The numbers a fold reaches

Straightedge and compass solve quadratics; a single fold solves cubics, and that one-step difference settles two problems Greek geometry could not and leaves a third exactly as impossible as it was. It is a fact about one fold, and it is where most accounts of the subject stop.

The fact that makes folding a mathematics rather than a collection of clever manoeuvres is a different one, and it is about all the folds at once.

The property that makes a theory possible #

Mark two lengths on a sheet. Call them a and b. Then a folder can produce:

a + b, by transferring one along the other with a single crease. a − b, by the same crease the other way. a × b and a ÷ b, by a pair of parallels and a unit, which is the similar-triangles construction every draughtsman knows. √a, by bisecting a right angle onto a mean. And ∛a, by the fold that is tangent to two parabolas at once.

Every one of those results is itself a length the folder can mark. So the set of reachable lengths is closed under all six operations — and that is a field, in the ordinary arithmetic sense, closed under square roots and cube roots as well.

Why that matters is worth spelling out, because it is easy to nod at and pass over. It means a folder can use the output of one construction as the input to another without checking whether the result is still constructible. Halve a length that was itself obtained by trisecting an angle that was itself obtained by a cube root, and the answer is reachable — not because anybody worked out how to reach it, but because the set has no edges to fall off.

A collection of tricks has no such guarantee. Every combination would need its own argument, and there would be no reason to expect the answer to be yes.

There is a small piece of bookkeeping worth doing, because it makes the closure concrete rather than abstract. Take the two lengths ∛2 and √2, both reachable. Their product is 2 raised to the five-sixths, and it satisfies x⁶ = 32. That number is nowhere in anybody’s list of famous constructions; nobody has ever needed it; and a folder can mark it, by multiplying two lengths with a pair of parallels. The set is much larger than the constructions people have names for, and it is the closure that says so.

The price, written as a degree #

Closure is generous and it is not unlimited, and what limits it is arithmetic rather than geometry.

Every one of the six operations either leaves a number where it is or extends the arithmetic it lives in by a degree of two — for a square root — or of three, for a cube root. Composing them multiplies the degrees. So a number a folder reaches satisfies some rational equation whose degree is a product of twos and threes, and nothing else.

√2 has degree two. ∛2 has degree three. The fourth root of two has degree four, which is two twice. The product ∛2 · √2 has degree six, which is a two and a three — and it is in the set for exactly the reason the closure says it should be, since it is a product of two things that are.

2 cos(2π/7) has degree three, which is why the regular heptagon is a fold away and a compass cannot reach it.

And the fifth root of two has degree five. Five is neither a two nor a three nor a product of them, and no tower of twos and threes has five in it. So the fifth root of two is not reachable, at any number of folds, by any construction anybody will ever invent.

The condition is necessary rather than obviously sufficient, and the distinction matters. A degree that is a product of twos and threes does not by itself guarantee a construction: it says only that nothing in the arithmetic rules one out. For the classical cases in the table — square roots, cube roots, the cosines of the constructible polygons — a construction is known and the condition is met, so the two agree. The general statement is a theorem of field theory and is quoted rather than proved.

How the degrees are checked #

The degree of a number is the sort of thing that is easy to assert and worth not asserting.

Each number here is given with a rational polynomial it satisfies — which is arithmetic anybody can check by substitution, and which is checked by substitution. That establishes an upper bound on the degree and nothing more: a number satisfying a degree-six polynomial might satisfy a degree-two one as well, and then it would be in the set after all.

So the lower bound is established by exhaustion. Every integer polynomial of every lower degree with coefficients up to six is evaluated at the number, and none of them vanishes. That is a search rather than a theorem — a number could satisfy a polynomial with a coefficient of seven — and it is the honest kind of evidence: a bounded exhaustion, reported as one.

The theorem behind the degree condition — that an extension built by adjoining square and cube roots has degree a product of twos and threes, and that a number of degree five cannot live in one — is field theory, it is a hundred and fifty years old, and it belongs to algebra rather than to this subject. It is used here and not re-derived.

Where the closure is doing the work #

It is worth pointing at three places in this collection where the field property is what makes the argument go, because in each of them it is invisible.

The regular polygons. Which n-gons folding constructs is decided by the degree of a cosine, and the decision procedure works because a composition of constructions is a construction. Without closure, “the 7-gon is constructible” would be a statement about one particular sequence of folds rather than about the number.

The exact divisions. Crossing the anti-diagonal with a line through a mark already in hand lands on the next unit fraction exactly, and the ladder can be continued indefinitely because each rung’s output is a legitimate input to the next. That is closure under division, used without comment.

The impossibilities. The eleven-sided polygon nobody can fold is impossible because a degree of five is not a product of twos and threes, and that argument is only available because every construction lands in the same field. If different constructions reached different sets, there would be nothing to take a degree in.

There is a fourth place, and it is the one where the closure is most nearly visible. Two folds from a bare square reach a half, a third, a quarter, a fifth, a sixth, an eighth and a twelfth — and the reason those are all fractions is that two rounds of the elementary axioms perform only additions, subtractions, multiplications and divisions on the rationals, and a field closed under those four contains the rationals and nothing more. The moment a bisector of a right angle enters, square roots arrive, and the set stops being the rationals in a single fold.

The other closure, and why it is a different subject #

There is a neighbouring statement that sounds identical and is not: the numbers a straightedge and compass reach are closed under the same operations except cube roots, and their degrees are powers of two.

Which toolkit reaches which numbers is a question about operation sets — what a fixed collection of operations can and cannot build — and it is a question with a long history in which folding is a late and interesting entrant. What this site owns is the folding half: what a fold reaches. The compass’s half, and the general theory of constructibility under a stated operation set, is somebody else’s and is linked rather than developed.

