proportion-family is one function. Everything below was drawn by it when the site was
last published, at settings 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.
With nothing specified
The A-series is one member of a family A rectangle whose sides are in the ratio √n divides into n rectangles of exactly the same shape, and A4 is the case n equals two. The others are just as real and just as foldable: √n is the diagonal of a rectangle one by √(n−1), so the whole family comes off a square one fold at a time. 1 : √3 = 1.7321 cut into 3, each part is 1.7321 — the same rectangle the family 1 : √2 = 1.4142 → 2 parts, 1 folds to build 1 : √3 = 1.7321 → 3 parts, 2 folds to build 1 : √4 = 2.0000 → 4 parts, 3 folds to build 1 : √5 = 2.2361 → 5 parts, 4 folds to build 1 : √6 = 2.4495 → 6 parts, 5 folds to build A0 is printed at 1.413793 and halves into 1.414634, which is a different rectangle every ratio here is checked against a square root the construction never takes
ns: [2, 3, 4, 5, 6], show: 3The A-series is one member of a family A rectangle whose sides are in the ratio √n divides into n rectangles of exactly the same shape, and A4 is the case n equals two. The others are just as real and just as foldable: √n is the diagonal of a rectangle one by √(n−1), so the whole family comes off a square one fold at a time. 1 : √3 = 1.7321 cut into 3, each part is 1.7321 — the same rectangle the family 1 : √2 = 1.4142 → 2 parts, 1 folds to build 1 : √3 = 1.7321 → 3 parts, 2 folds to build 1 : √4 = 2.0000 → 4 parts, 3 folds to build 1 : √5 = 2.2361 → 5 parts, 4 folds to build 1 : √6 = 2.4495 → 6 parts, 5 folds to build A0 is printed at 1.413793 and halves into 1.414634, which is a different rectangle every ratio here is checked against a square root the construction never takes
ns: [2, 4, 8, 16], show: 2The A-series is one member of a family A rectangle whose sides are in the ratio √n divides into n rectangles of exactly the same shape, and A4 is the case n equals two. The others are just as real and just as foldable: √n is the diagonal of a rectangle one by √(n−1), so the whole family comes off a square one fold at a time. 1 : √2 = 1.4142 cut into 2, each part is 1.4142 — the same rectangle the family 1 : √2 = 1.4142 → 2 parts, 1 folds to build 1 : √4 = 2.0000 → 4 parts, 3 folds to build 1 : √8 = 2.8284 → 8 parts, 7 folds to build 1 : √16 = 4.0000 → 16 parts, 15 folds to build A0 is printed at 1.413793 and halves into 1.414634, which is a different rectangle every ratio here is checked against a square root the construction never takes
ns: [2, 3, 4, 5, 6, 8], show: 4The A-series is one member of a family A rectangle whose sides are in the ratio √n divides into n rectangles of exactly the same shape, and A4 is the case n equals two. The others are just as real and just as foldable: √n is the diagonal of a rectangle one by √(n−1), so the whole family comes off a square one fold at a time. 1 : √4 = 2.0000 cut into 4, each part is 2.0000 — the same rectangle the family 1 : √2 = 1.4142 → 2 parts, 1 folds to build 1 : √3 = 1.7321 → 3 parts, 2 folds to build 1 : √4 = 2.0000 → 4 parts, 3 folds to build 1 : √5 = 2.2361 → 5 parts, 4 folds to build 1 : √6 = 2.4495 → 6 parts, 5 folds to build 1 : √8 = 2.8284 → 8 parts, 7 folds to build A0 is printed at 1.413793 and halves into 1.414634, which is a different rectangle every ratio here is checked against a square root the construction never takes
What it checked while it drew
Collected while it drew, not written by hand.
The count is how many separate times that claim was put to the test.
all 5 of these rectangles divide into parts of their own shape, worst departure 4.4e-16 ×3
the printed A0 is 1.413793 and halves into 1.414634, so the shape is not preserved once the millimetres are rounded ×1
Where it is called
Every figure listed here is drawn by the same rule, so a mistake in the rule would be in all
of them at once, and so would a correction.
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 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.
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.
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.
All the figures ·
The axioms and construction field ·
The patterns a reader can fold