One cut belongs to the island that would cost most alone

Two islands are three numbers, and the proof does not survive them put a sea with two islands in it on the page and priced what an angle-keeping map pays for leaving it whole. The sea carries three numbers no such map can change, the best uncut map has to be searched for, and one cut to each island releases the freedom Chebyshev’s criterion needs. Keeping a sea with two islands whole costs a few per cent of scale variation.

It ended by asking whether the sheet’s edge could be aimed — moved so that a given pair of islands sat at the cheapest arrangement of the three numbers. That question turns out to be confounded. Moving a sea’s edge moves where its islands sit in the canonical picture, so the ground being mapped changes with the design choice, and a better price might be the edge’s doing or the islands’. Measured, moving the edge sideways by eleven degrees lowered the price from 1.032 to 1.007, and most of that was the cut map getting worse rather than the uncut map getting better — the islands had moved in the picture, and nothing separated what the edge did from what the moved islands did.

There is a design choice that leaves the ground exactly where it is. A cartographer who can afford one cut, not two, still has to decide where it goes: to one island’s shore, to the other’s, or between them. The sea, the islands and the three numbers do not move. Only the cut does.

A cut is a logarithm, and the cut chooses which one #

The measurement rests on the fact the earlier essay established about cuts, and it is worth restating because it makes the question sharp.

An angle-keeping map of the sea has a scale whose logarithm is the stereographic’s own plus a harmonic function, and the map is decided by choosing that function. A map that leaves the sea whole may not use the functions that carry $\ln|z - a|$ for a point $a$ inside an island — one such function per island — because each would make the map wind round its island without closing. A cut from an island to the open sea admits that island’s logarithm and nothing else. On a band of latitude the coefficient the fit puts on it is the conformal conic’s cone constant less one, to six decimals, which is how the earlier measurement checked that a cut really is this.

So spending one cut is choosing which island’s logarithm to admit. A cut between the two islands is a third choice and a weaker one: a loop drawn round both islands together crosses no cut, so it may not wind either, and what that cut admits is only the difference of the two logarithms. And the path a cut takes does not matter to the scale at all, only which boundaries it joins — so a cut can always be routed along the shortest way across the water, and its length is simply the gap it has to cross.

Each choice is scored the same way: the scale spread of the best map it allows, and from that its share of what two cuts buy. The share is measured in logarithms of the spread — how far the one-cut map has come from the uncut one, over how far the two-cut map has come. A share of one would be a cut that does everything two would; zero, a cut that does nothing.

The central island, not the large one #

Twenty arrangements of the same two islands — the large one at four distances from the sea’s centre, the small one at five, always on opposite sides — give a clear first answer. Size matters and position matters more.

Where the large island is near the centre its cut wins everywhere, by a wide margin once the small island is pushed towards the shore: at 6° against 22°, the large island’s cut buys 93 per cent of what two cuts buy and the small island’s 2. Push the large island out and the advantage passes to the small one wherever it is more central. With the large island at 18°, its outer edge six degrees from the coast, its own cut buys between 9 and 28 per cent and the small island’s between 53 and 78 — for every placement of the small island except the one furthest out.

Where the two are equally far from the centre, the large island wins at 6°, 10° and 14°, and loses at 18°. At that distance the large island’s outer edge is six degrees from the shore and the small island’s is nine: the large island has nearly become a dent in the coast, and the channel beside it is too narrow to make it much of a hole. A ring can be drawn whole, and only one way is the limiting case in the other direction — a single island exactly at the centre, the most hole a sea can have — and there the whole price of staying uncut is carried by that island’s logarithm.

An island’s price alone names the cut #

A rule is available that does not need the two-island fit at all. Put each island alone in the sea and ask what leaving the sea whole would cost then: the best uncut map’s spread over the best cut map’s. That lone price is a number about one island, its size and its position, and it is cheap — a one-hole problem.

The island with the higher lone price is the one whose cut buys more, in eighteen arrangements of twenty. The two exceptions are near-ties by both measures at once. At 14° and 14° the lone prices are 1.0107 and 1.0108, equal to a part in ten thousand, and the shares 0.42 and 0.36. At 10° and 6° the small island’s lone price is higher, 1.0264 against 1.0212, and the large island’s cut wins by 0.64 to 0.61. Nowhere does a clear difference in lone price go with a clear difference the other way.

The lone prices themselves show why size is the weaker variable. At 6° from the centre the large island would cost the sea 3.14 per cent alone and the small one 2.64: doubling the radius adds half a point. At 14° the two cost the same, 1.07 and 1.08 per cent, and at 18° the large island is the cheaper, 0.23 against 0.44, because an island that large and that far out has nearly filled its channel to the coast. Moving an island from 6° to 18° takes nearly three points off the large one and more than two off the small. Position moves the lone price by several times what size does, and size does not even keep its sign.

The reason the rule works is the structure of the fit. A cut admits one island’s logarithm, and how much that term is worth depends mostly on the island it belongs to: how much winding the uncut map would otherwise have to be bent to avoid round it. The other island modifies that — its own presence changes how bad the uncut map is and how much room the admitted logarithm has — and that interaction is where the near-ties live.

The coefficient is not the worth #

It would be natural to read a cut’s worth off the size of the logarithm it admits — the coefficient the fit puts on that island’s term, which on a band of latitude is the cone constant less one and which what a standard parallel buys describes as deciding where a conic’s distortion vanishes. The coefficient is easy to read, and it ranks the islands the wrong way.

