What two points can build

A blank page with two dots on it has more in it than it looks.

There is nothing else on the page and nothing else is needed. The distance between the two dots is the unit, because there is nothing to measure it against, and every length that ever appears is built from it.

Two operations and no others #

Greek geometry allows exactly two instruments, and their permissions are narrower than the instruments themselves suggest.

A straightedge draws the line through two points that are already there. It has no marks on it, so it cannot carry a length from one place to another, and it cannot be laid against a figure and slid until something lines up.

A compass draws the circle centred at one existing point and passing through another. It does not remember a radius — in the strictest version it collapses the moment it leaves the page — so a length cannot be picked up and put down elsewhere either.

New points come from one place only: where two drawn objects cross. Two lines meet in a point, a line meets a circle in up to two, and two circles meet in up to two. Nothing else creates a point, so nothing else creates a length.

That is the whole system. It is worth stating flatly because the usual presentation buries it under a hundred worked constructions, and the constructions are consequences rather than rules.

Which makes it a closure, not a craft #

Once the rules are written that way, a construction stops being a sequence of clever moves and becomes something duller and far more useful: a set that grows.

Start with a set of points. Form every object the rules allow — every line through two of them, every circle centred at one through another. Take every intersection of two of those objects. Add the new points to the set. Repeat.

The picture at the top of this page is one turn of that handle, starting from the smallest possible set. Two points allow one line and two circles. Those three objects cross in four places that were not already there: two on the line, at $-1$ and $2$, and two off it, at height $±\frac{\sqrt3}{2}$ over the midpoint.

The second of those is already remarkable. An irrational number has appeared, out of two dots and no measurement at all, and the reason is that the two circles have the same radius: their crossing points are one unit from both centres, so they sit at the apex of an equilateral triangle, and the height of that triangle is not a fraction. Nobody chose it. It fell out of the rules.

The second round is not a picture anybody would draw on purpose. That is the point of showing it: the reachable set grows very fast, and after a handful of turns it contains essentially everything a geometer ever needed. What matters is not how the points are reached but which numbers are reachable at all, and that turns out to be a question about arithmetic rather than about drawing.

How fast the handle turns #

The growth is worth putting numbers on, because it explains why nobody works with the closure directly and why the algebraic description is not merely elegant but necessary.

With $p$ points on the page there are $\binom{p}{2}$ lines available and $p(p-1)$ circles — a circle for each ordered pair, since the centre and the point it passes through play different roles. Every pair of those objects may cross, and a circle contributes up to two crossings rather than one.

At two points that is one line and two circles: three objects, three pairs, four new points. At six points it is fifteen lines and thirty circles: forty-five objects, nine hundred and ninety pairs, and something over a thousand intersections before duplicates are removed. A third round would be working with hundreds of points and hundreds of thousands of objects.

So the closure is never computed. It is described. The whole of the rest of this ladder consists of finding a description tight enough to prove that a particular number is not in a set nobody will ever list.

The compass that forgets #

One detail of the rules deserves a paragraph, because it looks like a serious restriction and is not.

Euclid’s compass is usually described as collapsing: it can draw a circle centred at one existing point through another, but the moment it is lifted it forgets the radius, so a distance cannot be carried from one part of the figure to another. That sounds crippling. Copying a length is the first thing anybody wants to do.

It is not crippling, and the proof is the second proposition of the Elements: given a segment and a point, a collapsing compass can construct a segment of the same length starting at that point. The construction goes through an equilateral triangle — the same equilateral triangle the first figure on this page produced without being asked — and once it is available, every later construction may pretend the compass remembers.

That is the shape of a great many results in this subject: an apparently weaker system turns out to reach exactly the same set, and the proof is a construction that simulates the stronger instrument. Nothing about the reachable numbers changes, which is exactly why the reachable numbers are the right thing to study.

Either instrument alone reaches the same set #

The collapsing compass is one instance of a pattern the section above names and then leaves, and the pattern has two much stronger instances. Both say the same thing: the reachable set does not depend on the instruments nearly as much as it appears to.

Take the straightedge away entirely. With a compass alone, no line can be drawn at all — and every point that straightedge and compass together can construct is still reachable. That is Mohr’s theorem, published in Danish in 1672, forgotten, and rediscovered by Mascheroni in 1797.

