The circle that will not square

The first three classical problems close by finding a number of degree three. The fourth closes by finding a number of no degree whatsoever, and the difference in kind is the whole subject of this essay.

The problem #

A circle of radius $1$ has area $\pi$. A square of the same area has side $\sqrt\pi$. Squaring the circle means constructing that side from the radius, with the two instruments, in finitely many steps.

Since the constructible numbers are closed under square roots, and closed under squaring too, $\sqrt\pi$ is constructible exactly when $\pi$ is. So the problem is: is $\pi$ a constructible number?

By the degree theorem it is enough to show that $\pi$ is not. But the previous three essays all did that by finding the degree and observing it was odd. Here there is nothing to find.

Two kinds of number #

Every number this ladder has met so far satisfies some polynomial equation with whole-number coefficients. $\sqrt2$ satisfies $x^2-2=0$. $\sqrt[3]2$ satisfies $x^3-2=0$. $2\cos\frac{2\pi}{7}$ satisfies $x^3+x^2-2x-1=0$. Such numbers are called algebraic, and each has a degree: the smallest degree of a polynomial it satisfies.

A number satisfying no such equation is called transcendental. It is not that its degree is large; it is that the question “what is its degree?” has no answer, because the set of polynomials it satisfies is empty.

Both facts about transcendental numbers are strange in opposite directions. Almost every real number is transcendental — the algebraic numbers are countable, since polynomials with whole-number coefficients can be listed and each has finitely many roots, while the reals are not, which is Cantor’s diagonal argument applied to exactly this. And yet no explicit transcendental number was known until 1844, when Liouville built one on purpose out of a rapidly converging series. The generic case had to be constructed by hand.

$\pi$ was shown transcendental by Ferdinand von Lindemann in 1882, and squaring the circle died with it: a transcendental number has no degree, so its degree is not a power of two, so it is not constructible.

Why no figure on this page proves it #

Here is the honest position, and it belongs near the front rather than buried at the end.

The impossibility proofs in the earlier essays are carried by an exhaustive search. The rational root theorem produces a finite list of candidates; every one is tested; none is a root; the conclusion follows. A reader can check the arithmetic and be finished.

Transcendence is not like that. The claim is:

For every polynomial with whole-number coefficients, $P(\pi) \ne 0$.

That is a statement about infinitely many polynomials, and no finite search touches it. A search can report that it did not find one, and that report is compatible with $\pi$ being algebraic of a degree just past where the search stopped.

The figure at the top of this page runs the search anyway, because what it does show is worth having. Every integer polynomial of degree at most four with coefficients between $-5$ and $5$ — a hundred and sixty-one thousand of them — is evaluated at $\pi$, and the closest approach to zero at each degree is reported. The numbers fall: $0.142$ at degree one, $0.031$ at two, $0.016$ at three, $0.005$ at four.

That falling sequence is exactly what would happen if $\pi$ were algebraic and the search were closing in. It is also exactly what happens because polynomial values are dense and a wider search gets nearer to anything. The table cannot tell those two situations apart, and that is the point of showing it.

The two controls are what make the first figure a measurement rather than a shrug. A machine that says “not found” to everything says nothing by saying it to $\pi$. This one finds $x^2-2$ and $x^3-2$, so its silence has content — but the content is evidence, not proof, and the essay says so rather than letting the picture imply otherwise.

How thin the algebraic numbers are #

The counting argument deserves more than the sentence it got, because it makes the situation genuinely peculiar.

List the polynomials with whole-number coefficients. There are countably many — order them by degree and by the size of their coefficients, and every one appears somewhere on the list. Each has at most as many roots as its degree. A countable list of finite sets is countable, so the algebraic numbers are countable.

The real numbers are not, and the proof is the diagonal argument: any list of reals can be beaten by a number differing from the $n$-th entry in the $n$-th place. So the transcendental numbers are not merely present, they are almost all of what there is. Pick a real number at random by any reasonable procedure and it is transcendental with probability one.

That makes the historical order absurd. The typical real number is transcendental; the first one anybody could name was constructed by Liouville in 1844 by writing down $\sum 10^{-k!}$ — a decimal expansion with runs of zeros growing fast enough that its rational approximations are too good for any algebraic number to allow. $e$ followed in 1873 and $\pi$ in 1882. Two of the three most familiar numbers in mathematics were the second and third examples ever produced of the generic case.

Liouville’s method is worth one more sentence because it connects directly to a subject this collection has already been through. An algebraic number of degree $d$ cannot be approximated by fractions much better than $1/q^{d}$ allows — that is a theorem, and it is a sharpening of the general statement about how close a fraction can get. Liouville’s number is built to violate that bound: its decimal has runs of zeros growing so fast that truncating it gives approximations far too good for any degree. So the first transcendental number ever exhibited was produced by approximation theory, which is not where anyone would have looked.

What is typical and what is nameable have very little to do with one another here, and constructibility is one more property where the typical number fails and the nameable ones are the exceptions.

What the proof actually uses #

Lindemann’s argument has no picture and it is worth saying what it does have, because the shape of it explains why no picture is coming.

It rests on a theorem of Hermite’s about the exponential function. Hermite showed in 1873 that $e$ is transcendental; Lindemann generalised the method to show that if $a$ is a non-zero algebraic number then $e^{a}$ is transcendental.

Now use the identity that ties the exponential to the circle,

$$e^{i\pi} = -1.$$

The right-hand side is algebraic — as algebraic as a number gets. So $i\pi$ cannot be a non-zero algebraic number, since then $e^{i\pi}$ would be transcendental and it is not. Therefore $i\pi$ is transcendental, and since $i$ is algebraic, $\pi$ is transcendental too.

