Twos and threes run out

The numbers a fold reaches establishes the boundary: the lengths a folder can mark are closed under addition, subtraction, multiplication, division, square roots and cube roots, so a number is reachable exactly when the degree of its minimal equation is a product of twos and threes. A compass supplies the twos and one fold supplies the threes. A fifth root is out of reach at any number of folds.

That is a characterisation, and a characterisation invites a count. How many degrees is it?

The answer sounds as though it ought to be “most of them”, because the set is infinite, because it strictly contains the compass’s set, and because the essays that state it are pleased about it. Counted, it is almost none — and the compass’s is almost none of that.

The count is a triangle #

Write a reachable degree as $2^a 3^b$. It is at most $N$ when

$$a \log 2 + b \log 3 \le \log N$$

which is a half-plane, and $a$ and $b$ are non-negative whole numbers, so the reachable degrees at most $N$ are exactly the lattice points of the first quadrant under a straight line. Each lattice point is one degree and no degree is two lattice points, because the factorisation into primes is unique.

Counting lattice points in a triangle is measuring the triangle, up to the boundary. The triangle has legs $\log N / \log 2$ and $\log N / \log 3$, so its area is

$$\frac{(\log N)^2}{2 \log 2 \, \log 3}$$

and that is the count, to within a term of order $\log N$ from the edges. The number of degrees a fold reaches grows as the square of a logarithm.

The picture makes the mechanism visible in a way the formula does not. Every reachable degree is one point of a lattice, the line that bounds them has a fixed slope of $-\log 2 / \log 3$, and raising the bound slides the line outward at a rate proportional to $\log N$ in each direction. Doubling $N$ moves the line by one square-root step. Squaring $N$ doubles both legs and so quadruples the count — which is the whole of what “grows as a logarithm squared” means, and it means the count is barely growing at all.

What that is as a share #

Twenty of the first hundred degrees. Forty of the first thousand. Sixty-seven of the first ten thousand, a hundred and one of the first hundred thousand, and a hundred and forty-two of the first million.

As a share: twenty per cent, four per cent, two-thirds of one per cent, a tenth of one per cent, and one part in seven thousand.

The formula’s prediction sits beside the count in the table because the two are different calculations. One lists the numbers and factorises each; the other is the area of a triangle. They agree to within a couple at every decade, which is the boundary term, and the agreement is what licenses reading the count as a statement about all $N$ rather than about the ones in the table.

A hundred and forty-two degrees out of a million. The reachable set is infinite and it is thin, in the sense the density makes precise: pick an integer at random below a million and the chance that a fold can settle an equation of that degree is under two parts in ten thousand. Pick one below a billion and it is worse again.

Where the closure comes into it #

The count above rests on a characterisation, and the characterisation rests on the closure, so it is worth setting the two side by side — because the closure is the part that makes the count mean anything.

Closure is what makes the degree multiplicative. A construction that reaches a length of degree three and then takes a square root of it has reached a length of degree six, and the six is a product because the tower is a composition of extensions rather than a single equation somebody solved. Without that, there would be no reason for the reachable degrees to be closed under multiplication, no lattice, and no triangle to measure — the reachable degrees would be some set with no structure and the count would be a different kind of question.

So the thinness is not an accident of which primes happen to be available. It is a direct consequence of the same property that makes the theory a theory. A set of degrees closed under multiplication and generated by finitely many primes is a multiplicative semigroup on $k$ generators, and the count of its members below $N$ is $(\log N)^k$ over a constant. The compass has one generator and folding has two. Everything else in this essay is that sentence with numbers in it.

The compass, priced the same way #

The compass’s reachable degrees are the powers of two, which is the lattice points on one axis: $\log N / \log 2$ of them, a count that grows as the logarithm itself.

So the comparison the whole of this field is built on has a number in it.