The trick is that a line is never needed as an object, only as a source of intersections. Two lines meet in a point, and that point can be found by circles; a line meets a circle in two points, and those can be found by circles. So a construction using lines can be rewritten step by step into one that never draws a line and produces exactly the same points, with the lines existing only in the description of what the points are.

Take the compass away instead, and the answer is nearly as good. With a straightedge alone almost nothing is reachable — no square roots, since nothing curved is available to intersect — but supply one circle, drawn once, with its centre marked, and everything comes back. That is the Poncelet–Steiner theorem, conjectured in 1822 and proved in 1833. One circle, for the whole of a construction of any length, and the straightedge does the rest.

Both results have the same moral and it is the essay’s own. The reachable set is a property of the arithmetic rather than of the drawing. A number is constructible because it can be reached by solving linear and quadratic equations, and any pair of instruments capable of solving those in some form reaches the same numbers. Which instrument does which job is a question about convenience; the boundary is fixed elsewhere.

That is why the answer to what can be drawn is stated as closed under the four operations and square roots rather than as a catalogue of permitted moves. A catalogue would have to be rewritten for the collapsing compass, again for the compass alone, again for the single-circle straightedge, and each rewriting would produce the same set. The algebraic description is the one that does not have to be rewritten.

It also sharpens what an impossibility proof has to do. Showing that a cube cannot be doubled is not showing that nobody has found the right sequence of circles and lines — it is showing that a certain number is not in a set closed under square roots, which rules out every instrument in the family at once, including ones nobody has thought of. A proof about the arithmetic covers a compass that has not been invented.

The instruments that do change the answer, like the marked ruler above, change it because they solve equations of a higher degree — and that is again a statement about arithmetic rather than about mechanism.

The one construction that decides the arithmetic #

Every schoolbook construction — bisecting a segment, dropping a perpendicular, copying an angle — is a shortcut through the closure, and none of them changes what is reachable. One construction does change it, and it is the reason the whole subject has an algebraic answer.

Lay a segment of length $1$ next to a segment of length $n$. Draw the semicircle whose diameter is the whole thing. Erect the perpendicular where the two segments meet, and it reaches the arc at height exactly $\sqrt{n}$.

The reason is the inscribed angle theorem: any point on the semicircle sees the diameter at a right angle, so the apex is the right-angle corner of a triangle whose hypotenuse is the diameter. The perpendicular drops that right angle onto the base, splitting the triangle into two smaller ones similar to it and to each other. Matching the sides gives $h/1 = n/h$, so $h^2 = n$.

Nothing in that construction is optional or clever. It is one circle, one line and one perpendicular, and it means that the reachable numbers are closed under square roots.

Together with the obvious facts — that sums, differences, products and quotients of reachable lengths are reachable, each by a similar-triangles argument of the same kind — this says something precise about the set of reachable numbers. It contains the rationals, and it is closed under the four arithmetic operations and under taking square roots of things already in it. Nothing else is claimed, and nothing else turns out to be true.

The lengths, listed #

It is worth writing down what that closure actually contains, because the description is short and the set is enormous.

A number is constructible when it can be written using whole numbers, the four operations, and square-root signs, nested as deeply as one likes. So

$$\sqrt{2},\qquad \frac{1+\sqrt5}{2},\qquad \sqrt{2+\sqrt3},\qquad \sqrt{5+2\sqrt{5}}$$

are all constructible; the second is the golden ratio and the last is a length in the regular pentagon.

What is not on the list is any number needing a cube root, or a fifth root, or anything that cannot be written with square roots at all. $\sqrt[3]{2}$ is not there. $\pi$ is not there. Neither is $\cos 20°$.

That the list stops there is not obvious from anything above, and proving it is the work of the next essay in this ladder. What is already clear is the shape of the argument: since each step of a construction solves either a linear equation, where two lines meet, or a quadratic one, where a circle is involved, the numbers cannot get more complicated than square roots allow.

A tower, not a heap #

The set of constructible numbers is not a shapeless collection. It is built in layers, and the layers are what make it countable and what make impossibility provable.

