The arcs that make the centre cheap

The centre a compass finds in six circles answered Napoleon’s problem for two points in general position on a circle whose centre has been lost: six circles, and no construction of five does it. That essay’s search was run at arcs of 37.3°, 71.9°, 103.7° and 141.1° between the two given points, chosen because nothing about them is special, and all four came out at six with the same construction.

Special arcs behave differently, and the difference is large. If the two points are 60° apart, the problem costs two circles. This essay runs the same exhaustive search at every arc that is a multiple of five degrees and maps the price of the centre as the arc varies. The map is not a curve. It is a scatter of cheap arcs among a sea of sixes, and the cheap arcs fall into classes, each class defined by how many steps of a compass carried round the circle land on a point 60° from where they started.

The price, arc by arc #

The search is the one described before: from the drawn circle and the two points, try every circle that can be drawn, then every circle that can be drawn after that, depth by depth, until the centre appears. At each arc the first depth at which it appears is the price. Thirty-four arcs from 10° to 175° were searched, and for the cheap ones the lower bound — that nothing cheaper works — is re-run every time the figure is drawn.

Of the thirty-four, fifteen cost the generic six. The other nineteen are cheaper:

• two circles at 60°;
• three at 30° and 150°;
• four at 15°, 20°, 75°, 100°, 105°, 140° and 165°;
• five at 10°, 45°, 50°, 70°, 90°, 110°, 130°, 135° and 170°.

Two features stand out. The cheap arcs are spread across the whole range, not clustered near some favourable size. And neighbours are unrelated: 60° costs two and 55° and 65° each cost six. The chart has no slope to it, because the price is not decided by how large the arc is but by whether it is one of a particular set of fractions of a turn.

The thirty-four arcs are also far from typical, and it is worth being clear how far. Every construction of five circles or fewer is one of finitely many sequences of moves, and for each such sequence the condition that it lands exactly on the centre is an equation in the arc. The equation is not satisfied by every arc, since the arc of 37.3° defeats every short sequence, so it holds at only finitely many arcs between nought and 180°. Finitely many sequences, each cheap at finitely many arcs: the set of arcs that cost less than six is finite. An arc chosen at random costs six with certainty. Among the multiples of 5°, by contrast, nineteen of thirty-four are cheap, and the average price is just over five. The grid of round numbers is exactly where the coincidences live, because round numbers of degrees are the arcs whose multiples can land on 60°, 45° or 90°.

The dial also shows that the price is not symmetric about 90°. An arc of 20° costs four circles and an arc of 160° costs six, although the two configurations are related: two points 20° apart are 160° apart the long way round, but the constructions that exploit the short arc do not transfer to the long one, because what the compass can do depends on the distance between the two points — the chord — and the chords for 20° and 160° are different. The prices of 30° and 150° happen to agree, and those of 15° and 165° too, for reasons the next section makes plain.

A chord equal to the radius #

The cheapest case explains the rest. If the two given points $A$ and $B$ are 60° apart, the chord $AB$ equals the radius of the circle: the triangle $OAB$ is equilateral. The circle centred $A$ through $B$ then has the same radius as the given circle and passes through its centre, and so does the circle centred $B$ through $A$; the two cross at the centre. Two circles, and nothing cheaper is possible, since a single circle centred at a point on the circle meets nothing that could be the centre.

Every other cheap arc finds a way to manufacture such a chord. The mechanism is easiest to see by stepping the compass round the circle.

A circle centred at $B$ with radius $AB$ meets the given circle again at a point one arc further on; a circle centred there with the same opening reaches one arc further still. So the compass, kept at the opening $AB$, steps round the circle an arc at a time. If after $k$ steps it reaches a point 60° from $A$ — or 60° the other way — then that point and $A$ are a radius apart, and the two-circle trick finishes the job. The condition is that $k$ times the arc is $\pm 60°$ round the circle: $ka \equiv \pm 60° \pmod{360°}$, a congruence of the kind numbers that wrap studied.

The classes #

Sorting the cheap arcs by the smallest such $k$ accounts for almost all of them.

