The centre a straightedge cannot find

One circle and a straightedge proved that a straightedge, which on its own can barely bisect a segment, reaches every point a compass and straightedge reach together — provided one circle has been drawn and its centre is marked. That essay stated, and left undrawn, the other half: a circle without its centre is not enough. Detlef Cauer proved it in the years just before the First World War, and the proof is worth seeing, because it turns out to be a statement about a geometry that is not Euclid’s at all.

The difficulty in proving that something cannot be constructed is that constructions are not a list. Any sequence of lines through any points already found is allowed, the sequences can be as long as anyone likes, and there is no way to check them one by one. What works instead is to find a transformation that every construction must respect, and that moves the thing being sought.

The figure below is that transformation at work. On the left, a circle whose centre is not marked, six points on it, five chords and the points where the chords cross — a small straightedge construction. On the right, the same construction after a map that takes every straight line to a straight line and the circle to itself. Chord for chord and crossing for crossing, the two drawings match. And the map has put the centre somewhere else.

What a straightedge actually sees #

A straightedge does one thing: it draws the line through two points. Every point it adds to a figure is the crossing of two such lines, or a crossing of a line with a curve already drawn. So a straightedge construction is a record of incidences — which points lie on which lines — and of nothing else. It never measures a length, an angle or a distance.

That makes it blind in a precise way. Suppose some map of the plane takes every straight line to a straight line, and takes the given figure to itself. Run any construction on the figure, then apply the map to the whole drawing. Every line of the construction goes to a line, every crossing to the crossing of the image lines, and the given figure goes to itself — so the image is another valid construction on the same figure, step for step. Whatever the construction produced, the map carries it to what the image construction produces. If a construction could find the centre, the image construction would find the image of the centre; but both are constructions on the same circle, so both find the same point, and the map must have fixed the centre.

So one map that keeps the circle and moves its centre is a complete proof that no straightedge construction can find the centre. The hero is that map.

A map that keeps the circle and moves the centre #

The maps that take straight lines to straight lines are the projective transformations — the maps a slide projector makes when the screen is tilted. Written in homogeneous coordinates, where a point $(x, y)$ is the ratio $(x : y : 1)$, they are $3 \times 3$ matrices, and the unit circle is the set where $x^2 + y^2 - z^2 = 0$.

A matrix that preserves the quantity $x^2 + y^2 - z^2$ preserves the circle. The rotations do, and they fix the centre. So do the matrices

$$\begin{pmatrix} \cosh b & 0 & \sinh b \\ 0 & 1 & 0 \\ \sinh b & 0 & \cosh b \end{pmatrix},$$

for the same reason that $\cosh^2 b - \sinh^2 b = 1$. These are the matrices that, in relativity, change the frame of an observer moving at speed $\tanh b$; here they act on the plane. The centre $(0 : 0 : 1)$ goes to $(\sinh b : 0 : \cosh b)$, which is the point $(\tanh b, 0)$. With $b = 0.62$, as in the hero, the centre lands at $(0.551, 0)$.

The figure checks everything the argument needs rather than trusting the algebra: every one of the six points on the circle lands on the circle, and every crossing of two chords lands exactly on the crossing of the two image chords. The rest is the paragraph above. Cauer’s proof was this, in the language of his time.

The plane in which the centre is ordinary #

The maps that keep a circle form a group, and it is a famous one.

Take the inside of the circle as a plane in its own right, its chords as the straight lines of that plane, and the circle-preserving maps as its rigid motions. That is the Beltrami–Klein model of the hyperbolic plane, the geometry in which Euclid’s parallel postulate fails — through a point off a chord there are infinitely many chords that do not meet it. The boosts are its translations, the rotations its rotations about one point, and the triangle’s twenty-eight copies are twenty-eight congruent hyperbolic triangles. They look smaller near the rim only because the model squashes an infinite plane into a disc.

In the hyperbolic plane every point is like every other: there is a rigid motion taking any point to any point. The Euclidean centre of the disc is just one point of the hyperbolic plane, with nothing geometric to single it out. A straightedge working on a bare circle is doing hyperbolic geometry without knowing it, and in hyperbolic geometry there is no centre to find.

