Reachable is not cheap

The numbers a fold reaches makes two claims and rests everything on the second. The first is that one fold solves a cubic, which is one fact about one fold. The second is that the lengths a folder can mark are closed under addition, subtraction, multiplication, division, square roots and cube roots — so constructions can be built out of constructions, and that is what turns a collection of tricks into a theory.

The closure is stated as a licence. It is also a bill. A construction is a sequence of folds with a length, and composing two constructions composes their lengths as well as their results, so a number reachable in principle may take a great many creases to get to.

That is the rung the ladder’s own closing section names and does not make. Making it turns up something the degree does not predict.

What a tower is, and why it has a height #

The reachable set is a field, and a number in it sits at the top of a tower of extensions. Each step of the tower is a single algebraic operation that the tool can perform: a square root, which the compass supplies and which several axioms give, or a cube root, which comes from the fold that produces a common tangent to two parabolas.

A step of degree two or a step of degree three, and nothing else — which is what the enumeration behind the seven axioms leaves a folder able to do. That is the whole of what the closure says, and it has a consequence for the degree that is usually stated the other way round.

A number’s degree is the product of the degrees of the steps in its tower, and what each axiom is worth is a statement about which steps each one supplies. So the reachable degrees are exactly the products of twos and threes — which is why a fifth root is unreachable at any number of folds, and why the boundary of the reachable set is an arithmetic condition on a single number.

Read for the cost rather than for the boundary, the same statement says something else. If a degree is 2^a · 3^b then the shortest tower to it has a + b steps, because each step contributes exactly one factor and the factorisation is unique. The height is not an estimate: it is the number of prime factors of the degree, counted with multiplicity.

The two orderings, and where they come apart #

Now the two columns can be compared, and the comparison is the point.

Degree measures how far out of reach a number is for a weaker tool. A compass reaches degrees that are powers of two; a fold reaches degrees with no factor above three; a number of degree five is out of reach for both. Ordering numbers by degree orders them by how much tool is needed.

Height measures what the construction costs. Ordering numbers by height orders them by how many operations have to be composed.

These are different orderings and they disagree. The clean case is a pair of pure radicals:

• 2^(1/8) has degree 8 = 2³, so its tower is three square roots: three steps.
• 2^(1/9) has degree 9 = 3², so its tower is two cube roots: two steps.

The ninth root is of higher degree and is reached in fewer steps. A compass cannot get near either; a folder gets to the further one sooner.

That is not a curiosity about two particular numbers. It happens whenever a degree with more threes in it is compared with a smaller degree made of twos, and it happens because three is a bigger step than two while three and two are the same number of steps. A tool with a cube root available buys height cheaply, and height is what the compass does not have.

Two certificates, and why there are two #

A table of degrees is only worth as much as the argument that each degree is right, and this one uses two arguments because one of them runs out.

Exhaustion. For the smaller degrees the claim “this number satisfies no rational polynomial of lower degree” is checked by trying them: every integer polynomial of every lower degree with coefficients in a bounded range is evaluated at the number, and none may vanish. That is a genuine check and a number arrived at by a slip would fail it.

It is also expensive, and the expense is the reason for the second certificate. At degree six the search is over some millions of coefficient tuples; at degree nine it is over tens of billions, and the check stops being affordable long before the numbers stop being interesting.

Eisenstein at two. For a polynomial xⁿ − 2 the classical criterion settles irreducibility in one pass: two divides every coefficient but the leading one, and four does not divide the constant. That is a proof rather than a search, it costs nothing, and it applies to exactly the numbers the exhaustion cannot reach.

Both are used, each row says which applies to it, and the figure refuses to draw a row that has neither. The division is worth stating because it is the honest shape of the evidence: the small degrees are checked by brute force and the large ones by a theorem, and neither is checked by being asserted.

The polygons have heights too, and the order is startling #

The second column is free wherever the first one is computed by factorising, and this ladder’s neighbour computes the first column for every regular polygon. So the heights come out with no new argument at all, and they are worth having because they invert the ladder’s own story.