Everything in that argument is about the exponential function, its power series, and estimates on integrals of it. Nothing in it is about circles, compasses, areas, or lengths. The connection to the geometric problem is the single identity above — which is itself the deepest fact in the relationship between circles and waves — and once that identity is in hand the geometry is over.

That is why this essay’s figures are searches rather than diagrams. The proof lives in a part of mathematics with no diagrams in it, and drawing something suggestive would be drawing something else.

What squaring the circle would have needed #

It is worth seeing the construction that is missing, because it is one step away from constructions that work.

Given a segment of length $\pi$, the square is one semicircle away: lay $\pi$ next to $1$, draw the semicircle, and the perpendicular at the join has length $\sqrt\pi$. The hard part is not the square. It is that a segment of length $\pi$ never appears, no matter how the two instruments are used.

The relationship between $\pi$ and the circle’s own measurements is as direct as it could be — it is the ratio of circumference to diameter and the area of the unit disc. What cannot be done is transfer: the circumference of a drawn circle cannot be laid out straight, because straightening it is exactly producing a segment of length $2\pi$.

That operation has a name — rectification — and it is barred by the same theorem. A curve can be drawn and its length cannot be constructed, which is a distinction with no analogue among straight figures and is one of the more counterintuitive consequences of the whole business.

The lunes, which can be squared #

The problem is not that curved regions are beyond squaring. Some of them are, and knowing which is what made the circle look approachable for so long.

Hippocrates of Chios, around 440 BC, squared a lune: the crescent between two circular arcs. Take a right-angled isosceles triangle, draw the semicircle on the hypotenuse, and draw the semicircle on one of the legs bulging outward. The crescent trapped between them has exactly the area of half the triangle — a region bounded entirely by arcs, whose area is a rational multiple of a triangle’s, and therefore squarable with the two instruments.

The reason it works is that the $\pi$s cancel. Areas of circular segments are proportional to $\pi$ times the square of a radius, and the particular arrangement makes two such terms subtract exactly. There are five squarable lunes altogether; Hippocrates found three, and the classification was completed only in the twentieth century.

That is a genuine and slightly cruel near-miss. It shows a curved region being squared, by the permitted instruments, with the $\pi$ vanishing from the arithmetic — and it gave two thousand years of geometers a reason to believe the circle itself would fall to a cleverer arrangement. It never could, because in the circle’s case there is nothing for the $\pi$ to cancel against.

After 1882 #

Transcendence settled the mathematics and did nothing whatever to the flow of proposed solutions, which is a fact about people rather than about circles.

The most celebrated episode is the Indiana General Assembly’s Bill 246 of 1897, which proposed to enact a circle-squaring method as state law. The bill implied several mutually inconsistent values for $\pi$, among them $3.2$. It passed the House unanimously and was indefinitely postponed in the Senate after a mathematician who happened to be in the building explained the situation. Legislating a value for $\pi$ is impossible for a duller reason than transcendence: $\pi$ is a ratio, and ratios are not subject to legislation.

The steadier lesson is the one this field keeps returning to. An impossibility result closes a question and does not close the asking of it, because the result is about a stated operation set and most people asking do not have the operation set in mind. A construction using a marked ruler, an infinite process, or a curve that is not a circle is not a counterexample; it is a different question with a different and often affirmative answer.

The one thing the pictures on this page do #

Every figure here is a report of a computation and none is a proof. That is unusual for this collection and it is worth stating flatly rather than leaving the reader to infer it.

The searches show that the machinery which located $\sqrt2$ and $\sqrt[3]2$ locates nothing at $\pi$ within the range it can afford. That is a fact about the range, and it is exactly as much as an exhaustive method can contribute to a statement quantified over an infinite set.

The site’s standing claim is that a figure can be a proof. Here it cannot, and the honest thing is the same thing the incompleteness essay had to do one field earlier: draw the part that is finite, state which part is not, and let the argument that closes the gap live in the prose where it belongs.

The distinction that grid draws is the one worth carrying out of this ladder. Both questions are settled; both settlements are impossibility results; and only one of them has a picture. What decides which is not the difficulty of the mathematics but whether the claim is quantified over a finite set. The polygons are a claim about ninety-eight integers, and ninety-eight things can be checked. Transcendence is a claim about every polynomial there is.

The approximations, and why they are not the point #

There is an enormous literature of very good approximate quadratures, and they are worth separating from the false proofs, because most of them were never claims to have solved the problem.

Kochański’s construction of 1685 produces a segment of length $3.141533$ against $\pi$’s $3.141593$ — accurate to four decimal places, using two circles and a straight line. Ramanujan gave several, one of them accurate to eight places. Every one of these is a genuine and elegant piece of geometry, and none is in tension with Lindemann.

The reason is the same as for the trisections: constructible numbers are dense, so an approximation to any accuracy exists and finding a short one is a craft problem. The theorem forbids exactness in finitely many steps, and exactness is not a limit of accuracy. It is a different property, and the difference is the same one that separates a converging sum from its value.

What this means in practice is that “squaring the circle” as a phrase for a hopeless task is slightly wrong. The task is not hopeless; it is only impossible to do exactly, with those instruments, in finitely many steps. Drop any of the three qualifiers and it becomes routine.

Where this ladder ends #

Four problems, four closures, and two different shapes of argument. The cube, the angle and the polygons all fall to a degree count, and every one of those counts is a finite search a reader can redo. The circle falls to a theorem about the exponential function, and no search will ever reach it.

The ladder stops here because the operation set has been fully described: the constructible numbers are the ones reachable from the rationals by finitely many square roots, and every classical question about the compass is a question about that description. What comes next in this field is not a bigger compass but a different operation set entirely — a finite alphabet, an arithmetic that closes, and the question of what can be checked rather than what can be built.