What each axiom is worth

The seven axioms are not seven useful folds somebody collected; they are every fold there is, and the proof is a counting argument that takes about a minute. A fold line is determined by two numbers, each alignment of a point or a line onto a point or a line costs one, and the enumeration of ways to spend two closes the list.

That argument establishes completeness. It is silent about everything else, and in particular about whether the seven are seven of anything.

Two questions with two answers #

Is the list complete? Every fold is one of the seven. Answered by counting degrees of freedom, and the answer is yes.

Is the list independent? Does each member do something no combination of the others does? That is a different question, it has a different kind of proof, and the counting argument contributes nothing towards it.

The two get run together constantly, and the reason is that complete sounds like it ought to imply minimal. It does not, in this or any other setting: a list can be complete and full of duplicates, and finding out requires a different measurement.

The measurement #

Take the four elementary axioms — the fold through two points, the fold placing one point on another, the fold placing one line on another, and the fold through a point perpendicular to a line. Those are the four that need only what is already on the paper, and they are what a folder has at the start.

Put them on a bare square, which has four corner points and four edges. Ask each axiom for every fold it specifies, and count how many distinct lines result.

Thirty-eight folds are specified. Twelve distinct lines are drawn. Two thirds of what the axioms name is something another axiom has already named.

And the unique contributions are lopsided beyond anything a reader would guess. The bisector of two lines supplies four lines nothing else supplies. The fold through two points supplies none. The fold placing a point on a point supplies none. The fold perpendicular to a line through a point supplies none.

It is worth being precise about what “unique” means here, because the claim is easy to overstate. An axiom contributes uniquely when it names a fold line that no other axiom names on that configuration. Contributing nothing does not mean the axiom is useless; it means that everything it offers is already on the table. On a bare square, the fold through two points gives the four edges and the two diagonals — six perfectly good lines, every one of which the bisector also gives.

Why the square is so degenerate #

The reason is worth having, because it explains why the result is not an accident of one shape while also explaining why it is not the general case.

A square’s four corners and four edges are related to one another in every way a configuration can be. The line through two adjacent corners is an edge, so the fold through two points gives back a line already present. The perpendicular bisector of two adjacent corners is a midline, which is also the bisector of the two opposite edges — so axiom 2 and axiom 3 give the same line by different routes. The perpendicular to an edge through a corner is the adjacent edge.

So a square is a configuration in which almost every alignment coincides with almost every other, and that is precisely why folders start with squares: the symmetry is what makes the first few folds land where they are wanted. The price is that at the start, most of the toolkit is doing nothing.

There is a nice consequence for the very first thing anybody is shown. A folder demonstrating axiom 1 on a square folds corner to corner and produces a diagonal; a folder demonstrating axiom 2 folds corner onto corner and produces the same diagonal; a folder demonstrating axiom 3 folds edge onto edge and produces a midline, which axiom 2 also produces. Three demonstrations, two lines, and the impression that the operations are much of a muchness. The impression is correct about the square and wrong about the axioms.

The inversion #

Do a round of folding — take the twelve lines, keep every crossing that lands on the paper, and now there are nine points and twelve lines. Ask the same question.

Three hundred folds specified; ninety-two distinct. The bisector supplies fifty-six unique lines, the fold through two points eight, the fold point-onto-point eight, and the perpendicular none again.

Do another round. Now there are 565 points and 92 lines, and the answer is a different answer.

378,614 folds specified; 274,300 distinct. The fold through two points supplies 121,054 unique lines. The fold point-onto-point supplies 142,649. The bisector of two lines — which carried the first two rounds by itself — supplies 4,994, less than two per cent of the total. And the perpendicular, which had supplied nothing at all twice running, supplies 1,661.

The ranking has completely inverted. The axiom that was everything is now almost nothing, and the two that were nothing are now almost everything.

What is actually going on #

The mechanism is simple once it is stated and it generalises past folding.