A regular n-gon is constructible when cos(2π⁄n) is, and that number has degree φ(n)⁄2 where φ is Euler’s totient. Factorise the degree and count:

• The heptagon. φ(7) = 6, so the degree is 3, so the height is one. A single cube-root step.
• The nonagon. φ(9) = 6, degree 3, height one. Also a single step.
• The thirteen-gon. φ(13) = 12, degree 6 = 2·3, height two.
• The seventeen-gon. φ(17) = 16, degree 8 = 2³, height three. Three square roots.

Set those beside what the tools can do. A compass reaches the seventeen-gon and not the heptagon — that is the famous fact, and it is what the constructible-polygons ladder is about. Read the second column and the order reverses: the heptagon is a one-step construction and the seventeen-gon is a three-step one.

So the polygon that took two thousand years and a nineteen-year-old to construct is, in the folder’s terms, three times the tower of the one the compass cannot reach at all. The difficulty that made the seventeen-gon famous is entirely in the first column, and in the second column it is the dearest of the four.

That reading is available to anybody who has both facts and is not, as far as this collection can find, ever made. The reason is presumably that the two columns come from the same arithmetic and are therefore easy to conflate: the totient’s factorisation decides reachability and counts the steps, and having used it for the first it is natural to think it has been used up.

When a degree factorises several ways #

A degree of six is 2·3 and it is also 3·2, and a construction could take the square root first or the cube root first. It is worth being clear that this does not make the height ambiguous.

The height is the number of prime factors counted with multiplicity, and that number is the same for every ordering because the factorisation into primes is unique. Six is two factors however they are arranged; twelve is 2·2·3 and is three factors whichever order they come in.

What the ordering does change is the intermediate quantities. A tower through √2 and then a cube root passes through a different field from one that takes the cube root first, so the two constructions have different intermediates on the paper even though they have the same number of steps and the same top.

That matters for the fold count and not for the height, which is the distinction this rung keeps having to make. It is also where a real construction can be cheaper than another of the same height: an ordering whose intermediates happen to be points the sheet already carries needs no folds to produce them.

So the height is exactly the part of the cost that does not depend on how the construction is arranged, and the part that does depend on it is not bounded by anything here.

What this does to the closure’s own argument #

The ladder’s first rung ends on the closure as the thing that makes composition safe. A folder may compose constructions without ever asking whether the result is still constructible, and that is true and is the reason the subject has a theory.

This rung says the safety has a price and names it. Composition is safe in the reachable set and costly in the fold count, and the two are not related in the way a reader would guess: composing a two-step construction with a three-step one gives a five-step one, so heights add while degrees multiply.

That difference in how the two compose is the whole of the disagreement between the orderings. Degrees multiply and heights add. A quantity that multiplies and a quantity that adds cannot order a set the same way unless every element is a power of the same thing, and the reachable degrees are not.

Which has a practical reading for anybody building a construction. The question “is this number reachable” is answered by factorising its degree, and the answer is yes or no. The question “how dear is it” is answered by counting the factors, and the answer is a small integer that a large degree does not predict.

The height is a lower bound and not a fold count #

It would be easy to over-read the height, so it is worth being precise about what it does and does not say.

The height is the number of extension steps, and every step needs at least one fold. So the height is a lower bound on the number of folds a construction takes — a two-step tower cannot be built with one crease.

It is not the number of folds. A single extension step may take several folds to perform: the cube-root step is one axiom, but placing the two parabolas’ defining points and lines is itself construction work, and a step that needs an intermediate reference costs the folds that produce it. So the true fold count of a construction is the height plus whatever the bookkeeping comes to, and the bookkeeping is not bounded by anything in this table.

And the shortest tower is not necessarily the cheapest construction. A tower of two cube roots has two steps; a longer tower through different intermediates might use fewer folds overall if its intermediates are ones the sheet already carries. Nothing here rules that out, and the reachable-points ladder has measured how quickly a sheet accumulates intermediates — cheaply at first, and then not.

