A crease that does not exist yet

Seven, and then twenty-two sets the catalogue of simultaneous-fold operations against the number of coincidences a pair of hands has to achieve in the same instant — two at one fold, four at two, six at three — and the comparison invites a reading in which multifolds are hard because hands are clumsy.

Once every alignment is attached to the fold it constrains, that reading stops fitting. Most simultaneous operations do not have to be simultaneous. And the ones that do are not asking for steadier hands; they are asking for something a hand cannot supply at any steadiness, which is knowledge of a crease that does not exist yet.

Three kinds of simultaneity, and only one of them is any #

Each fold needs its own two puts an arrow from every fold to every fold that its own alignments name, and reads the operation off the resulting graph.

An operation with no arrows is two single folds that happen to be made together. There are 28 of them at two folds, and a folder can make either first, or both, or one on Tuesday.

An operation whose arrows form no cycle can be sorted. One fold names nothing and can be made first; the other names it and can be made second. It is a sequence, and though it may be convenient to make both creases at once, everything it constructs is constructible one fold at a time. There are 49 of these, nearly half the catalogue.

An operation whose arrows form a cycle cannot be sorted. Each fold is described in terms of the other, so neither is available first, and there is no order to put them in. There are 28 — a quarter of the catalogue and the whole of what simultaneity means.

That division is invisible to any count that pools its constraints, because a pooled count has no folds for the arrows to run between. It is also not the division the coincidence argument makes: a folder making two creases together has the same number of alignments to hold in all three cases, and in two of the three there was no need to do it together at all.

What a cyclic operation asks for #

Take the simplest cyclic operation available. The first fold passes through a named point and carries a second named point onto the crease the other fold is making. The second fold does the same in reverse. Four conditions, four freedoms, and each fold’s description contains the other.

A pair of hands cannot execute that description. Not because it is delicate, but because the instruction refers to something not yet present: a folder asked to bring a point onto a crease must be able to see the crease, and at the moment of asking neither crease exists. The instruction is not difficult; it is not yet an instruction.

A mechanism is in a different position, and the difference is worth stating precisely. A mechanism holding two adjustable fold lines does not need to know the creases in advance. It needs to reach a state in which all four conditions hold, and it can look for that state by trying: set the second line anywhere, solve the first against it, re-solve the second against the answer, and repeat.

Guess, solve, re-solve #

Solving one fold against a known line is the fifth axiom — through a named point, carrying a named point onto a named line — and it has at most two answers. The loop is that solve twice, alternately, each pass using the line the last pass produced.

The first guess is visibly wrong and the second is nearly right. By the fifth pass the creases have stopped moving to the eye, and by the tenth they have stopped moving to eleven decimal places.

The straight line is the whole mechanism. Each pass multiplies the error by the same factor, about 8 per cent here, so accuracy costs a fixed number of passes per decimal place — a little under one pass each. Ten passes take a guess that was half a radian out to eleven decimal places, which is a great deal better than any hand and better than the paper deserves.

That is what a machine’s advantage over hands actually is in this subject. Not steadiness, and not the ability to hold four alignments: the ability to be wrong and then less wrong, which a folder making a crease in paper cannot be, because a crease once made cannot be un-made.

Where the loop goes when it does not converge #

A contraction is a local statement and the figure’s straight line is one configuration’s behaviour from one starting guess. Sweeping the guess through every direction gives a less tidy picture and a more honest one.

Of sixty starting angles spread evenly across every direction, twenty-nine run to the solution and thirty-one stall — on a pass where the fifth-axiom solve has no answer at all, because the parabola it needs a tangent to does not reach the point it must pass through. None runs to a different solution.

So the loop is not a method. It is a method, with a basin: from inside, it converges geometrically; from outside, it fails in one step and says so. A mechanism built on it needs a way of choosing where to start, or a fallback when the step has no answer — a joint solve on all four conditions at once, which is a root-find in four variables rather than two root-finds in one.

That is a real limit and it is the interesting kind, because it is not about the operation. The operation has its solution; the loop is one way of finding it, and roughly half of the ways in are ways that do not work.

Twenty-eight, and what they might be worth #

Twenty-eight operations is not many, and the question is whether they are worth building a mechanism for. Two things say they might be.

The first is that a single cyclic operation was already known to be worth a great deal. Two creases at once finds the hendecagon on the far side of the two-fold boundary, and counting operations is not counting power locates why: two folds settle a quintic where one settles a cubic, and eleven needs a fifth root. A quarter of the catalogue carrying the whole of the extension is the same shape as the single-fold case, where one axiom of the seven carries the cube root and the other six do not.

Why the list stops at seven is the single-fold version of the same accounting, and reading the two together is what makes the shape visible: a catalogue’s power tends to sit in a small, identifiable part of it, and the part is identifiable by a structural property rather than by trying every entry.

The second is that the advantage compounds. At three folds the cyclic operations number 1,705 of 3,042 — more than half — because a graph on three nodes has many more ways to contain a cycle than a graph on two. A mechanism’s share of the catalogue grows with the number of lines it can hold, and it grows past half somewhere between two folds and three.

The sequence that looks simultaneous #

The middle column deserves more than a sentence, because forty-nine of a hundred and five is the largest of the three and the easiest to misread.

A sequential operation is one where some fold names nothing and the other names it. Performed, it looks exactly like a multifold — two creases going in together, two alignments held at once — and it is in every respect a multifold except the one that matters for constructibility. Everything it reaches, a folder reaches by making the first crease, looking at it, and making the second.