That is the surprising connection this essay turns on. A question about drawing instruments, posed in the Renaissance and settled by Cauer, is answered by the non-Euclidean geometry discovered in the 1820s — and the answer is the fact that made that geometry homogeneous.

A distance a straightedge can measure #

There is a sense in which the straightedge is not quite blind. It cannot see Euclidean distance, but it can see the hyperbolic one, and the reason is a single projective invariant.

Take two points inside the circle and draw the chord through them. The chord meets the circle at two more points, and the four points on one line have a cross-ratio — the one number four collinear points keep under every projective map. In the Klein model, the hyperbolic distance between the two inside points is half the logarithm of that cross-ratio. Every ingredient is something a straightedge can construct or a projective map preserves, so the distance is too. The circle-preserving maps are exactly the maps that keep it, which is why they are the hyperbolic motions and not merely a family of maps that happen to keep a circle.

So a straightedge with a bare circle can compare hyperbolic lengths, bisect hyperbolic segments, and drop hyperbolic perpendiculars — all by incidences, all invariant. What it cannot do is prefer one point of the disc to another, because the hyperbolic plane offers no preference. Two worlds that both obey the rules drew the same plane as Poincaré’s disc, where the straight lines are circular arcs meeting the rim at right angles; Klein’s version straightens them into chords, and chords are what a straightedge draws. That is why it is Klein’s disc, and not Poincaré’s, that a straightedge inhabits.

What a straightedge can still do with a bare circle #

The blindness is total for the centre and partial for everything else. A bare circle still gives a straightedge a good deal of work to do, all of it projective.

For any point, the straightedge can draw its polar — the line that, for a point outside the circle, joins the two points where the tangents from it touch. Two lines through the point cut the circle in four points; the two ways of pairing those four points up make two crossings, and the polar is the line through them. The construction uses only incidences, so it survives every circle-preserving map, as it must. Tangents, poles, polars, the harmonic conjugates that every straightedge construction runs on: all of them are available. So is Pascal’s theorem, which says that the three crossings of opposite sides of any hexagon inscribed in the circle lie on one line — six points on a conic, and the line they share — and which lets a straightedge given five points of any conic find as many more of its points as it likes.

The centre is the pole of one particular line — the line at infinity, which in homogeneous coordinates is $z = 0$. A straightedge could find the pole of that line if it could draw the line. It cannot, because the line at infinity is exactly what the circle-preserving maps move: the boost sends it to the vertical line $x = 1/\tanh b$, outside the circle on the right. The centre is lost because infinity is lost, and marking the centre hands back exactly the missing piece.

With the centre marked, every diameter is a segment with a known midpoint, and a segment with its midpoint lets a straightedge draw parallels to it — which is to say, find points on the line at infinity. The rest of the Poncelet–Steiner theorem follows from there. The comparison between this figure and the polar figure is the whole of this essay in two pictures: without the centre, only what the hyperbolic motions preserve; with it, everything.

Two circles, apart #

If one bare circle is not enough, perhaps two are. The answer depends on whether they meet, and the case where they do not has a map as explicit as the boost.

For two circles of equal radius $r$ with centres at $\pm d$, $d > r$, put $s = d^2 - r^2$ and take the map $(x, y) \mapsto (s/x,\; \sqrt s\, y/x)$. A short calculation — substitute and simplify — shows it takes each circle to itself, and the figure checks it at seventy-two points. It is projective, since it is a ratio of linear expressions with a common denominator, so it takes lines to lines. It sends each centre $(d, 0)$ to $(s/d, 0)$, which is not the centre. So no straightedge construction can find the centres of two separate circles of equal size. The same exchange, described in the next paragraph, works for separate circles of any sizes; only the formula is less tidy.

The map also shows where the argument’s teeth are. It carries the line at infinity to the vertical line $x = 0$ between the circles — their radical axis, the line of points with equal tangents to both. Every pair of circles meets, in the complex projective plane, in four points: two of them are the imaginary “circular points at infinity” that every circle passes through, and for separate circles the other two are an imaginary pair on the radical axis. The map simply exchanges the two imaginary pairs. A straightedge, which sees only real incidences, cannot tell one pair from the other, and so cannot tell infinity from the radical axis.