An axiom that takes lines as input is limited by how many lines there are, and lines grow slowly: 4, then 12, then 92. An axiom that takes points is limited by how many points there are, and points grow explosively, because every pair of lines that crosses on the paper makes one: 4, then 9, then 565.

So the bisector of two lines has of order the square of the line count to work with, and the fold through two points has of order the square of the point count. At the start the two counts are equal and the bisector wins on the symmetry; three rounds in, one has grown by a factor of twenty-three and the other by a factor of a hundred and forty.

An axiom’s worth is therefore not a property of the axiom. It is a property of the axiom and the configuration, and since the configuration is what a folder has built so far, it changes as the folding proceeds.

There is a second reading of the inversion that a designer of tools would find useful. An operation’s inputs are its currency, and a toolkit is balanced when its operations spend currencies that grow at similar rates. This one is not: three of the four elementary axioms spend points and one spends lines, and points and lines do not grow alike. A toolkit like that is unbalanced by construction, and its unbalance is invisible at the start and unmistakable three rounds in.

Where the crossover is, exactly #

The mechanism is stated as points growing faster than lines, and the crossover can be located rather than described, because each axiom’s capacity is a formula in the two counts.

The fold through two points spends a pair of points, so its capacity is $\binom{P}{2}$. So does the fold placing one point on another. The bisector spends a pair of lines and names two folds for each pair, so its capacity is $2\binom{L}{2}$. The perpendicular spends one of each, so its capacity is $PL$.

Put the three rounds’ counts in. At the start, four points and four lines: six, six, twelve and sixteen. The bisector has twice the capacity of either point-pair axiom, which is why it carries the first round on its own. After one round, nine points and twelve lines: thirty-six, thirty-six, a hundred and thirty-two and a hundred and eight — the bisector still ahead. After two, five hundred and sixty-five points and ninety-two lines: a hundred and fifty-nine thousand against eight thousand.

Set the two capacities equal and the crossover is at $P^2 \approx 4L^2$, which is $P = 2L$. Four against eight at the start, nine against twenty-four after one round, and five hundred and sixty-five against a hundred and eighty-four after two.

So the inversion happens somewhere between the second round and the third, and it happens when the point count passes twice the line count. That is a condition a folder could check on the paper in front of them.

Which makes the ranking predictable rather than measured #

Having the crossover as an inequality rather than as an observation turns the finding into something usable on a configuration nobody has enumerated.

A folder who knows how many marks and how many creases are on the sheet knows which operation is currently the productive one, without counting a single fold: below twice the creases, the bisector; above it, the point-pair axioms. And since every crossing of two creases adds a point while adding no crease, the count moves in one direction only — once a configuration has crossed the line it never crosses back.

That also explains the perpendicular’s peculiar career. Its capacity is $PL$, the geometric mean of the other two, so it is never the largest and never the smallest — and it is squeezed out at both ends by whichever of the other two is ahead. An operation whose currency is the product of two others’ is an operation that is always second, which is a poor place to be in a marginal-contribution census and says nothing at all about whether it is worth having.

The redundancy, and where it goes #

The other quantity worth watching is how much of what the axioms specify is duplicate.

At the first round it is 68 per cent — thirty-eight folds naming twelve lines. At the second it is 69 per cent. At the third it has fallen to 28 per cent: three hundred and seventy-eight thousand folds naming two hundred and seventy-four thousand lines.

That is the same story from the other side. Duplication happens when two alignments coincide, coincidence is what symmetry produces, and a configuration built out of a square starts extremely symmetric and gets less so with every crossing added.

Which means the redundancy a folder actually experiences is highest at the beginning — the moment when the axioms are being learned and demonstrated, and the moment at which they look most interchangeable.

The one that never wakes up early #

The perpendicular deserves its own paragraph, because it is the odd one out twice over.

