Twenty-two is a floor

Seven, and then twenty-two derives the axiom count rather than remembering it. A fold line in the plane has two degrees of freedom; alignments cost one constraint or two; enumerate the ways to spend two constraints on five kinds of alignment, discard the combination that determines nothing, and seven comes out without any of the seven being named.

Run the same enumeration at m simultaneous folds — 2m constraints from one pool — and it gives seven, twenty-two, fifty, ninety-five, a hundred and sixty-one.

The essay that does this says, in the machinery underneath it and in its own closing section, what the enumeration leaves out. An alignment may refer to a crease being made at the same instant, and none of those is counted. So the numbers are a floor.

This rung puts them back.

What the omission is #

The five alignments the enumeration uses are these. A fold may pass through a named point, may send a named point onto a named line, may be square to a named line — each of which costs one constraint — and may send a named point onto a named point or a named line onto a named line, each of which costs two.

Every one of those names something that is on the paper before the fold is made. That is what makes the enumeration well-defined, and for a single fold it is not a restriction at all: there is nothing else for a lone crease to refer to.

For several folds made at once it is a restriction, and a substantial one. Two creases being made in the same instant are two lines in the plane, and one of them may perfectly well be aligned to the other. A point sent onto the crease its neighbour is producing is a condition worth one constraint, exactly as a point sent onto a printed line is. A fold square to a simultaneous crease is another. One simultaneous crease landing on another is worth two.

Three more alignments, on the same cost rule as the rest, available only when there is more than one fold.

What admitting them does #

Run the enumeration over eight alignments instead of five, with the three new ones admitted only from two folds up, and the table changes at every row but the first.

Seven, eighty-six, two hundred and ninety-six, seven hundred and ninety-one, one thousand seven hundred and ninety-two.

The first entry does not move, and that is the check that the extension has not invented anything. A single fold has no neighbour, the new alignments have nothing to name, and the enumeration gives seven exactly as before. An extension that produced an eighth axiom would be an extension that had made a mistake, and the figure refuses to draw if it does.

The second entry nearly quadruples. Twenty-two becomes eighty-six, and sixty-four of those eighty-six use at least one of the new alignments — three quarters of the two-fold catalogue cannot be described in terms of the paper at all.

That is the finding, and it is worth putting plainly. The gap between one fold and two is not the gap between seven operations and twenty-two. It is the gap between seven and eighty-six, and most of the difference is operations that have no single-fold analogue whatever.

Why the new count is still a floor #

The essay would be dishonest if it presented eighty-six as the answer, and it is not the answer for a reason that is easy to state and hard to fix.

The enumeration spends 2m constraints from one pool, without tracking which fold each alignment attaches to. For alignments that name things already on the paper that is harmless: a condition “through this point” is the same condition whichever fold satisfies it, and a multiset of alignment types determines the operation.

For alignments that name another simultaneous crease it is not harmless. The condition “a point onto the crease being made at the same time” means something different depending on which crease, and with three folds there are two candidates and with four there are three. A combinatorial model that records only the type undercounts by whatever the number of distinct assignments comes to.

So eighty-six is a lower bound on a number nobody has computed, and the honest statement is that twenty-two was a floor, eighty-six is a higher floor, and the ceiling is unknown. The figure says so on its own face rather than in a footnote, because a count presented without its status is a count somebody will quote.

Getting the true number needs a different combinatorial model — one in which an operation is an assignment of alignments to folds together with a graph of which fold refers to which — and that is a genuinely harder enumeration than this one. It is the piece of work this anchor now owes.

Where the growth goes #

The two columns grow at different rates and the rates are worth extracting, because they say what kind of object each catalogue is.

The paper-only count runs 7, 22, 50, 95, 161, 252, 372, 525, 715. Difference it three times and the result is constant at four, so the count is a cubic in the number of folds — which is what a multiset enumeration over a fixed alphabet with a budget linear in m gives.

The extended count, from two folds up, runs 86, 296, 791, 1792, 3612, 6672, 11517, 18832, 29458. Difference it five times and the result is constant at sixteen, so it is a quintic. The alphabet grew from five alignments to eight and the degree rose by two, which is the arithmetic of the generating function: an alignment type available without limit contributes a factor to the growth and a type usable once does not.