The separating test is easy to apply. Every argument in this essay names a fold. Nothing in it needs a compass to exist.

What a folder gets from it #

The practical content is a permission and a prohibition, and both are useful.

The permission. Any sequence of folds may be composed with any other. A folder who has established a construction for one length may use it as a step in a construction for another, and does not have to ask whether the composite is still reachable. That is what makes technique cumulative in this subject — a construction published in 1936 can be a step in one published in 2020 — and it is the reason the historical constructions this site re-runs go on being usable rather than being curiosities.

The prohibition. A number whose degree has a prime factor other than two or three is out of reach, and no ingenuity closes the gap. That is worth knowing before spending an evening on a construction: the question “is this foldable” has an arithmetic answer that can be worked out in a minute, and working it out first is cheaper than not.

What a compass has that a fold does not #

The comparison is nearly always made in one direction — folding reaches more — and it is worth turning round once, because the asymmetry is not total.

A compass draws a circle, which is an infinite set of points obtained in one operation; a fold draws a line. So a compass user can find an intersection of two circles, which is two points, where a folder must use an axiom that reaches the same points by a different route. The reachable sets are what they are and the difference is in the number of operations, not in what is available.

That is why the field is the right object to compare. Two toolkits with the same reachable field construct the same things, whatever the shape of the individual operations, and the entire content of “folding beats the compass” is that one field is a subfield of the other.

The degree test is not the whole condition #

The essay is careful to say the degree condition is necessary rather than obviously sufficient, and the gap between the two is worth naming, because it is not a technicality — there are numbers the degree test passes and no fold reaches.

The exact statement replaces the degree by the size of the number’s Galois group — the symmetry group of its minimal polynomial’s roots. A number is reachable by folding exactly when that group’s order is divisible by no prime except two and three, which is the algebraic form of built by a tower of square and cube roots.

The degree divides the group’s order, so degree a product of twos and threes is necessary. It is not sufficient, because the group can be much larger than the degree and can pick up other primes on the way.

A degree-six number is the smallest place it happens. A polynomial of degree six chosen without any special structure has the full symmetry group on six roots, whose order is seven hundred and twenty — and seven hundred and twenty is sixteen times nine times five. Five is neither a two nor a three, so such a number is unreachable, and its degree is six, which is two times three.

Which is why the table’s agreement is not a coincidence #

That changes how the table of degrees should be read. Every entry in it — the square roots, the cube roots, the cosines of the constructible polygons — has a small Galois group as well as a small degree, because each is built out of radicals by construction and a radical tower is exactly what keeps the group’s order to twos and threes.

So the table’s rows agree with the degree test for a reason, and the reason is that they were all obtained by folding in the first place. Reading the agreement as evidence that degree decides the question would be reading a sample of constructible numbers as though it were a sample of numbers.

The practical rule survives with a caveat attached. A degree with a five or a seven in it settles the question outright and costs a minute; a degree that is a product of twos and threes settles nothing, and the honest verdict there is not ruled out. For anything anybody has actually wanted to construct the two coincide, because the things people want are the roots of equations that arose from a construction — and the numbers where they part company are the ones nobody has ever had a reason to name.

Where the model stops #

The degrees are checked by a bounded search. The lower bounds above exhaust integer polynomials with coefficients up to six. That is evidence and not proof; the proofs exist, they are standard, and they are not this subject’s.

Closure is demonstrated by construction, not derived. Each of the six operations is exhibited as a fold. That the set is closed follows, and the formal statement — that the reachable numbers form a subfield of the reals closed under square and cube roots — is quoted from algebra.

One fold at a time. Everything above is about sequences of single folds. Folding two creases at once reaches further, and a multifold’s numbers are a larger field with a different degree condition, which is not computed here.

Nothing about how many folds. The set of reachable numbers says what is reachable and not how expensive it is to reach. How many folds a given number costs is a separate and much harder question, and cost in every form belongs to a different subject.

Why it is not usually put this way #

Most accounts of folding’s power lead with the cubic, because the cubic is the striking fact and doubling the cube is the striking demonstration. Leading with the closure would be leading with something that sounds like bookkeeping.

But it is the closure that makes the cubic useful. A fold that solves one cubic is a curiosity; a toolkit in which every result can be fed back in is a construction system, and the difference between the two is the difference between a trick and a subject. The same thing happened to straightedge and compass two thousand years earlier and is equally invisible there: nobody states that Euclidean constructions compose, and every Euclidean proposition uses it in its first line.

The shape of the set #

It is worth trying to say what the reachable numbers look like, because “closed under six operations” is a description of a rule rather than of an object.

They are countable, which is immediate: every one of them is the result of a finite sequence of operations on the rationals, and there are countably many such sequences. So almost every real number is unreachable, in the same measure-theoretic sense that almost every crease pattern fails to fold — the reachable ones are a set of measure zero, and the entire subject lives inside it.

They also contain all the rationals, all the square-root towers a compass reaches, and a great deal besides — π and e are not among them, for reasons that have nothing to do with folding and everything to do with those numbers not satisfying any rational equation at all. That is the third kind of impossibility in the subject and it is the cheapest to state: a number that is not algebraic is not reachable by anything.

Where the ladder goes next #

The obvious continuation is upward: what a multifold reaches. Alperin and Lang showed that folding several creases simultaneously reaches higher degrees, so the field grows, and the natural question is which degrees are available at each number of simultaneous creases — and whether the fifth root ever arrives.

The other direction is downward, into what the closure costs. Every operation above is a fold sequence with a length, and composing constructions composes their lengths; a number reachable in principle may need a great many creases. What the shortest construction of a given number looks like is a question this collection has asked about fractions and never about algebraic numbers in general.