With $k = 1$ the arc is 60° and the price is two. With $k = 2$ — $2a \equiv \pm 60°$ — the arcs are 30° and 150°, and the price is three: one step, then the two circles. With $k = 3$ the arcs are 20°, 100° and 140°, and with $k = 4$ they are 15°, 75°, 105° and 165°; both classes cost four. With $k = 6$ the arcs are 10°, 50°, 70°, 110°, 130° and 170°, and they cost five. Every arc in each class has the same price, which the figure checks; the price is a function of $k$ and not of the arc.

The prices do not simply grow by one per step. Naive stepping would cost $k - 1$ circles to walk round and two more to finish, $k + 1$ in all: two, three, four, five and seven for $k = 1, 2, 3, 4, 6$. The search beats that for $k = 4$ and $k = 6$ by lengthening the stride: a circle centred at the point $j$ steps on and passing through $A$ has the chord of $j$ steps as its radius, and it meets the circle again $2j$ steps on, so each such circle doubles the distance walked. That is why $k = 4$ costs the same as $k = 3$, and $k = 6$ costs five rather than seven. The search finds these shortcuts without being told about them; it is how a reader can tell that the classes are not a hypothesis being confirmed but a pattern the search turned up.

The multiples of 45° #

One cheap class is not explained by stepping to 60°. Arcs of 45°, 90° and 135° each cost five circles, and none of them satisfies $ka \equiv \pm 60°$ for any $k$. The coincidence they exploit is of a different kind. Two points a quarter-turn apart span a chord of $\sqrt2$ times the radius, the diagonal of a square on the radius, and the arcs of 45° and 135° are halves and three-halves of that quarter-turn. Compass constructions produce lengths in the ratio $\sqrt2$ and $\sqrt3$ very cheaply — $\sqrt3$ appears in the first two circles of almost every construction, as the distance between the crossings of two circles through each other’s centres — and the constructions the search finds for these arcs combine such lengths to reach the radius in five circles. No stepping argument explains them, and the details differ from arc to arc.

So the cheap arcs are of two kinds, those that reach a chord equal to the radius by stepping, and those that reach the radius through the square’s diagonal. Arcs at multiples of 5° that belong to neither class — 25°, 35°, 40°, 55°, 65°, 80°, 85°, 95°, 115°, 120°, 125°, 145°, 155°, 160° and 175° — all cost the generic six, and for those the cheapest construction is the inversion of the previous essay, which needs no coincidence at all.

Stepping round a circle is drawing a polygon #

Carrying a fixed compass opening round a circle is the oldest construction there is. With the opening equal to the radius it steps six times and closes, drawing the regular hexagon — the construction with which the straightedge buys nothing began its compass-only arguments. With any other opening it draws the vertices of a polygon inscribed in the circle, a regular polygon if the arc divides the full turn exactly and a never-closing spiral of points if it does not.

The cheap classes are therefore statements about polygons. An arc of 20° is the side of the regular 18-gon, and the third vertex of that 18-gon is a vertex of the hexagon; an arc of 15° is the side of the 24-gon, whose fourth vertex is a hexagon vertex. In general an arc $a$ is cheap by stepping exactly when the polygon it generates shares a vertex with the hexagon on the same circle, and the price measures how far round the polygon that shared vertex lies. Which polygons can be drawn asked which regular polygons are constructible at all, a question about the arithmetic of the number of sides; this map asks a finer one about polygons that are given, and it is answered by the arithmetic of their angles against 60°.

An arc that is an irrational fraction of a turn — 37.3° is not, strictly, but it behaves like one for the first few dozen steps — generates points that never meet a hexagon vertex, and stepping never helps. Then the inversion construction is the only way, at its fixed price of six. The dial figure’s sea of sixes is the set of arcs whose polygons stay clear of the hexagon for longer than a cheap construction could afford to walk.

Shortcuts nobody designed #

The doubling stride and the constructions for 45°, 90° and 135° were not put into the search; it found them. That is worth noticing, because it is the most useful thing an exhaustive search does. A designer looking for a cheap construction of the centre at 20° would think of stepping three times from $A$ and finishing with two circles of the right radius. The search’s construction is subtler and uses only four circles because it steps in both directions at once: the circle centred $A$ through $B$ meets the given circle again 20° on the far side of $A$, the circle centred $B$ through $A$ meets it 40° from $A$ on the other side, and those two new points are exactly 60° apart — a chord equal to the radius, after two circles instead of three. Two circles of that radius, centred at the two points, cross at the centre.