So both catalogues are polynomial and the extension raises the degree by two. That is worth stating for anybody tempted to read the multifold as an unbounded resource: allowing more folds at once buys polynomially many operations, not exponentially many, and the practicality argument the ladder below makes — that the coincidence count grows linearly and hands cannot keep up — is not outrun by the catalogue.

Which sharpens what the multifold’s power actually consists of. It is not that there are enormously many more operations; it is that the kind of condition available changes, from conditions on one unknown to conditions between unknowns. A cubic count and a higher-degree cubic count are not different in nature. A condition between two unknown lines is different in nature from a condition on one, and that is where the reach comes from.

The pattern of an omission #

It is worth asking why this was left out, because the answer is structural rather than careless and it is the sort of thing that recurs.

The five alignments were written down while thinking about a single fold. For a single fold they are complete — there is provably nothing else a lone crease can be aligned to, which is the whole content of the argument that the list stops at seven. So the alphabet was correct, and correct for a reason, in the case it was derived in.

Generalising an enumeration usually means changing the budget and keeping the alphabet, because the budget is where the parameter obviously lives: a fold line has two freedoms, m folds have 2m, spend them the same way. That is exactly what the generalisation did, and the alphabet came along unexamined because it had been proved complete.

The proof of completeness was a proof about the case, and it was read as a proof about the alphabet. That is the shape of the mistake, and it is not peculiar to this subject. An argument establishing that a list is exhaustive establishes it relative to what the objects in the list may mention, and generalising the setting can add things to mention without anybody revisiting the list.

This collection has met the same shape before, one field over. A test imported without its hypothesis records a rule about layer orderings taken from a literature where the sheet is a disc with finitely many panels, applied for years to patterns whose whole interest is that they repeat. The rule was proved, the proof was relative to a hypothesis, and the hypothesis travelled less well than the rule did.

What a pair of hands has to do #

The other column in the ladder’s own table is what a folder must achieve in the same instant, and the new alignments change what that column means as well as how large it is.

The count of coincidences is 2m, because that is how many constraints are being satisfied — and the restriction to one fold was a rule somebody imposed rather than a property of paper and each is a coincidence somebody has to bring about simultaneously. Two folds, four coincidences; three folds, six. That is the number the rung below this one uses to argue that multifolds are impractical, and it is unchanged by anything here.

What changes is the kind of coincidence. An alignment to a printed line asks a folder to bring a point onto something that is sitting still. An alignment to a simultaneous crease asks them to bring a point onto a line that is itself moving, and to have both arrive together.

Those are not comparable difficulties. The first is a matter of watching two things; the second requires the two motions to be coupled, which a pair of hands has no mechanism for. So the sixty-four new two-fold operations are not merely more of the same — they are the ones that a hand cannot attempt at all, however patient.

A jig can. A device holding two adjustable fold lines has both of them as objects it controls, so a condition between them is a constraint in its linkage rather than a coincidence in somebody’s fingers. That is a real distinction between a hand and a machine, and it says the multifold’s extra power is available to one and not the other — which is the opposite of the usual reading, where a machine is a way of doing accurately what a hand does approximately.

What this does to the reach #

The natural next question is whether the extra operations reach further, and the honest answer is that this rung does not establish it.

An operation is a specification of a fold, and what a set of operations reaches is a question about the algebraic degrees the specifications produce. Two creases at once establishes that two simultaneous folds reach the quintic and the sextic, which is the result the multifold literature turns on, and that result is about the operations somebody wrote down rather than about the whole catalogue.

More operations could mean more reach or could mean many redundant ways to specify the same folds. Deciding which needs the degrees, not the count — and the counting argument here is deliberately silent about them.

What can be said is a direction. The alignments that name simultaneous creases produce conditions between the unknowns rather than between an unknown and a constant, and a system of that shape is where higher degrees come from: two unknown lines constrained to each other give a resultant of higher degree than either constrained to a fixed object. So the new operations are the right kind of thing to increase the reach, and whether they do is a computation this collection has not made.