Two circles that meet #

If the circles meet at two real points, that exchange is impossible: no real map can swap a pair of real points with a pair of imaginary ones. Every circle-preserving map then keeps the line at infinity, the centres stop being movable, and the argument fails — which is the signal to look for a construction.

The first steps are visible. The tangents to each circle at the crossing points $P$ and $Q$ are polars, drawn by the construction above. The two tangents of one circle meet on that circle’s line of symmetry through $P$ and $Q$, which is the line of centres, and the same is true of the other circle; so the two meeting points $K_1$ and $K_2$ fix the line of centres. That line crosses the common chord $PQ$ at the chord’s midpoint. A segment with its midpoint gives parallels to $PQ$ — one point at infinity.

The second point at infinity, the one along the line of centres, comes from a theorem of Girard Desargues: the circles through $P$ and $Q$ cut any line in pairs of points that are partners in a single involution, and the degenerate “circle” made of the line $PQ$ together with the line at infinity is one of them. On the line of centres the two given circles supply two partner pairs, which fix the involution; the partner of the midpoint is then the point at infinity, and a straightedge can construct it. With two points at infinity the whole line at infinity is available, and the centre of each circle is its pole. Two circles that meet give their centres away; two that do not, keep them.

Why a surveyor would care #

Constructions with the straightedge alone were not invented as puzzles. A line of sight is a straightedge: two stakes in a field determine a line, a third stake can be put on it by eye, and where two sighted lines cross can be marked without measuring anything. Johann Heinrich Lambert studied what can be done this way in the 1770s, in a book on perspective, and Charles Brianchon wrote a memoir in 1818 for surveyors and military engineers working where lengths could not be taken but lines could be sighted — across a river, or up to an enemy wall.

In that setting the questions about instruments are practical. A circle already on the ground — a round pond, a circular earthwork — is a gift, and whether its centre can be found by sighting alone decides whether the gift is any use. Cauer’s answer is that it cannot, and the reason is not a lack of ingenuity in the surveyor: the field itself, seen only through lines of sight, has the geometry of the hyperbolic plane inside the pond, and in that geometry the pond’s centre is not a special place.

The answer for two ponds is the one drawn above, and it has the same practical reading. If they overlap, sighting finds both centres. If they do not, no amount of sighting will, however large the field and however many stakes are used.

What the drawings leave to the argument #

Each figure here checks its map on the points it draws: the six points and crossings of the hero, twelve points of the circle for each of twenty-eight maps, seventy-two points of the separate circles. That is evidence that the maps do what the algebra says, and the algebra is what proves they do it everywhere.

The deeper limitation is in the logic of the proof. A map that moves the centre proves that no construction finds the centre; it does not say what constructions can find, and the positive half — the polar construction, the tangents of meeting circles, Desargues’s involution — has to be supplied separately. The two halves are different kinds of argument. One exhibits a symmetry, the other exhibits a sequence of lines, and a symmetry can show a construction does not exist but never that one does.

And the pictures are real. The argument about separate circles turns on intersections that happen only among complex points, which no figure can draw; the map exchanging them is drawn, and the points it exchanges are not.

What does a straightedge reach if it can also carry a length? #

Every instrument met so far has turned out to reach exactly the constructible points or strictly less, and the dividing line has been the same each time: whether the instruments can recover the line at infinity, and with it parallels and midpoints. A compass alone can; a straightedge with a centred circle can; a compass stuck at one opening, with a straightedge, can; a straightedge with a bare circle cannot, and neither can a straightedge with two separate ones.

There is one more classical instrument that answers differently, and the answer is neither everything nor nearly nothing. A pair of dividers that only carries a length — sets it off along any line from any point, without drawing a circle — combined with a straightedge, can find midpoints, draw parallels and drop perpendiculars, so the line at infinity is no obstacle at all. And yet it reaches strictly fewer points than a compass does. What it misses cannot be seen by any symmetry of the plane: the dividing line is arithmetic, and it runs through the conjugates of the numbers being constructed rather than through the geometry of the page. That is the question of the lengths dividers cannot reach.

The price of a construction counted the cost of constructions that exist, and this essay has been about one that does not. The next is about a construction that exists for some lengths of a segment and not for others that look no harder, which is a stranger kind of impossibility than either.