Two creases at once

The seven axioms are complete and the completeness is a real theorem: there are exactly seven ways of specifying a fold by bringing points and lines into coincidence, and the proof is a degrees-of-freedom count that takes about a minute.

What the theorem is complete about is easy to miss. It is complete about one fold at a time. That restriction is in the statement, it is doing enormous work, and it is not a fact about paper.

What one fold buys, and why #

A fold is specified by a line in the plane, and a line in the plane has two parameters. Each alignment condition — this point onto that point, this point onto that line, this line onto that line — removes one of them.

Two parameters, so two conditions determine a fold, and the seven axioms are the seven ways of choosing two conditions from the available kinds. That is the completeness proof in a sentence.

The interesting one is the sixth, which asks for a single fold placing each of two points onto each of two lines. Two conditions, both of the point-to-line kind, and the algebra that comes out is cubic.

So a single fold extracts cube roots as well as square roots, and everything reachable by folding is reachable by a tower of quadratic and cubic extensions. That is the whole of folding’s advantage over the compass and it is exactly one degree wide.

What a second simultaneous fold buys #

Now lift the restriction.

An operation that makes two creases at the same time has four parameters — two lines — and can therefore satisfy four alignment conditions at once. Nothing about paper forbids it: the folder slides the sheet until two conditions hold together, and creases both.

More parameters allow more conditions, and more conditions allow higher-degree algebra. The result, which is Alperin and Lang’s and is quoted here rather than derived, is that two simultaneous folds solve the quintic and the sextic, and that with enough simultaneous folds any algebraic number is reachable.

The multifold axioms are therefore not an extension of the seven in the way an eighth axiom would be. They are the same operation with the one-at-a-time restriction removed, and the seven turn out to have been describing the smallest case of a hierarchy.

The polygons, which is where it can be checked #

The reachability of regular polygons is the cleanest place to see the difference, because it turns on a single arithmetic quantity.

The regular n-gon is constructible by a tool exactly when the degree of the minimal polynomial of its central angle’s cosine can be reduced by the extractions that tool performs — which comes down to the prime factorisation of φ(n), Euler’s totient.

• A compass extracts square roots, so it needs φ(n) to be a power of two. That is the Gauss–Wantzel condition.
• A single fold extracts cube roots as well, so it needs φ(n) to have no prime factor above three. That gives the Pierpont condition and the heptagon, where the two answers first differ.
• Two simultaneous folds reach the quintic, so fives are permitted too.

Everything in the table is computed from that: φ(n) by trial division, its factors by trial division, and three predicates asking which primes appear. The degrees each tool reaches are quoted from the literature and nothing here re-derives them.

The table, counted #

The census in the figure is worth reading as totals, because the totals say how much each rung of the hierarchy is actually worth.

Of the twenty-two regular polygons from the triangle to the 24-gon, the compass reaches twelve — the ones whose totient is a power of two: 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24.

One fold adds seven more, the ones whose totient admits a three: 7, 9, 13, 14, 18, 19, 21. Nineteen of twenty-two.

Two simultaneous folds add exactly two: 11 and 22, whose totients are $2 \cdot 5$. Twenty-one of twenty-two.

So the first rung is worth seven polygons in this range and the second is worth two, which is the shape of the whole hierarchy: each extra simultaneous fold admits a larger prime, and larger primes divide fewer totients.

And the 23-gon is out of reach of both #

One polygon in the table is reached by none of the three, and it is the useful one to name because it shows the hierarchy does not stop where this essay does.

$\varphi(23) = 22 = 2 \cdot 11$. The obstruction is an eleven, and no arrangement of quadratics, cubics or quintics produces one — so the 23-gon is beyond the compass, beyond a single fold, and beyond two simultaneous folds alike.

That is the honest statement of what lifting the restriction buys. It does not make everything foldable; it moves a threshold, from primes above three to primes above five, and the polygons stranded on the far side of the new threshold are simply rarer. Removing the one-at-a-time restriction is not a door but a step, and the 23-gon is standing on the next one.

The hendecagon changes sides #

This site already has an essay whose title is the eleven-sided one nobody can fold. Its argument is exact and its conclusion needs one word added.

φ(11) is 10, which factors as 2 · 5. The five is the obstruction: a fold solves cubics, no arrangement of cubics produces a five, and the hendecagon is out of reach.