So the honest statement is a bound in a stated direction. The height is what a construction cannot cost less than, and the disagreement between the orderings is a disagreement about lower bounds, which is enough for the finding and not enough for a recipe.

Which theorem was checked and how #

Every stated polynomial is evaluated at its number and required to vanish, to a part in a billion. A polynomial written down wrongly fails this before anything else runs.

Every degree carries a certificate and the figure refuses a row without one. Below degree seven the certificate is exhaustion over integer coefficients; from degree eight up it is Eisenstein at two, checked as an arithmetic property of the coefficients rather than quoted.

The factorisation is checked by multiplying it back. Each reachable degree’s twos and threes must multiply to the degree, which catches a factoriser that dropped a factor.

And the inversion is asserted rather than pointed at. The figure searches every pair of reachable numbers for one where a larger degree carries a shorter tower, and refuses to draw if none exists. That is the essay’s claim in the form that a wrong height would break.

Where the model stops #

The tower is a tower of quadratic and cubic steps and nothing else. That is what one fold at a time gives. Allow two creases to be made simultaneously and higher-degree steps become available, so the height of a given number falls — and by how much is a question the multifold ladder owes.

The degree of a product is not the product of the degrees. ∛2·√2 has degree six and its two factors have degrees three and two, which happens to multiply correctly; that is not general, and the table states each degree from its own minimal polynomial rather than composing.

The heights are computed for the reachable numbers only. A number of degree five has no tower at all, and the table marks it rather than assigning it an infinite height, because an unreachable number is not expensive — it is absent.

And nothing here is a claim about any published construction. No folding sequence is analysed. What is computed is a lower bound on what a sequence must contain, from the arithmetic of the degree.

What the picture cannot show #

The table draws heights and cannot draw towers. Which intermediate quantities a given construction passes through, and whether two constructions share them, is not in it — and sharing is exactly what would make a real construction cheaper than its height suggests.

Nor can it show the cost of the bookkeeping. Every step in the table is drawn as a single unit and the folds that set a step up are invisible, so the picture systematically understates what a construction costs while correctly ordering what it cannot cost less than.

The most conspicuous absence is the fold sequence itself. This ladder has never drawn one, and the reason is worth saying: a construction of a number of degree nine has no canonical sequence, several published ones would differ, and drawing one of them would present a choice as though it were the answer.

The idealisation, named #

A fold is exact and a crease is a line. Both matter here and the second matters more than it usually does.

An exact fold means the tower is a tower of algebraic extensions and the composition is exact, so a two-step construction lands on the number rather than near it. A real fold does not, and exactness and accuracy are different quantities — a fact this ladder’s neighbours spend a rung on.

A crease with no width means a construction may use as many intermediate references as it likes without the sheet filling up. It does fill up: references crowd, and a construction whose intermediates land within a crease’s width of one another cannot be executed however few steps it has.

So the height is a count in an idealised model, and both idealisations bite in the same direction — a real construction of a tall tower is dearer than its height, not cheaper.

Where the ladder goes next #

The closure has been priced and the field it closes has not been questioned, which is the next rung.

Origami numbers are defined on the plane. A point is reachable if some sequence of axioms specifies it, and the axioms specify lines, and a line is infinite. A folder has a piece of paper: a crossing of two fold lines is a reference only where there is paper under it, and the numbers the sheet’s edge throws away turn out to be most of them — seventy-two per cent at two rounds on a square, counted rather than estimated.

Sideways from here, the two-column reading of the degree is worth carrying to the polygons. Which regular polygons a fold reaches is a condition on the totient’s factorisation, so every polygon has a height as well as a verdict — and the heptagon and the hendecagon differ in the first column rather than the second, which is not the way that ladder tells it.

The habit worth carrying is about what a boundary condition can be reused for. A criterion that decides membership by factorising something can usually be read again as a cost, by counting the factors instead of testing them. That is a free second result from an argument already made, and it is almost never taken.