Which theorem was checked and how #

The single-fold count must come out at seven in both enumerations. That is the check that the extension is an extension rather than a change: the new alignments are admitted only from two folds up, and a version that admitted them at one would produce an eighth axiom and the figure would refuse.

The two-fold count must strictly exceed twenty-two, or the omission was empty and the essay has no subject.

More than half the two-fold operations must use a simultaneous crease. A version in which the new alignments contributed a handful would make the finding a footnote, and the figure states the threshold rather than leaving the reader to judge.

And both counts must grow strictly with the number of folds, which is what makes them catalogues rather than artefacts of the recursion.

Where the model stops #

The cost rule is the ladder’s own and is not re-derived here. Passing through a point costs one; landing a point on a point costs two; being square to two lines is excluded because it is one condition twice. All of that comes from the rung below and this rung changes none of it.

The three new alignments are the three that seem right and are not a proof of completeness. There might be others — a fold making a stated angle with a simultaneous crease, for instance, if angles are allowed to be named — and the enumeration is only as complete as its alignment list.

Nothing here is about which operations are useful. A catalogue includes operations that specify folds nobody wants, and the single-fold case is the precedent: seven axioms, of which two are used constantly and one hardly at all.

And the count is a count of specifications rather than of folds. Two operations may specify the same crease on a given configuration, exactly as several of the seven axioms coincide on a square, so the catalogue’s size is an upper bound on the distinct folds available at any particular moment.

What the picture cannot show #

A table of counts cannot show what any of the operations is. The single-fold view can draw its eight combinations because there are eight; eighty-six is past the number a page can carry, and the ones that matter most — the ones naming a simultaneous crease — are also the hardest to draw, because drawing one means drawing two creases that are constrained to each other and neither of which exists yet.

That difficulty is not incidental. The operations the enumeration was missing are exactly the ones a diagram cannot show, since a diagram shows a fold against a sheet and these are folds against each other. It is a plausible explanation for the omission surviving as long as it has: an operation nobody can draw is an operation nobody lists.

Nor can the figure show the true count, which is what a reader most wants. Both columns are floors and the picture says so on its face; the gap between the second floor and the ceiling is not drawn because it is not known.

The idealisation, named #

A fold is a line, the plane is unbounded, and every alignment named is achievable. The third is the one doing work here and it is worth separating from the physical question about hands.

The enumeration counts specifications. Whether a given specification has a solution on a given configuration is a separate matter — an axiom may name no fold, and the fold that sends a point onto a line through a point has two solutions, one or none depending on a distance. So a catalogue of eighty-six operations is a catalogue of eighty-six things one could ask for, and how many of them a particular sheet answers is smaller and configuration-dependent.

That is the same relation the seven axioms have to a real sheet, so nothing new is being assumed. What is worth noticing is that the new operations are more likely to have no solution than the old ones, because a condition between two unknown lines is a condition on a system with fewer degrees of freedom left over.

Where the ladder goes next #

This anchor owes the true count and the true count needs a different model, which is the rung after this one.

An operation at m folds is an assignment of alignments to folds together with a reference graph saying which fold names which. Enumerating those is a labelled-structure count rather than a multiset count, it is a standard piece of combinatorics once the model is written down, and it would give the number of which eighty-six is a floor. Whether it is a hundred or a thousand is not guessable from here.

The other direction is the machine. Nothing in the ladder’s argument about hands applies to a device, and the question of what a mechanism holding two adjustable fold lines can construct has a definite answer that nobody appears to have asked. It is the counterpart of what a machine that folds one crease at a time can reach, run at the construction end rather than at the folding end — and this rung says the machine’s advantage is larger than it looks, because three quarters of the two-fold catalogue is unavailable to hands in principle rather than in practice.

The habit worth carrying is about enumerations generally. An enumeration is only as complete as the list of things its elements are allowed to refer to, and that list is usually written down while thinking about the simplest case. Here the simplest case had nothing else to refer to, so the omission was invisible exactly where it was made.