Out of reach of one fold at a time. Two simultaneous folds reach the quintic, the five stops being an obstruction, and the eleven-sided polygon becomes constructible.

That is the finding this rung exists for, and it is a good illustration of a habit worth having: an impossibility result is always an impossibility for a stated set of operations, and the honest version of the sentence names them. “Nobody can fold the hendecagon” is a claim about a folding convention; “no tower of quadratic and cubic extensions contains cos 2π/11” is a claim about arithmetic, and only the second is permanent.

Where two folds stop #

The census runs far enough to find the next wall.

The first n whose totient has a prime factor above five is 23: φ(23) is 22, which is 2 · 11. The eleven in that factorisation is a genuine obstruction to two simultaneous folds, and the twenty-three-sided polygon needs more of them.

That is the shape of the hierarchy: each additional simultaneous fold admits another prime, and the impossible polygons retreat rather than disappear. There is no n that is unreachable by every multifold operation — with enough simultaneous folds any algebraic number is constructible — but for any fixed number of folds there is always a smallest polygon out of reach.

The retreat is slow, and the table shows how slow. Between eleven and twenty-three there are twelve polygons and every one of them is reachable by two folds; the second wall is more than twice as far out as the first. That is a consequence of totients being smooth much more often than not: a random number has small prime factors, and it takes an n whose totient contains a large prime to defeat a tool, which is a rare event that gets rarer.

There is also a small surprise in the middle of the table and it is worth pointing at. The polygons that defeat one fold and not two are eleven, twenty-two and twenty-five — and twenty-two is eleven doubled, which contributes nothing new, while twenty-five is five squared and gives a totient of twenty containing two fives. So between eleven and twenty-three only two genuinely new obstructions appear, and both of them are the same prime showing up in a different way.

Every wall in this subject is one prime. The compass stops at three because three is not a power of two. One fold stops at eleven because ten contains a five. Two folds stop at twenty-three because twenty-two contains an eleven. It is the same sentence three times with a different number in it.

What the single fold’s extra degree buys is worth having in view before another one is added.

The count, stated properly #

The parameter count above is worth doing carefully once, because it is the whole reason the hierarchy exists and it is short enough to check.

A line in the plane has two degrees of freedom — a direction and an offset, or any equivalent pair. Bringing a specified point onto a specified line, or onto another specified point, is one equation in those two numbers. Two equations therefore pin the line down, generically to a finite set of solutions rather than to one, which is why several of the axioms produce more than one fold and axiom six can produce three.

So a single-fold operation is: two unknowns, two equations, and a solution set whose size is the degree of the resulting polynomial. The degree three of axiom six is not an accident of that particular alignment; it is what two point-to-line conditions on one line produce.

Now do it for two lines. Four unknowns, so four conditions can be imposed, and the conditions available include ones that relate the two lines to each other — a point landing on the second line after being reflected in the first, for instance, which is a condition a single fold cannot express at all. That is the real gain: not simply more equations, but equations of a kind that require a second crease to state.

The degree that comes out of four such conditions is where the quoted results take over. The count says the system is richer; it does not by itself say the answer is a quintic, and this essay does not pretend the arithmetic above is a proof of anything beyond the parameter budget.

Three simultaneous folds have six parameters, and so on. Each additional crease adds two unknowns and admits conditions coupling it to all the earlier ones, and the reachable degree climbs — which is why “with enough folds, anything algebraic” is the eventual statement rather than a surprise.

Which theorem was checked, and how #

The division of labour in this essay is deliberate and the check is on this repository’s half.

Computed here: φ(n) for every n in the census, by trial division; the prime factorisation of each totient, also by trial division; and the three predicates. The generator refuses to draw if no polygon in its range defeats a single fold, if none defeats two, or if the first polygon that defeats one fold is not reached by two — that last refusal being the case the essay is about, so a census in which it did not occur would be a census that had missed the point.

The site’s fold check asserts the specific numbers: that φ(11) is 10 and factors as 2 · 5, that the first single-fold failure is at eleven, that two folds reach it, that the first two-fold failure is at twenty-three, and that the heptagon remains the first place a fold beats a compass.

