.8%38.5%48.1%4100.0%70.7%66.7%83.3%567.4%51.4%48.4%60.5%669.6%61.2%57.7%72.2%772.9%53.5%50.5%63.1%882.8%58.6%55.2%69.0%975.1%54.4%51.3%64.1%1075.3%57.4%54.2%67.7%1176.3%54.8%51.6%64.5%1280.4%56.8%53.6%67.0%best after the square: square 8 · A series 6 · 3 : 2 6 · 6 : 5 6each polygon at its own best rotation on each sheet · the square uses all of a square and 70.7% of the next sheet along

The square is in the answer

The largest regular polygon a square sheet holds is not increasing in the number of sides, and the octagon's win is the striking part: it uses 82.8% of the paper against the twelve-gon's 80.4% and the hexagon's 69.6%. Run the same census over rectangles and the octagon's advantage is gone — on every proportion tried the hexagon leads, and the order among the even-sided polygons reverses outright.

construction · Optimal constructions the same fold, on five sheetsa corner brought to the midpoint of the far edge, and what comes outsheetleft edgeright edgethe crossingwhat happenedsquare1.000 × 1.0003/87/82/3a third, exactlyA series, tall1.000 × 1.4147/1611/162/7a number, and not a thirdA series, wide1.414 × 1.000—1.2500—the crease leaves the paper3 : 2, tall1.000 × 1.5004/92/31/4a number, and not a third3 : 2, wide1.500 × 1.000—1.3438—the crease leaves the paper2 sheets answer and are wrong; 2 refuse — and the difference between the two is a right anglethe alignment does not know what shape the paper is, and neither does the folder following it

A construction assumes its sheet

Haga's fold gives exactly two thirds on a square. Run the same alignment on an A-series sheet held tall and it gives exactly two sevenths, with the crease meeting the vertical edges at seven sixteenths and eleven sixteenths — every one of them a clean fraction, none of them what the recipe promised. Turn the same rectangle through a right angle and the crease leaves the paper instead, which is the loud failure rather than the quiet one.

construction · Paper proportion seven constructions on five sheetswhere each construction's point lands, as a fraction of the sheet, against the squaresquareA series, tallA series, wide3 : 2, talldouble, widehalve it, edge onto edgea fold along the stretch1/2, 0the samethe samethe samethe samea third, from two crossing linescrossings only1/3, 1/3the samethe samethe samethe samea fifth, by repeated crossingscrossings only1, 1/5the samethe samethe samethe samea third, by Fujimoto's halvingsfolds along the stretch0.333, 0the samethe samethe samethe sameHaga's fold, corner to midpointa fold across a slant1, 2/31, 2/7off the paper1, 1/4off the papera corner halved, edge onto edgea fold across a slant1, 11, 0.7070.707, 11, 2/31/2, 1a corner onto the opposite cornera fold across a slant1, 0off the paper3/4, 0off the paper5/8, 0a stretch along the edges keeps crossings, midpoints and folds along the edges; it does not keep a fold across a slant

A stretch keeps crossings

A rectangle is a square stretched along its edges, and a stretch along the edges keeps straight lines straight, crossings as crossings, midpoints as midpoints and the fraction a point divides a segment into. So a construction made only of those — halve an edge, cross two lines — lands at the same fraction of every rectangle, and four standard constructions do. A fold across a slanted line is a reflection the stretch does not keep, and every construction that uses one — Haga's, a corner halved, a corner brought to its opposite — returns a different point on some rectangle, or none.

construction · Paper proportion creases to mark each fraction of an edgeportable: every rectanglesquare onlyA-series sheet only0123456creases11112112313124131234515/2/3/4/5/6/7/8/9/10/11/12the portable column is one number for every rectangle; each sheet's column is true of that sheet alone

What the square saves

A construction made only of crossings, midpoints and folds along the edges lands at the same fraction of every rectangle, so its cost is one number for all of them. Counted crease by crease against every fold the first four axioms allow, that portability costs about a crease and three quarters a fraction up to twelfths — and the square is not the cheapest sheet to give it up for. An A-series sheet, where Haga's fold goes silently wrong, marks two sevenths in two creases; the square needs three, and any sheet at all needs five.

construction · Paper proportion

Named alongside it

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

ConstructionRational divisionSheet shapeExact divisionReference pointClosureGluingHaga's theoremOptimalityThe Möbius bandSymmetryA-series

All concepts