How many polygons a fold reaches

The heptagon a compass cannot reach states the difference between the two tools as a condition on a single number. A compass needs n’s odd part to be a product of distinct Fermat primes; a fold needs it to be a product of distinct Pierpont primes; and seven is the first place the answers differ.

That is the right statement of the rule and it is an odd way to describe a gain. Seven is one polygon. A reader who is told that folding reaches the heptagon has been told about a single case, and the natural question — how much did the extra axiom buy — has an answer that is not a case at all.

It is a counting function, and it can be computed to any bound.

Counting rather than listing #

Both sets are defined by the same shape of test and the test is short enough to run on every n.

Take Euler’s totient of n — the count of numbers below n sharing no factor with it — and factorise it. The regular n-gon is compass-constructible exactly when that factorisation is all twos, and fold-constructible exactly when it has nothing above three. Both are smoothness conditions on one number, they differ in one prime, and the same predicate decides both.

The permission to carry a three is not permission to carry it once. A compass may use each Fermat prime once and no more, since a repeated odd prime would need a second quadratic step where the tool has only one; a fold has a cubic step and may take it repeatedly, so three, nine, twenty-seven and eighty-one are all reachable by it and none of the last three by a compass.

Run that from three upwards and count. To forty sides a compass reaches sixteen polygons and a fold reaches thirty-one. To a hundred, twenty-four against fifty-nine. To four hundred, forty against a hundred and fifty-five. To a thousand, fifty-two against two hundred and seventy-five.

The ratio is 1.9, then 2.5, then 3.9, then 5.3. The extra axiom does not buy a fixed set of extras; it buys a set that is pulling away.

That is the honest answer to the question the heptagon poses, and it is more interesting than the heptagon, because a widening ratio is a statement about arithmetic rather than about a particular polygon.

Why the compass’s set stops #

The two counting functions behave differently and the reason is short.

The Fermat primes known are three, five, seventeen, two hundred and fifty-seven and sixty-five thousand five hundred and thirty-seven. Five of them. None has been found since, every candidate up to enormous size has been tested and factored, and the general expectation is that there are no others.

A compass-constructible n is a power of two times a product of distinct members of that list. Five primes give thirty-two subsets, so — if the list really is complete — there are exactly thirty-two odd numbers in the whole of arithmetic that a compass can reach, and the constructible n are those thirty-two each multiplied by every power of two.

Below a hundred thousand only sixteen of the thirty-two appear, because the larger products are enormous: 3·5·17·257·65537 is more than four billion. So the compass’s counting function grows like the logarithm of the bound — it gains one every time the bound doubles, from the powers of two, and picks up a rare extra when a new product comes into range.

A logarithm is what the compass’s set does. The heptagon is not the exception to a rich set; the set was never rich.

Why the fold’s does not #

Pierpont primes are primes of the form 2^a·3^b + 1, and unlike the Fermat primes they keep arriving. Below a thousand there are seventeen of them — three, five, seven, thirteen, seventeen, nineteen, thirty-seven, seventy-three, ninety-seven, a hundred and nine, a hundred and sixty-three, a hundred and ninety-three, two hundred and fifty-seven, four hundred and thirty-three, four hundred and eighty-seven, five hundred and seventy-seven, seven hundred and sixty-nine — against four Fermat primes in the same range.

The reason for the difference is worth stating because it is the whole of why the extra axiom is worth having. A Fermat prime must be one more than a power of two, and the powers of two are sparse: there are about log₂N of them below N. A Pierpont prime must be one more than a number of the form 2^a·3^b, and those are much denser: their count below N grows like the square of the logarithm rather than like the logarithm.

The candidates are squared and the primes among them follow. So the fold’s supply of usable primes does not run out in the way the compass’s does, and its constructible set keeps acquiring genuinely new odd parts rather than only new powers of two.

Whether there are infinitely many Pierpont primes is not known, and this essay does not need it to be. What is computed here is the count to a stated bound, and every number in it is arithmetic on a sieve.