So nearly half the two-fold catalogue is notation. It is a way of writing down a two-step construction as one step, and there are reasons to want that — it is shorter, it is what a diagram would show, and a folder working quickly does perform such a pair together. What it is not is new reach, and twenty-two is a floor counted these alongside the cyclic ones with nothing to tell them apart.

This is also where the earlier coincidence argument is still exactly right and now applies to a smaller set. A sequential operation performed as one motion genuinely does demand that several alignments be held together, and a folder who cannot hold them can fall back on the order. A cyclic one offers no fallback. Dexterity is optional for forty-nine of them and irrelevant for twenty-eight, and the coincidence count could not see which was which.

What a mechanism would have to be #

Nothing above describes a machine, and the gap between an iteration and a device is where most of the engineering is.

The iteration adjusts two lines and evaluates four conditions. A device doing that must hold the sheet without creasing it — every pass is a trial position, and paper that has been trialled is paper that has been marked. So the search happens somewhere other than in the paper: in a model, in an optical measurement of the sheet’s own reference points, or in a mechanism that positions two straightedges and only then presses.

That separation is the honest version of the machine’s advantage. A folder’s every attempt is permanent; a solver’s attempts are free until the last one. The advantage is not in the folding at all — it is in being allowed to compute before folding, and any device or person with that permission has it. A folder who works out both creases on paper first, by any means, and then makes them, is performing a cyclic operation, and the mechanism is only the automation of that.

Which means the trichotomy is less about machines than it first looks. It is about when a construction can be described in the order it is performed, and the cyclic operations are exactly the ones that cannot. The fold a machine can make asks the one-crease version of that question; this is its counterpart at the construction end, and the answer has the same shape — the limitation is not on the folding but on the describing.

What the loop cannot show #

The figures draw one configuration of one cyclic operation and a sweep of starting guesses, and each of those is narrower than the claim they support.

They cannot show what the other twenty-seven cyclic operations do. Nothing here surveys them, so whether they all admit a contracting loop, or whether some are stiff in a way that needs a joint solve everywhere, is unmeasured. The one drawn was chosen because it is the simplest, not because it is representative.

They cannot show where the basin’s boundary is. The sweep counts sixty starting angles and reports how many arrive; it does not describe the set of good starts, which for all this says could be an interval, several intervals or something worse.

They cannot show whether the operation constructs anything unavailable elsewhere. A cyclic description is not a guarantee of new reach — it guarantees only that this description has no order, and the same points might be reachable by some entirely different sequence. Fifty years in the wrong language is the standing reminder that a construction can exist under another name.

And they cannot show how many solutions the operation has. The loop finds one fixed point and the sweep found no other, which is evidence and not a count — the system’s solutions are the roots of a polynomial and a numerical loop finds only the ones it is attracted to. A complex root or a repelling real one would be invisible to every figure here.

What the model assumes #

A fold line is exact and a trial costs nothing. The loop’s whole advantage is that its intermediate positions are free, which is true of a computation and of a positioning mechanism, and false of a folder’s paper.

The alignments are the same ones the catalogue uses, priced the same way, with a crease being made in the same instant treated as an ordinary line once it is provisionally known.

The branch is chosen by continuity. Each solve has up to two answers and the loop takes the one nearest the last, which is the choice a mechanism tracking a moving line would make and is not the only possible one. A different branch rule is a different loop with a different basin.

And the sheet is unbounded for the purpose of solving. A solution whose fold line misses the paper is still a solution of the system; the figure checks that the drawn one crosses the sheet, and the sweep does not check that for the starts it counts.

How the numbers were checked #

Every solved fold is verified by folding. The two creases the loop settles on are used to reflect their own points, and each reflected point is required to land on the other crease to within a part in ten billion — so the fixed point is checked against the conditions rather than against the solver.

The contraction is measured, not assumed. The ratio between consecutive steps is taken over the first five passes and required to be under a half; a loop that wandered or diverged would fail there before any figure was drawn.

The last step is required to move nothing at all, which is the statement that the loop has settled rather than merely slowed.

And the sweep is required to find at least one start that arrives. A loop nothing reaches would not be a way of finding anything, and the count of stalls is reported rather than hidden, because half of them stalling is the result.

Still open: whether every cyclic operation has a loop #

The one measurement this account most obviously owes is the survey it did not make.

Twenty-eight cyclic operations at two folds, and one of them has been solved. Each of the others is a different pair of conditions, and the alternating loop is available for any of them whose one-fold solves are well posed. The questions are whether each contracts, at what rate, and from how much of the space of starts — and the answer is a table of twenty-eight rows that would say whether “a mechanism can do the cyclic ones” is a theorem or an anecdote.

The harder version is the one the stalls point at. A pass fails when the fifth-axiom solve has no real answer, which is a statement about where the other line happens to be, so the failures are an artefact of the method rather than of the operation. A joint solve on all four conditions at once has no such failure mode, and comparing the two would separate what is hard about these operations from what is hard about this way of attacking them.

Sideways from here, the same distinction belongs beside the reach. Counting operations is not counting power finds three counts growing at three speeds, and the cyclic share is a fourth: it is the part of the catalogue that is not already inside the single-fold theory, and it is the right denominator for any claim about what simultaneity buys. Whether the 28 and the 1,705 add anything to the field — rather than merely to the list — is the question what each axiom is worth asks one level down, and it has not been asked here.

The habit worth carrying is about instructions that refer forward. When a procedure names something it has not yet produced, ask whether the naming forms a cycle. If it does not, there is an order and the procedure is a sequence; if it does, no order exists and the only way through is to solve — which means the difficulty was never in the doing.