Nobody would be surprised by that construction once shown it, and few would think of it unprompted. The search finds it because it tries everything, and it finds the comparable shortcuts at every other cheap arc without being told what to look for.

The same was true of the fewest moves to draw it, where the cheapest constructions of familiar targets were sometimes not the textbook ones, and of the lengths dividers cannot reach, where an instrument that only carries lengths turned out to reach more than expected. Constructions are short programs in a strange language, and the shortest program for a task is rarely the one a person writes first. For problems small enough to search, the search is the only way to know.

What the map says about construction problems #

The usual way to state a construction result is uniform: “the centre of a circle can be found with the compass alone”, true for every circle and every pair of points. The price is not uniform, and that is typical. The number of moves a construction needs depends on the data, sometimes heavily, and the dependence is arithmetic rather than geometric — here, on which fractions of a turn the arc is. The price of a construction counted Lemoine’s operations for constructions done once, in general position; the map above is what that count looks like when the position is allowed to vary.

The same phenomenon appears in the fewest moves to draw it, where the start was two points a unit apart and the targets were fixed; there the data could not vary, and every answer was a single number. With an arc in the data, each answer becomes a function, and the function is a record of coincidences. A designer of constructions — Lemoine’s practitioner, or Mascheroni drawing with a compass — would naturally choose convenient data, and the chart says how much that choice is worth: from six circles down to two.

What the search covers and what it does not #

The prices are established for arcs that are whole multiples of five degrees between 10° and 175°, at the search’s resolution and with its conventions — circles centred at known points through known points, crossings more than twenty radii away discarded. The lower bounds for the cheap arcs at 15°, 20°, 30°, 45°, 60°, 90° and 150° are re-run whenever the chart is drawn; the generic sixes rest on the exhaustive failures at depth five recorded when the search was first run, and on the replay of the six-circle construction at each arc.

Arcs below 10° are not on the chart, because there the search runs out of memory before it finishes depth six: with the two points close together the circles through them nearly coincide, their crossings crowd together, and the number of distinct configurations explodes. Whether a very short arc costs more than six — whether the generic construction fails when the points are too close and something longer is needed — is not decided here. There is also a question of what counts as the same construction. The search treats two constructions as one when they draw the same circles, so mirror images count separately, and the prices it reports are unaffected; but the drawings show one cheapest construction each, and for most arcs there are several, sometimes many, of equal length, of which the search reports the first it meets. And arcs that are not multiples of five degrees are covered only by the classes: an arc with $ka \equiv \pm 60°$ for some small $k$ is predicted to be cheap, and the search confirms the prediction wherever it was run, but the chart does not test arcs like 12° or 36° directly.

Still open: a formula for the price #

The classes suggest a formula — price as a function of the smallest $k$ with $ka \equiv \pm 60°$, with the $\sqrt2$ class as a second family — and the search supports it on every arc it has examined. But a formula would need two things the search cannot give. It would need a proof that no arc outside the classes is cheap, which means a lower bound that is not a search; and it would need to know what happens for large $k$, where stepping is long and the search cannot reach. Does the price keep growing like the logarithm of $k$, as the doubling shortcut suggests, or does it hit the generic six and stop, since the inversion construction works for every arc? The second must be true eventually — six is always available — so the classes with large $k$ all cost six, and the question is only where the cheap classes end.

More generally, the price of a construction as a function of its data has been studied very little. The classical results say what can be constructed; Lemoine’s geometrography and the searches say what a single construction costs; the space in between, where the data vary and the cost moves with them, has hardly been mapped for any problem beyond this one.

Coincidences, priced #

A generic arc gives the compass nothing to work with but the circle itself, and the inversion construction pays six circles to turn the circle into a line. A special arc gives it a shortcut — a chord that is, after a few steps, exactly the radius — and the price falls to five, four, three or two. The chart is a list of those shortcuts, and the remarkable thing about it is how arithmetic it is. Geometry asks where the centre is; the cost of answering depends on whether a multiple of the arc is 60°, which is a question about fractions of a turn, and which the compass, stepping round the circle a chord at a time, answers by trying.