Origami & Folding
what one sheet can do
Generator

The sheet that is the same shape after it is folded

A generator in the axioms and construction library, called 26 times across 6 essays. Below: what it draws at its defaults and at the arguments the essays give it, what it checked while drawing, and everywhere it is used.

paper-proportion is one function. Everything below came out of it during this build, at arguments taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and when the generator changes, this page changes with it.

At its defaults

The sheet that is the same shape after it is foldedOne rectangle halved repeatedly across its long side, drawn nested, at two starting proportions. On the left the shape alternates between two rectangles and comes back to itself on every second fold. On the right the proportion is √2 and every nested rectangle is the same shape as the sheet it came from, which is what the A series is for.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

view: "assumes"

A construction assumes its sheetThe classical corner-to-midpoint fold, run on five proportions of paper. On a square it puts the crease at three eighths and seven eighths of the vertical edges and crosses the far edge at exactly two thirds. On other sheets it does one of two things: the crease leaves the paper, so the construction refuses itself; or it stays on the paper and returns a different clean fraction with nothing to indicate that anything has changed. Which of the two happens depends on which way round the rectangle is held.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

view: "assumes", sheets: [1,1,square, 1,1.2,6 : 5, tall, 1.2,1,6 : 5, wide, 1,1.8,9 : 5, tall, 1.8,1,9 : 5, wide]

A construction assumes its sheetThe classical corner-to-midpoint fold, run on five proportions of paper. On a square it puts the crease at three eighths and seven eighths of the vertical edges and crosses the far edge at exactly two thirds. On other sheets it does one of two things: the crease leaves the paper, so the construction refuses itself; or it stays on the paper and returns a different clean fraction with nothing to indicate that anything has changed. Which of the two happens depends on which way round the rectangle is held.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, exactly6 : 5, tall1.000 × 1.2000.413190.760420.42017a number, and not a third6 : 5, wide1.200 × 1.000—1.0400—the crease leaves the paper9 : 5, tall1.000 × 1.8000.461420.615740.16722a number, and not a third9 : 5, wide1.800 × 1.000—1.7150—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

view: "assumes", sheets: [1,1,square, 1,1.1,11 : 10, tall, 1,1.4142135623730951,A series, tall, 1,2,double, tall, 1,1.3,13 : 10, tall, 1.3,1,13 : 10, wide]

A construction assumes its sheetThe classical corner-to-midpoint fold, run on five proportions of paper. On a square it puts the crease at three eighths and seven eighths of the vertical edges and crosses the far edge at exactly two thirds. On other sheets it does one of two things: the crease leaves the paper, so the construction refuses itself; or it stays on the paper and returns a different clean fraction with nothing to indicate that anything has changed. Which of the two happens depends on which way round the rectangle is held.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, exactly11 : 10, tall1.000 × 1.1000.396690.809920.52083a number, and not a thirdA series, tall1.000 × 1.4147/1611/162/7a number, and not a thirddouble, tall1.000 × 2.0000.468750.593750.13333a number, and not a third13 : 10, tall1.000 × 1.3000.426040.721890.34722a number, and not a third13 : 10, wide1.300 × 1.000—1.1338—the crease leaves the paper4 sheets answer and are wrong; 1 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

What it checked while it drew

Collected by running this generator with a listener on the assertions, not written here. The count is how many separate times this build put that claim to the test.

Where it is called

Changing this generator changes every figure on this list, which is what makes the list worth publishing rather than keeping in a check script.

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.

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.

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.

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.

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.

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.

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