Start with the rationals, which are reachable with no drawing at all. Take a square root of something in there — say $\sqrt2$ — and everything expressible with it forms a larger system, closed under the four operations, in which every element has the form $a + b\sqrt2$ with $a$ and $b$ rational. Take another square root, and the system grows again.

The table on the right of that figure is the whole argument in miniature. A system of the stated size is only a system if it is closed under multiplication, and the check is finite: multiply every basis element by every other, and see that the answer stays inside. Four elements, sixteen products, all of them accounted for. $\sqrt2 \cdot \sqrt3 = \sqrt6$, which is on the list; $\sqrt6 \cdot \sqrt2 = 2\sqrt3$, which is a whole multiple of one that is.

Each rung doubles the size. That doubling is where the impossibility proofs come from, and it is why the next essay is about counting rather than about drawing.

What the picture cannot show #

The figures on this page draw one round of the closure and then a cloud. They cannot draw the closure itself, because it is infinite — every round adds points, and no round is the last.

That matters more than it usually does, because the interesting claims about the set are all claims about all of it. “Every constructible number has a certain form” is not visible in any finite picture; neither is “$\sqrt[3]{2}$ is not in the set”, which is a statement about infinitely many rounds none of which contains it.

So this essay’s figures establish the rules and nothing more. They show that the operations are exactly three kinds of intersection, that a square root costs one semicircle, and that the layers really are layers. The claims about what is unreachable are made in the essays that follow, and they are made by arithmetic, because arithmetic is the only thing that can speak about all the rounds at once.

The same limitation appears wherever a subject makes a claim about every member of an infinite family, and the standard response is the one used here: draw the step rather than the whole, and check that the step is what it is claimed to be. That is why the first figure marks each new point and states that it was verified to lie on two of the drawn objects. The verification is the only part a picture can carry honestly, and the picture carries it. What follows from the step, applied forever, has to be argued in symbols — as it is in the square that cannot shrink, where an infinite descent is exactly one step plus the observation that it never stops.

Why the restriction was ever worth keeping #

A modern reader’s first question is why anyone would tie their hands like this. Marked rulers exist. So do protractors, and paper that can be folded.

Part of the answer is historical: the two instruments were the ones whose behaviour could be stated exactly, and Greek geometry was built on stating things exactly. A marked ruler has a length written on it that came from somewhere, and where it came from is a question the system could not answer.

The better answer is the one this whole field is about. A restriction with a stated boundary is more informative than a tool that always works. Allowing more operations makes more things constructible and makes the question less interesting; allowing exactly these two makes the reachable set an object with a description, and an object with a description can be shown not to contain something.

It is also worth being precise about what a wider tool buys, because the comparison is what gives the restriction its meaning. A ruler with two marks on it — a neusis, in the Greek term, used by sliding it until the marks land on two given curves — trisects any angle and doubles any cube. So does a sheet of paper folded along a crease that brings two points onto two lines at once. Both reach every number satisfying an equation of degree three or four, which is strictly more than square roots allow and still not everything.

So the boundary this ladder maps is not a boundary of geometry. It is the boundary of one stated operation set, and moving the boundary is a matter of stating a different one. That is why the failures — the cube, the angle, the circle — are the famous part. They are not curiosities about drawing. They are the first theorems anybody proved about the limits of a computation, two thousand years before there was a word for one, and they were proved by finding the right way to describe what the operations could reach.

That grid is the payoff and it is four essays away. The pattern in it — 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, and then a long silence around 7, 9, 11, 13 — is not a fact about drawing at all. It is a fact about which numbers are one less than a power of two, arrived at by counting the size of a tower like the one above.

Where this ladder goes #

The closure is defined. What remains is to describe it well enough to prove that something is outside it, and then to find the somethings.

The next rung turns the doubling into an exact statement about degrees, and shows that a constructible number satisfies a polynomial equation whose degree is a power of two. After that the impossibilities come quickly, because each one is the observation that some number satisfies an equation of degree three and nothing else. The cube and the angle are both that observation. The polygons are a more careful version of it. And the circle is different in kind, because $\pi$ satisfies no polynomial equation with whole-number coefficients at all — a much stronger statement, proved much later, and the one place in this field where no picture helps.