It contributes nothing at the first round and nothing at the second, which no other axiom manages. It is also the only one of the four that takes one point and one line rather than a pair of the same thing, so its currency is the product of the two counts rather than the square of either — which puts its growth between the other two and explains why it wakes up last and modestly.

There is a small irony in that. A perpendicular through a point is the most classical of the four — it is the first construction in any compass-and-straightedge course — and folding, which beats the compass by exactly one degree, has the least use for it at the start. What folding is good at from the first fold is bisecting, which is the operation a compass finds awkward.

The habit this is an instance of #

The general form is worth stating because it applies well beyond folding.

A toolkit’s operations are usually compared on what they can do in principle: this one reaches cube roots, that one only quadratics, and the comparison is made once and quoted forever. That is the right comparison for a question about the outer limit of a toolkit and it is the wrong one for a question about how the toolkit behaves in use.

In use, an operation’s value depends on the state it is applied to, and the state changes as the work proceeds. So a toolkit has no fixed ranking, and any ranking presented as fixed is a ranking taken at some particular state — usually the empty one, because that is the state a demonstration starts from.

Where the model stops #

Four axioms, not seven. Only the elementary four are measured. Axioms 5, 6 and 7 fold a point onto a line, which is a different kind of operation with a different cost, and the closure that includes the fifth is affordable once and not twice. Their marginal contributions are not computed and the answer would be different in kind.

Unique means unique at that round. An axiom contributing nothing at a round has not been shown to be derivable in general. It has been shown that on that configuration every line it names is named by something else — which is a fact about the paper in front of the folder and not a theorem about the axiom.

One starting shape. A bare square. A rectangle would be less degenerate, a triangle differently degenerate, and a sheet with a mark already on it different again. The measurement is of the case every folder starts from.

Nothing about cost. How expensive it is to enumerate the third round, or how a folder would search it, is a question about algorithms and is not asked here. What is counted is what exists.

What it is good for #

Two things, and the second is the one worth carrying away.

For teaching. The axioms are usually introduced on a square, one after another, with an example of each — and three quarters of those examples are producing lines that the previous axiom would also have produced. That is not wrong but it does make the operations look more alike than they are, and the difference between them shows up only after there is something on the paper.

For a folder counting folds. Two folds from a bare square reach a half, a third, a quarter, a fifth, a sixth, an eighth and a twelfth and no seventh at all, which is a statement about what the reachable set contains. The measurement here is about which operation put each of them there, and at the first two rounds the answer is very nearly always the same one.

For anyone assessing a toolkit. The general form is that the marginal value of an operation depends on the state it is applied to, so a ranking of operations measured at one state is not a ranking. That is obvious when said out loud and is routinely ignored: the impulse is to ask which axiom is the powerful one, and the honest answer is at which point in the folding.

A caution about the third round #

The numbers at the third round are large and it is worth saying what they are and are not.

They are counts of distinct fold lines specifiable from a configuration of 565 points and 92 lines. They are not folds anybody would make: a folder making 274,300 creases has ruined the paper, and most of those lines are nowhere near any reference a person wants. The enumeration refuses to go past this depth precisely because the candidate count grows faster than anything can carry.

So the inversion is a statement about availability rather than about usefulness. What it establishes is that the relative productivity of the operations is not fixed, and that a measurement taken at the round everybody demonstrates on gives exactly the wrong ranking for every round after it.

Where the ladder goes next #

The obvious continuation is the one this measurement is one step short of: whether an axiom that contributes nothing at a round is contributing nothing ever — that is, whether any of the seven is genuinely derivable from the others. That is a question about the operations rather than about a configuration, it has been raised in the literature, and settling it needs a different kind of argument from anything counted here.

The other direction is the folder’s version. If the useful axiom changes as the folding proceeds, a folder who reaches for the same operation throughout is leaving something on the table — and a sequence chosen to exploit the shift might reach a given reference point in fewer folds than the systematic routes this site has measured. Whether it does is a search, and it has not been run.