Quoted, not computed: which degrees each tool extracts roots of. That a single fold solves cubics is Beloch’s; that two simultaneous folds solve the quintic and the sextic, and that multifold reaches arbitrary degree, is Alperin and Lang’s. This repository does not prove any of them and does not implement a multifold construction.

The distinction matters because the table would look identical either way. A reader is entitled to know which columns are arithmetic anybody can repeat and which are theorems being cited.

There is a way of reaching an unreachable number that does not need a second simultaneous crease.

What it would take to actually do #

An honest account has to say what a two-fold operation is like as a physical act, because it is not the same kind of thing as the seven.

A single-fold axiom is a manipulation with an obvious procedure: bring the point to the line, hold it, crease. Two of the seven require sliding the paper until an alignment is achieved, which is already fiddlier than it sounds.

A two-fold operation requires the sheet to be manipulated until four conditions hold simultaneously, with two creases then made without disturbing anything. It is a real operation and it has been performed, and it is nobody’s idea of a comfortable one. The gap between “constructible” and “practicable” is much wider here than it is for the seven axioms, and the multifold hierarchy is chiefly a statement about what folding is rather than a technique anybody uses.

That is not a criticism of the result. Straightedge and compass constructions are also idealisations of an operation nobody performs at scale, and their interest was never practical either.

Why the restriction was invisible #

It is worth asking why nobody noticed for thirty years that the one-fold rule was a rule.

The most likely reason is that the axioms were written down as a description of what folders do, and folders make one crease at a time because that is what hands do. A person holds the sheet, brings two things together, and creases. Making two creases at the same instant is not a natural motion, so it never appeared in the descriptive list, and the descriptive list then became the definition.

The completeness theorem sealed it. Once there is a proof that the list of seven is complete, the natural reading is that folding has been characterised, and the qualifier “for one fold at a time” reads as a technical precondition rather than as the whole of the scope. A theorem that is complete about a restriction makes the restriction harder to see, not easier.

That is a general hazard rather than a criticism of anybody. A characterisation theorem is the most convincing possible statement that a subject has been mapped, and it is exactly as complete as its hypotheses — which is why the useful question to ask of one is always complete about what.

The axioms were written down as a description of what folders were already doing.

Who established what, and when #

The seven axioms are Huzita’s and Hatori’s, from the 1990s, with the completeness proof following — and with the sixth axiom’s cubic traceable to Margherita Beloch in the 1930s, decades before the axiom list existed. Fifty years in the wrong language is this site’s account of that gap.

The multifold axioms are Roger Alperin and Robert Lang’s, published in 2009, with related work by Jorge Nishimura on solving quintics by two-fold operations. Their contribution is the systematic one: not that a clever multifold construction exists for some particular problem, but that the whole hierarchy can be described and its reach characterised.

The pattern in the history is the one this site keeps finding. The restriction to one fold was never argued for; it was simply how everybody was folding, and it took thirty years for somebody to notice that it was an assumption rather than a fact.

The clearest illustration that repetition buys quantity and not reach is a division.

What the hierarchy is a hierarchy of #

It is worth being precise about what varies along it, because “more folds” is ambiguous in a way that matters.

The hierarchy is not about how many folds a construction uses. A construction with fifty single folds is still a single-fold construction, and everything it reaches is reachable by a tower of quadratic and cubic extensions however long it goes on. Repetition buys nothing at all in terms of reachable degree.

What varies is how many creases are made simultaneously, which means how many alignment conditions are satisfied at the same instant by one manipulation of the sheet. That is a different resource entirely, and the reason it buys something is the parameter count: two lines have twice the freedom of one, and freedom is what conditions are imposed on.

The distinction has a clean analogy in the older subject. Straightedge and compass allows any number of steps and reaches only square roots; adding a marked ruler — one operation that satisfies two conditions at once — reaches cubics. In both subjects the extra power comes from a richer single operation and never from more operations.

That is why the completeness theorem for the seven axioms remains true and remains interesting. It is complete for its operation set, exactly as Gauss and Wantzel’s result is complete for theirs, and the multifold results do not contradict it any more than a marked ruler contradicts Euclid.

Where the ladder goes next #

The obvious next question is what an optimisation looks like when the tool is a fold. The largest equilateral triangle a square holds turns out to be reachable by a quadratic and therefore by a compass, which is a mild disappointment and a useful calibration; which of the largest inscribed polygons are foldable at all is where the two subjects meet properly.