What one fold can refer to, and what two can
reference-closure 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
view: "rounds", depth: 2
view: "growth", depth: 2
view: "numbers"
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.
- at three folds, 94.4 per cent of the marks the axioms reach have another mark within 0.2 mm of them on a 150 mm sheet ×2
- 3 of the four specify nothing the others have not already specified at that round, and 3 of those become indispensable later ×1
- 565 references from the linear axioms and 16890 once the conic one is admitted, each priced by the angle its own folds cross at ×1
- a mark is where two fold lines cross, and the sheet's four edges are fold lines nobody had to make ×1
- adding the fold that bisects an angle takes the same second round to 565 references, of which 432 have a coordinate that is no fraction at all ×1
- at the first round from a bare square the four axioms specify 38 folds and draw 12 distinct lines ×1
- every proportion is compared at the same area, so a sheet with more marks on it is not simply a bigger sheet ×1
- one fold from a bare square reaches exactly nine references — the four corners, the four edge midpoints and the centre — and every coordinate among them is nought, a half or one ×1
- round 2 specifies 300 folds and only 92 of them are different creases, so 69 per cent of the axiom list's output on that configuration is a fold somebody has already made ×1
- the conditioning is read off the crossing angles the closure actually produces rather than modelled from a folder's hand ×1
- the crossings are thinned to the marks a folder could tell apart at a third of a millimetre, because two crossings a thousandth of a millimetre apart are one mark on paper ×1
- the denominators two linear folds reach are 1, 2, 3, 4, 5, 6, 8, 10, 12, 16, 20 and nothing else, and the busiest of them is 8, which 19.5 per cent of the coordinates use ×1
- the first fold puts 4 of its 5 new references on the edge of the paper and the second puts 48 of 556 — the edge is where the early ones are and it fills up ×1
- the first of those is 2 − √2 on the sheet's own edge, 16 references carry it — more than any other irrational — and what it leaves beyond it is √2 − 1, the tangent of the diagonal's half-angle ×1
- the reachable set goes 4, 9 — four corners, then the halves, then everything the halves can be crossed against ×1
- the reachable set goes 4, 9, 565 — four corners, then the halves, then everything the halves can be crossed against ×1
- the same second round without the bisector leaves 133 references and with it 565, of which 432 are past the fractions ×1
- the shallowest crossing goes from 36.9° to 6.3° when the conic axiom is admitted — an error multiplier of 9.1 against 1.7 ×1
- the two sets agree about the typical reference — 68° against 63° at the median — so what the conic axiom adds is the tail and not a general worsening ×1
- two folds that never bisect an angle reach 133 references, every coordinate among them a fraction, and a third is already one of them ×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.
A hole is an edge
A folder's first fold has to be specified by aligning things that are already there, and what is already there is the sheet's outline. Cut a square hole in the middle and the outline doubles: one round of alignments reaches nine references on a plain square and two hundred and twelve on a holed one — more than the plain square reaches in two rounds.
A reference on a sheet with no corner
Every construction in this subject begins from the sheet's own boundary: two edges meet at a corner, a corner is a point, and a point is what an axiom takes as input. A cylinder has two circles of edge and no corners at all, so a construction on one has nothing to start from and the seam is not a mark.
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.
Closer than a crease is wide
One fold from a bare square leaves nine marks, seventy-five millimetres apart. Two folds leave five hundred and sixty-five, the closest pair half a millimetre apart. Three folds — using one axiom of the seven — leave half a million, and ninety-four per cent of them have another mark within a fifth of a millimetre. What bounds a folder is not what the axioms reach; it is what the paper can tell apart.
Dividing a loop into n
Fujimoto's method divides a strip into any number of equal parts by folding badly and then folding the error in half, over and over. It converges because each step halves what is left over. On a closed loop there is no edge for the leftover to sit against, and what replaces the edge is the loop's own closure.
How far from the nearest reference
One fold puts nine reference points on a square sheet and two folds put five hundred and sixty-five. That is sixty-three times as many points, and it brings the worst-covered spot on the paper from a third of a sheet away to a twelfth — four times closer. A count of references is not a measure of what a fold buys, because a set of points can be arbitrarily crowded and still leave most of the sheet out of reach.
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 edge was there first
A folder's first fold has nothing to align to but the sheet's own outline, and it shows in where the marks land: four of the five references the first fold adds are on the paper's edge. By the second fold the edge holds forty-eight of five hundred and fifty-six new ones. The rim is where references are cheap and it fills up, because an edge is a line a fold can cross twice while two folds inside the paper cross once each.
The field has no edge
Origami numbers are a field on the unbounded plane and a folder has a piece of paper. A fold line runs forever; a crossing of two of them is a number in the field wherever it lands, and it is a reference somebody can put a finger on only where there is paper under it. Counted rather than assumed, two rounds of the four linear axioms on a square put seventy-two per cent of their crossings off the sheet.
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 third fold cannot be listed
Two folds from a bare square reach five hundred and sixty-five reference points. The third round specifies three hundred and seventy-eight thousand folds, of which two hundred and seventy-four thousand are distinct — and the crossings of those with each other run to the tens of billions. The closure stops being computable at exactly the depth a folder starts working at, and what can be said instead is a bound rather than a list.
What buys the reach costs the accuracy
A reference is a crossing of two creases, and a crossing transmits a folding error multiplied by one over the sine of the angle the creases make. Measured across the whole closure on a square, the four linear axioms never produce a crossing shallower than thirty-seven degrees — and the conic axiom, the one that sends a point onto a line and reaches the heptagon, produces crossings under seven.
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.
Every generator · The axioms and construction field · The patterns a reader can fold