The seven ways to specify a fold
axiom-set 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
which: [1, 2, 4]
sub: "axiom-solutions", view: "count", trials: 4000
sub: "axiom-solutions", view: "onpaper", trials: 4000
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.
- 2965 of 4000 random line pairs leave both of the axiom's folds on the paper, and the axiom does not say which is meant ×3
- 4,000 alignments drawn at random inside the sheet, and the number of folds each axiom names counted rather than assumed to be one ×2
- a3 names creases that never touch the paper: 13.3% of the folds it specifies ×2
- 3009 of 4000 inputs give two folds; the angle between them runs from 0.6° to 90.0° with a median of 68° ×1
- a fold is a line, and a line the paper does not reach is a fold nobody can make — so every named fold is clipped to the sheet before it is counted ×1
- a5 names no fold at all in 24% of them, and two folds in most of the rest ×1
- both folds are drawn from the axiom's own arithmetic rather than sketched, so the right angle between them can be measured off the screen ×1
- the fifth axiom's two answers are a circle's two crossings of a line, so a single answer is the tangent case and a random draw does not find it ×1
- the seven ways to specify a fold are listed with the degree each reaches, exactly one of them cubic, and the four solved here in closed form return 1, 1, 2, 1 folds — the pair at axiom 3 being the two bisectors of a crossing ×1
- the two folds the third axiom names are the two angle bisectors, and a bisector pair is square by construction — checked to machine precision on every input here ×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.
A fold needs something to align
Every axiom names things that must already be on the paper — a point to fold onto a point, a line to bring to a line. So what a folder can build is bounded by what they can refer to, and that set is finite at every depth: nine references after one fold, several hundred after two, and every one of them computable in advance.
An axiom may name no fold
The seven operations are stated about points and lines in a plane, and a plane has no edges. On a square, three of them always name exactly one fold and always land it on the paper; placing a line on a line names two, and 13.3% of the folds it specifies are creases the sheet never reaches; and placing a point on a line through a second point names two folds, one, or — 23.8% of the time — none at all.
Cheap where it reaches
Two folds from a bare square put marks at a half, a third, a quarter, a fifth, a sixth, an eighth and a twelfth — and at no seventh, ninth or eleventh at all. A rule that reaches every fraction takes n folds to reach one nth. The systematic route and the short one disagree everywhere, and neither of them knows about the other.
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.
One crossing, and then another
Folding a strip into thirds by Fujimoto's method halves the error at every fold and never reaches a third. There is a construction that arrives instead: cross the square's diagonal with a line through the mark you already have, and the crossing lands on the next fraction exactly — one fold per step, all the way down.
One fold at a time, and there are exactly seven of them
A fold is specified by bringing points and lines into coincidence. There are seven ways to do that, the list is provably complete, and one of the seven does something no compass can.
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 largest triangle in a square
The biggest equilateral triangle a square sheet holds is tilted by exactly fifteen degrees and uses 46.4% of the paper. Both numbers come out of a quadratic — which means a compass reaches this optimum too, and folding's advantage is not needed here at all.
The numbers a fold reaches
Folding solves cubics, which is one fact about one fold. The reason the subject has a theory rather than a bag of tricks is a second fact about all of them: the lengths a folder can mark are closed under addition, subtraction, multiplication, division, square roots and cube roots. Constructions can therefore be built out of constructions — and no tower of them ever arrives at a fifth root.
What each axiom is worth
The list of seven folds is complete, and the proof of that says nothing at all about whether its members are independent or equal. Measured on a bare square, one of the four elementary axioms supplies every fold the others cannot and the other three supply nothing. Two rounds later the ranking has inverted, and the one that carried the first round is the least productive of the four.
Which of the seven survive
The seven axioms are the complete list of ways one fold can be specified by aligning marked things. Every one of them names points and lines on a sheet, three of them quietly assume that a line has two sides, and on a closed sheet a line need not — so the list is complete for a disc and shorter for anything else.
Why the list stops at seven
The seven axioms are not seven useful folds somebody collected. They are every fold there is, and the proof is an exercise in counting degrees of freedom that takes about a minute.
Every generator · The axioms and construction field · The patterns a reader can fold