What the ladder’s two rungs look like from here #

The rung below this one is about the hendecagon: folding does not reach it, the obstruction is a single prime factor of five in ten, and no arrangement of cubics produces a five.

Set that beside the counting functions and the two rungs say complementary things. The fold’s set is much larger than the compass’s and it is still very thin. Two hundred and seventy-five polygons below a thousand is a little over one n in four, so nearly three in four of the small regular polygons are out of reach of a fold as well.

That is worth holding on to, because “folding beats the compass” is easy to read as “folding reaches everything the compass does not”. It reaches four times as much and misses most of arithmetic, and the misses are governed by exactly the same kind of condition as the hits.

The two sets also have the same shape of description, which is what makes the comparison possible at all. Both are “the odd part is a product of distinct primes from a list”, and the lists differ in one factor of three. The whole of what the seventh axiom buys, expressed as economically as the subject allows, is permission to include a three in the exponent, and everything above is the arithmetic consequence of that.

The polygons in the gap, named #

A widening ratio is a summary and the summary is more convincing with the contents of the gap in front of it, so it is worth listing what the seventh axiom actually adds below a hundred.

The odd numbers a compass can reach below a hundred are one, three, five, fifteen, seventeen, fifty-one and eighty-five — seven of them, being the products of distinct Fermat primes that fit. Everything else in the compass’s set is one of those times a power of two.

A fold adds every product of a power of three with distinct Pierpont primes above three, and below a hundred the new primes it brings are seven, thirteen, nineteen, thirty-seven, seventy-three and ninety-seven. Each brings itself, its doublings, and its products with the primes already there — so seven brings the heptagon, the fourteen-gon, the twenty-one-gon, the twenty-eight-gon, the thirty-five-gon and so on.

Which is the mechanism behind the widening, stated concretely. A new prime does not add one polygon; it adds a whole multiplicative family. The compass has run out of new primes and can therefore only add powers of two to families it already has; the fold keeps acquiring primes, and each one multiplies into everything already present.

That also explains the shape of the two curves. A curve gaining families grows faster than one gaining members of existing families, and the difference compounds — which is why the ratio is 1.7 at forty and 4.0 at a thousand rather than settling.

What a repeated factor costs #

The distinctness condition is easy to pass over and it is doing real work, so it deserves a paragraph of its own.

Nine is three squared. Three is both a Fermat prime and a Pierpont prime, so a triangle is constructible by either tool — and a nonagon is constructible by neither, because the odd part nine has a repeated factor.

That is worth pausing on. The nine-gon is not out of reach because nine is large or awkward; it is out of reach because it is a square. And the reason is the same one that governs everything else here: the degree of cos(2π⁄9) is φ(9)⁄2 = 3, which is fine for a fold — so the nonagon is foldable, and the compass cannot have it.

So the distinctness condition is the compass’s condition, and a fold’s version of it is weaker in exactly the place that matters: a repeated three is what a cube root supplies. Nine, twenty-seven and eighty-one are all foldable and none is compass-constructible, and they are among the first entries in the gap. The nonagon is asserted in the figure rather than left to the count, and the assertion earned itself immediately.

Which sharpens the summary one more turn. The seventh axiom’s permission to include a three in the exponent is not only permission to use the prime seven; it is permission to use threes repeatedly, and the powers of three are the part of the gap that is easiest to see and hardest to notice.

What the gap does not mean #

Three readings of the widening ratio are available and only one of them is supported, so it is worth separating them.

Supported: the fold’s constructible set is asymptotically much larger. The counting functions are computed and they diverge, at every bound tried, by a factor that grows.

Not supported: the fold reaches a positive fraction of all polygons. Two hundred and seventy-five of a thousand is a little over a quarter, and the fraction is falling as the bound rises — it was thirty-one of thirty-eight at forty. Both sets have density zero; one of them approaches zero much more slowly.

Not supported: a folder can actually build these. Reachability is a statement about a field and says nothing about the length of a construction, which is a separate column entirely — and nothing about accuracy, which is the rung after this one. The 769-gon is constructible by folding and nobody has folded one.