The ratio rises at every decade and never settles, because it is $\log N / (2 \log 3)$ — a logarithm, which grows without bound and grows slowly. Folding beats the compass is true at every scale and the margin is not a constant factor; it widens, and it widens at the rate a logarithm widens, so it is a factor of seven at a million and would take another ten decades to reach fourteen.

That is a more interesting statement than either “folding is stronger” or “folding is much stronger”. It says the two instruments differ in the exponent of the logarithm, which is the same kind of difference as between a linear and a quadratic algorithm and is invisible in any comparison made at one scale. At degree four they are the same instrument. At degree six they are not, and the gap has been opening ever since.

The one axiom that is doing it #

Splitting the reachable degrees the other way says where the difference comes from.

Of the twenty reachable degrees below a hundred, seven need no cube root and thirteen do: sixty-five per cent. Below a million it is a hundred and twenty-two of a hundred and forty-two, eighty-six per cent, and the share rises towards one because the count of powers of two is a logarithm and the count of everything is a logarithm squared.

So: almost everything a fold reaches, it reaches because of the single axiom that produces a cubic. Two creases at once shows that allowing simultaneous folds raises the degree further, and the enumeration of what the axioms are shows how the seven come about — but within the single-fold theory, six of the seven axioms produce quadratic conditions and one produces a cubic, and that one is carrying five-sixths of the reachable degrees at a million and more of them at every larger bound.

A construction that never uses the conic axiom is a compass construction with extra steps. That is not a figure of speech; it is what the count says. The whole of folding’s advantage over two thousand years of Greek geometry is the availability of one alignment, and its share of the advantage grows.

Two generators and the exponent #

The exponent in $(\log N)^k$ is the number of primes available, and that is the quantity worth watching, because it is the only thing in this arithmetic that can change.

Adding a larger prime to the list raises the exponent, and the exponent is what eventually decides everything. A machine that could also extract fifth roots would have degrees $2^a 3^b 5^c$, and the count would be $(\log N)^3$ over $6 \log 2 \log 3 \log 5$ — at a million, 507 against 142, which is three and a half times as many. So the third prime is worth more than the second one was, in the only sense the count recognises: the ratio between two and three generators at a million is larger than the ratio between one and two.

What it is not worth is much, in absolute terms. Five hundred and seven degrees out of a million is still five parts in ten thousand, and a fifth-root machine would be reaching a vanishing share of the degrees exactly as the other two do. The constant in the denominator grows with each prime added — each new generator costs a factor of $\log p$ immediately — so the gain is deferred to scales nobody works at, and the share goes to zero whatever the exponent is.

Making the smallest prime available is what matters, and it is already available. The factor $1/\log 2$ is the largest of the three, which is why the reachable degrees are dominated by their powers of two — of the 142 degrees below a million, every one of them has $a$ at most 19 and $b$ at most 12, and the lattice is wider than it is tall.

So the interesting instrument is not one that reaches a new prime but one that reaches a new prime cheaply, and that distinction has no expression in the characterisation at all. The characterisation says which degrees; only the count says how many; and only the constant in the count says whether a new axiom is worth having.

The height column is the same lattice read radially. A point at $(a,b)$ has degree $2^a 3^b$ and height $a+b$, so the two orderings disagree exactly because the lattice’s two axes carry different weights in the degree and the same weight in the height. A ninth root is at $(0,2)$, height two, degree nine; an eighth root is at $(3,0)$, height three, degree eight. The disagreement is visible in the picture as two points on different sides of the line $a+b = 3$ and the same side of the hyperbola through nine.

Why the set felt large #

It is worth asking why the characterisation reads as generous when the count is this thin, because the answer is a habit rather than an error.

A set closed under six operations feels large. Closure is what makes the reachable numbers a field and makes constructions composable — the finding the first of these essays is about — and a set one can add, multiply, divide and take roots in without leaving is a set with no visible edges from the inside. But closure under operations says nothing about density in any ambient set. The rationals are closed under four of the six and are countable; the reachable numbers are closed under all six and are countable too.

