Generator

Axiom 6 is a common tangent

A generator in the axioms and construction library, called 20 times across 9 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.

beloch-fold 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

`cubic: [1, 0.5, -0.5, -0.125], name: "8x³ + 4x² − 4x − 1"`

`cubic: [1, 0, -2, 0], name: "x³ − 2x"`

`sub: "lill", cubic: [1, 0.5, -0.5, -0.125], name: "8x³ + 4x² − 4x − 1"`

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.

the cubic has 3 real roots in range, so there is a fold to draw ×2
8x³ + 4x² − 4x − 1 has 3 real roots, each of which the drawn path must find ×1
every fold line drawn is tangent to both parabolas at once, to 1e-9 — which is the whole content of Beloch's fold ×1
every ray drawn by Lill's method arrives exactly at the end of the coefficient path, to 1e-9 — so the picture solves the cubic rather than illustrating it ×1
the cubic has 1 real root in range, so there is a fold to draw ×1
the polynomial Lill's method is run on is genuinely a cubic, so the path has four segments and the method applies ×1
the trisecting fold found by Abe's construction lands inside the sheet rather than off the paper ×1
x³ + x + 1 has 1 real root, each of which the drawn path must find ×1
x³ − 2 has 1 real root, each of which the drawn path must find ×1
x³ − 3x + 1 has 3 real roots, each of which the drawn path must find ×1
x³ − x² − 2x + 1 has 3 real roots, each of which the drawn path must find ×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.

Fifty years in the wrong language

Margherita Beloch showed in 1936 that one fold solves a general cubic. The result was correct, published, and in a mathematics journal — and the subject that needed it did not find it until 1991. The cost of a paper nobody reads is measurable, and it is most of a century.

Folding beats the compass, by exactly one degree

Straightedge and compass solve quadratics. A single fold solves cubics. That one-step difference settles two problems Greek geometry could not, and leaves a third exactly as impossible as it was.

Seven, and then twenty-two

The seven axioms are not seven useful folds somebody collected; they are the number of ways to spend a fold line's two degrees of freedom, and the count can be derived. Run the same derivation for two folds made at once and it gives twenty-two, for three fifty, for five a hundred and sixty-one — while the number of coincidences a pair of hands has to achieve in the same instant goes two, four, six, ten.

The axiom that names two folds

Bring one line onto another and the operation is satisfied by either of two folds, always exactly perpendicular to one another, creasing the paper in completely different places. Over four thousand random line pairs on a square, both folds land on the sheet two thousand nine hundred and sixty-five times, and the statement of the axiom does not say which one is meant. The fifth axiom is worse: its two answers are at any angle at all, from half a degree apart to square.

The biggest one that can also be folded

Which regular polygon uses a square sheet best, and which of them a fold can actually construct, are two questions with completely different pedigrees. Answered side by side over sixteen polygons, they turn out to agree — and the reason is that both are questions about the arithmetic of the same number.