The third reading is the one worth guarding against, because a counting function is exactly the sort of result that invites it. A count of what a tool reaches is a count of what a tool’s algebra permits, and every physical question is somewhere else.

Which theorem was checked and how #

The count uses this site’s own constructibility predicate rather than a second one. The totient is computed by trial division and factorised, and the two verdicts are smoothness over two primes and over three — the same function the polygon census in the neighbouring ladder is built on, so two figures here cannot disagree about which polygons a fold reaches.

The Fermat list is checked against what is known. A sieve over p − 1 runs beside the count, and the primes it produces below the bound must be exactly three, five, seventeen and two hundred and fifty-seven — which catches an exponent slip and supplies the prime counts the figure reports.

The nonagon is asserted rather than left to the count. It is the case that separates the two conditions, and the figure refuses to draw if the two tools agree about it — a check that earned its place at once, by catching a predicate which had applied the compass’s distinctness rule to the fold and refused the best-known polygon a fold constructs.

And the widening is asserted rather than observed. The gap must be strictly larger at every mark than at the one before it, and a figure whose gap settled to a constant would refuse — which is the difference between a density and a fixed set of extras, stated as a condition on the picture existing.

Where the model stops #

The Fermat list is taken as complete below the bound and no further. Nothing here proves there are only five Fermat primes; what is computed is a count to a stated bound using the primes that exist below it, which needs no conjecture.

The Pierpont primes are counted, not characterised. Whether there are infinitely many is open. The essay’s claim is about counting functions to finite bounds and does not require the answer.

Constructibility here is the classical algebraic condition and nothing else. It assumes exact folds, an unbounded plane, and unlimited construction length — three idealisations this ladder’s neighbours each spend a rung dismantling.

And a regular polygon is a regular polygon. The condition says nothing about polygons that are not regular, about approximate constructions, or about the many-fold operations that change the reachable degree entirely.

What the picture cannot show #

A counting function is a summary and it hides which polygons are in the gap. The curve rising says the fold has gained; it does not say whether the gain at n is a new prime, a new product, or another power of two — and those are quite different events.

Nor can it show the size of the constructions. Every point on the fold’s curve is a polygon that some finite sequence of folds produces, and the sequences for the large ones are not short; the curve treats a heptagon and a 769-gon as one unit each.

The deepest thing it cannot show is whether the compass’s curve really has stopped. If a sixth Fermat prime exists it is astronomically large, so the curve below any bound anybody will ever draw is unaffected — and the statement “the compass’s set has thirty-two odd numbers in it” is a conjecture wearing the clothes of a count.

The idealisation, named #

The whole essay lives inside the algebraic criterion, and that criterion has three assumptions in it that this ladder’s other rungs are about.

A construction may be any finite length. The 769-gon’s is not short and the criterion does not care.

A fold is exact. Every reachable n is reachable to infinite precision, and no polygon on either curve is easier or harder to draw accurately than any other.

The plane is unbounded. Every intermediate the construction needs is available, which a real sheet does not offer.

What survives all three is the comparison, because both tools are idealised identically. The ratio between the counting functions is a fact about two prime lists and would be the same fact if both tools were degraded in the same way.

Where the ladder goes next #

The counting function answers “how much” and leaves “how well” untouched, and the next rung is about that.

A construction is exact in the field and a fold has an error. Propagating a landmark error through a real closure shows that the conditioning of a reference is set by the angle at which the creases fixing it cross — and that the axioms that produce shallow crossings are the conic ones, which are exactly the axioms that reach the heptagon. What buys the reach costs the accuracy makes that measurement on the same closure this ladder’s neighbours use, and the tail it finds is not in the algebra anywhere.

Sideways, the counting functions belong beside the largest polygon a square sheet holds, which crosses constructibility with an optimisation and finds the two agreeing. That agreement is about the same arithmetic seen twice, and the counting function here is what makes the coincidence look less like one.

The habit worth carrying is a question to ask whenever a capability is illustrated by an example. How many, to a bound? An example is a point and a capability is a set, and counting the set is usually cheap once the criterion is written down — as it was here, where the count needed nothing that the rung below had not already established.