Origami & Folding
what one sheet can do
Concept

Operation set — where it appears

The catalogue of folds a tool can specify, derived by counting the ways to spend a fold line's degrees of freedom on alignments. For one fold it comes to seven and is the axiom list; for several made at once it grows, and how far depends on what an alignment is allowed to name.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

the same count, with more than one fold made at a timefreedomsoperationsalignments at onceone fold272two at once4224three at once6506four at once8958five at once1016110the second column is what the algebra gains; the third is what a pair of hands has to hold

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.

construction · Multifold the count that was made, and the count that was notboth are floors: neither enumeration tracks which fold an alignment attaches tofreedomspaper onlywith simultaneous creasesneeding oneone fold277—two at once4228664three at once650296246four at once895791696five at once1016117921631at two folds the omission is 64 operations of 86 — 74% of them, and none can be described without naming the other creasea pair of hands cannot make a condition between two creases it is making; a jig holding two lines can

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.

construction · Multifold three enumerations of the same thinga fold line has two freedoms, so m of them have 2m — the question is whether the budget is one pool or m pursesfolds at oncepaper onlypooled, with crossingseach alignment attachedthe ratio1777× 1.022286105× 1.23502963042× 10.3495791145,211× 183.6516117929,782,771× 5459.1the last column is what tracking which fold an alignment names is worth, and it grows because the naming itself grows

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.

construction · Multifold first pointsecond pointthe sixth axiom, at its full countthe first foldthe second foldthe third foldthree folds, each carrying one point onto one line and the other point onto the other — the most any single fold can offer

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.

construction · Multifold throughthroughlands on the other creaseand so does this onea cyclic operation, solvedeach crease is described in terms of the other, so neither can be made first and no order exists

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.

construction · Multifold

Named alongside it

The objects these essays reach for when they reach for this one.

EnumerationMultifoldAxiomsDegrees of freedomConstructibilityThe Huzita–Hatori axiomsCubicQuintic

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