Generator

Eight combinations, seven of them a fold

A generator in the axioms and construction library, called 31 times across 6 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.

axiom-completeness 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: "machine", show: "loop"`

`view: "labelled", show: "three", exactTo: 2`

`view: "labelled", show: "kinds", folds: 2`

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.

and at 2 folds it has 14 in all, the extra 7 of them naming one of the 1 creases being made beside it ×2
64 of the 86 operations at two folds name a crease being made in the same instant — 74% of them cannot be stated in terms of the paper at all ×1
77 of the 105 operations at 2 folds are reachable by single folds in some order, and 28 are not ×1
a fold constrained only by the paper has the seven ways to spend its two freedoms that the axioms are ×1
above one fold the labelled count is larger at every size drawn — 105, 3042, 145,211 against 86, 296, 791 ×1
above one fold the labelled count is larger at every size drawn — 105, 3042, 145,211, 9,782,771 against 86, 296, 791, 1792 ×1
adding the alignments that name a simultaneous crease leaves the single-fold count at seven, because one fold has no simultaneous crease to name ×1
all 8 alignment combinations get a verdict from the enumeration, so none is silently unclassified ×1
and each of the three really does land both points on their lines, to 2.2e-16 — the solver's answers are checked by folding rather than trusted ×1
and it settles geometrically, the error falling to about 8 per cent of itself at every pass ×1
and only 28 of 105 — 27 per cent — actually have to be made at one instant; the rest are a sequence or two separate folds ×1
and the repair leaves the single-fold count at seven, so it has not invented an eighth axiom ×1
at one fold the seven operations offer at most three folds each, which is the cubic and is the whole of what a single fold adds to the compass ×1
each fold lands its own point on the other fold's crease, to 1.1e-16 — neither crease exists before the other and both conditions hold ×1
enumerating the alignments of points and lines gives exactly seven single-fold operations, which is why the axiom list stops where it does ×1
every one of the 105 two-fold operations falls into exactly one of three kinds: 28 that name nothing, 49 whose naming has no cycle in it, and 28 that do ×1
of 60 starting guesses spread across every direction, 29 run to this solution, 0 to another and 31 stall on a step with no answer ×1
so seven operations offer 11 constructions between them, and one of the seven offers three of those ×1
the 105 unordered pairs of fold specifications are the 105 two-fold operations, so every one of them is priced ×1
the catalogue outgrows the best an operation can offer at every step — 2.3, 11.7, 112.7, 1792.7, 40258.3 operations per solution ×1
the exact enumeration and Burnside's lemma agree at every fold count they can both reach — 7, 105 ×1
the exact enumeration and Burnside's lemma agree at every fold count they can both reach — 7, 105, 3042 ×1
the first 5 passes of the loop all put both creases across the sheet, so the whole approach is drawable ×1
the gap between the two counts widens at every fold count — 1.0, 1.2, 10.3, 183.6, 5459.1 times ×1
the loop settles on one pair of folds — 1.16438 and 0.26745 radians — with nothing moving in the last step ×1
the most any two-fold operation admits is nine folds — three from each half, and three is the most one fold can offer ×1
the paper-only enumeration gives 7, 22, 50 and the one that admits simultaneous creases gives 7, 86, 296 ×1
the paper-only enumeration gives 7, 22, 50, 95 and the one that admits simultaneous creases gives 7, 86, 296, 791 ×1
the paper-only enumeration gives 7, 22, 50, 95, 161 and the one that admits simultaneous creases gives 7, 86, 296, 791, 1792 ×1
the paper-only enumeration gives 7, 22, 50, 95, 161, 252 and the one that admits simultaneous creases gives 7, 86, 296, 791, 1792, 3612 ×1
the pooled count is below that at every size above one fold, so it was not a floor on the crossings — it was short of the case with no crossings in it at all ×1
the rule gives the seven axioms 1, 1, 2, 1, 2, 3, 1 solutions, which is what they are known to have ×1
the same enumeration gives 7, 22 operations at one to 2 simultaneous folds ×1
the same enumeration gives 7, 22, 50 operations at one to 3 simultaneous folds ×1
the same enumeration gives 7, 22, 50, 95 operations at one to 4 simultaneous folds ×1
the same enumeration gives 7, 22, 50, 95, 161 operations at one to 5 simultaneous folds ×1
the two enumerations agree at 3 fold counts, having built 17,576 structures to do it at the largest of them ×1
this point and line, and that point and line, admit exactly three folds carrying each point onto its own line — the sixth axiom at its full count ×1
two simultaneous folds admit at least 86 operations against the 22 the paper-only enumeration counts — so twenty-two is a floor and not the number ×1
with no crossings an m-fold operation is m single folds chosen from seven axioms, and the enumeration returns exactly the C(m+6, m) multisets — 7, 28, 84, 210, 462 ×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 crease that does not exist yet

Simultaneous folding is usually described as a problem of dexterity — several coincidences to be achieved in the same instant. The reference graph says otherwise: of the hundred and five two-fold operations, twenty-eight need no simultaneity and forty-nine can be done in an order, leaving twenty-eight whose folds each name the other. Those are not hard to hold. They are hard to know, and a loop that guesses and re-solves finds them at eight per cent of the error a pass.

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.

Counting operations is not counting power

The catalogue of simultaneous-fold operations runs from seven to nearly ten million between one fold and five. What a construction can reach does not: each fold admits at most three lines, because two parabolas have three proper common tangents and not four, so m folds admit at most three to the m — and the largest polynomial degree they actually settle is smaller again, at twice m plus one. Three counts of the same subject, growing at three speeds.

Each fold needs its own two

The enumeration that gives seven axioms spends a fold line's two degrees of freedom on alignments; run for m folds it spends 2m from one pool, and a pool can be spent three on one line and one on the other, which determines neither. Attaching every alignment to the fold it constrains repairs that, and two other things — and the two-fold count goes from twenty-two to a hundred and five, of which only twenty-eight have to be made at one instant.

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.

Twenty-two is a floor

The enumeration that gives seven single-fold axioms spends each fold line's two degrees of freedom on alignments to points and lines already on the paper, and its own account says what it leaves out — an alignment may refer to a crease being made in the same instant. Adding those back leaves the single-fold count at seven and takes the two-fold count from twenty-two to eighty-six, of which sixty-four cannot be stated in terms of the paper at all.

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