proportion-family 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 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 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.
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
Changing this generator changes every figure on this list, which is what makes the list
worth publishing rather than keeping in a check script.
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.
Every generator ·
The axioms and construction field ·
The patterns a reader can fold