And the examples are all small. Every worked construction in this subject settles a degree of one, two, three, four, six, eight or nine, and in that range the reachable degrees are most of them: seven of the first nine. A reader whose intuition is calibrated on the constructions anybody performs has calibrated it on exactly the region where the answer is different.

The same shape turns up wherever a set is defined by a multiplicative condition. Numbers with only small prime factors are called smooth, they are what the fastest factoring algorithms hunt for, and their density is thin for exactly this reason: a condition on every prime factor is a condition satisfied by lattice points under a surface, and lattice points under a surface of fixed dimension are polylogarithmically many. The degrees a fold reaches are the three-smooth numbers, and their thinness is not a fact about paper at all.

What a thin set of degrees does not mean #

Three things this count does not say, and the first is the one it would be easiest to take from it.

It is not a statement about the numbers, only about the degrees. Each reachable degree carries infinitely many numbers — every algebraic number whose minimal equation has that degree and whose tower is built from the right extensions — so the reachable numbers are not a hundred and forty-two of anything. What is thin is the set of degrees available, which is the set of shapes a construction can have.

Nor is it a statement about which numbers are interesting. Almost every number anybody has ever wanted to construct has small degree, and the count says the instrument is well matched to the demand. A tool that reaches a vanishing share of a space while reaching everything anyone has asked for is a good tool, and the thinness is a fact about the space.

And a reachable degree is not a performable construction. Reachable is not cheap prices a number in extension steps, and a degree of $2^a 3^b$ takes $a + b$ of them at best; the field has no edge and the sheet does finds seventy-two per cent of two rounds’ crossings falling off a square. A degree in the lattice is a permission, and two further arguments stand between it and a crease.

The count of lattice points is not exact. The area is the leading term and the boundary contributes a term of order $\log N$, which is why the table carries the listed count beside the predicted one. At a hundred the prediction is 18.4 against a true 20, and the discrepancy is the two points sitting on the line.

How the counting was done #

The reachable degrees are listed rather than derived. Every integer up to the bound is divided by two and by three until neither divides, and it is reachable when what is left is one. That is the definition of a three-smooth number applied directly, and it shares no arithmetic with the triangle’s area.

The two are then required to agree. A characterisation checked against a count is the arrangement used throughout these essays wherever a closed form and an enumeration are both available — the same arrangement the search and the parity count are held in — and the agreement at five decades is what turns a formula into a measurement.

The compass’s degrees are the same listing with three removed from the divisor list, so the ratio in the table is two counts made by one procedure and not a comparison of two arguments.

Still open: what the second axis is worth #

The lattice has two axes and this essay has treated them as interchangeable, which they are for counting and are not for folding.

A square-root step and a cube-root step cost different things at the paper. A square root is a bisection, available from axioms every folder uses; a cube root needs the conic alignment, which buys the reach and costs the accuracy. So the lattice points are not equally cheap, and a weighted count — lattice points under the line, each charged by its $b$ — would say what the reachable set is worth to somebody who has to fold it. The weighting is one line of arithmetic and the weight is a measurement nobody has made.

And multifold moves the line rather than the lattice. Simultaneous folds reach higher degrees, so the reachable degrees for a two-fold machine are the numbers made of twos, threes and whatever else those alignments settle — a lattice in more dimensions, whose count grows as a higher power of the logarithm. Which primes join the list at each number of simultaneous creases is the question the operation count approached from the side of how many operations there are, and this is the same question asked about what they settle. A third prime would take the count from $(\log N)^2$ to $(\log N)^3$, which is still nothing and is a different nothing.

The habit worth carrying is about characterisations that end in a condition. A set defined by a condition on every prime factor is thin, and the characterisation will not say so. “Twos and threes” reads as a generous rule because it names two primes rather than one, and the right response to a rule of that shape is to count what satisfies it before deciding how much it gives.