With the large island 18° out and the small one 6° out, the fit puts −0.034 on the large island’s logarithm when that is the one admitted, and −0.027 on the small one’s. The large island’s term is the bigger one. Its cut buys 9 per cent of what two cuts buy and the small island’s 78. A term can be large and nearly useless, because what matters is not how much the admitted logarithm bends the scale but whether it bends it where the scale’s extremes are. The spread of an angle-keeping map is set by its largest and smallest scale, and on a sea with islands those sit on the boundaries — the coast and the islands’ shores. A logarithm centred on an island far out is steep near that island and near the stretch of coast beside it, and nearly flat everywhere else, so it can pull down one extreme and leave the other where it was.

That is why the lone price is the better rule than any property of the term. It asks the whole question — how much the best map improves when this island’s winding is allowed — rather than a part of it, and a one-hole question is cheap enough to ask for every island in a sea. The same logic sits under two charts are enough, and one is not, where what an extra chart buys is read off what it does to the worst point rather than off how the chart is built.

A cut between the islands buys almost nothing #

The third way of spending a cut is the one a reader might expect to be best, because it joins the two holes and so seems to deal with both. It is the worst. Across the twenty arrangements a cut from one island to the other buys between nothing and 32 per cent of what two cuts buy, and in every arrangement at least forty points less than the better cut to the sea, which buys 42 to 93.

The arithmetic is the one set out above. A cut between the islands admits only the difference of their two logarithms, a single term that winds positively round one island and negatively round the other, and a map that needs to wind in the same direction round both, as a map of this sea does, gets almost nothing from it. It is worth most when the islands are far apart and unequal — one central and one near the shore — because then one of the two logarithms is doing almost all the work and the difference is nearly that one term alone. When the islands are placed symmetrically it buys exactly nothing: the difference of two logarithms of equal weight is useless to a map that wants them added.

Two cuts overlap near the centre and complete each other apart #

If a cut’s worth belonged entirely to its own island, the two single cuts’ shares would add to one. They do not, and the direction they miss in depends on where the islands are.

With both islands near the centre the shares add to as much as 1.61. Either cut alone buys most of what two cuts buy, because two central islands make a sea that is nearly a ring, and a ring needs one logarithm: whichever island is cut, its logarithm does much of the other’s job. That is the general form of what the earlier measurement found for two equal islands close together, where the first cut did 86 per cent of the work.

With the islands further out the sum falls below one, to 0.78: each single cut buys less than its share, and the two together buy more than the sum of each alone. There the two logarithms do separate jobs, and the second cut is worth more than it would be on its own, because the first has already moved the map to where the second’s term can be used fully. Fourteen of the twenty arrangements are complements in this sense. For a cartographer with one cut to spend, that is the case where spending it is the least satisfying: the second cut would have been worth more than the first.

Per kilometre, the long cut is the efficient one #

What a cut buys prices a cut in kilometres: a cut is a line on the map where the reader must jump, and the distortion it removes has to be set against its length. The choice here has an awkward feature on that count. The better cut goes to the more central island, and the more central island is the one further from the shore, so the better cut is the longer one — in fifteen of the sixteen arrangements where the islands sit at different distances.

It is still the better buy per kilometre, in fourteen of sixteen. The central island’s cut is longer by a factor of up to four, and it buys up to forty times as much. So there is no trade here between length and effect of the kind what a cut buys found for interrupted world maps, where each further kilometre of cut bought less than the last. The long cut is the efficient one, and a rule that minimised cut length would pick the wrong island.

What each number was compared against #

Two identical islands placed symmetrically must be worth the same cut. With both 12° across and 12° from the centre, the two single cuts buy 0.4518 and 0.4518 of what two cuts buy. A sea that could tell the two apart would be reporting something about the solver.

Every single cut must buy something and less than two cuts do. Every share across the twenty arrangements lies strictly between zero and one.

The cut between the islands must buy less than the better cut to the sea, everywhere. It does, by at least forty points of share; if it ever bought more, the comparison would be about something else.

The lone price must name the winner in most arrangements, and its misses must be near-ties. It names eighteen of twenty, and the two it misses differ by three points of share in one case and by a part in ten thousand of lone price in the other.

And the solver is the earlier measurement’s. The cut answers rest on the same harmonic fit, which reproduces the stereographic’s closed form for a sea with no island and the conformal conic’s for a band of latitude, to a part in ten thousand and better.

Where the sea is a model #

Islands are circles and the sea is a cap. The closed form that puts the sea in canonical position needs circles, and a real coastline is not one. What survives the assumption is the mechanism — a cut admits one logarithm, and its worth belongs mostly to its island — and the numbers are the circles’.

The uncut map is fitted, not proved. Every uncut spread is the best found in a stated family of harmonic functions, so it is an upper bound on the true best, and every share measured against it inherits that. The comparison between cuts is between fits made the same way, and the symmetric control bounds the solver’s own asymmetry at a part in ten thousand.

One sea, two island sizes. The sea is 60° across and the islands 12° and 6°, and nothing here says how the balance between position and size moves when the size ratio does. The finding that position outweighs doubling is a finding about a factor of two.

A cut costs nothing here but its length. Giving up continuity is the reminder that a reader pays for a cut in ways a kilometre count does not capture — a coastline split, a route broken — and Chebyshev’s criterion is what the two-cut map reaches and the one-cut map does not.

Still open: which two of three #

With two islands, a cartographer with one cut has a rule: cut the island that would cost the sea most alone. With three islands and two cuts the question has more shape. The lone prices would say to cut the two most expensive islands, but the substitutes-and-complements result says that two central islands’ cuts overlap, so the two most expensive islands may be exactly the pair whose cuts duplicate each other, and the better pair may include a cheaper island whose cut does a separate job.

Whether the lone-price rule survives a third island, whether the best pair of cuts is ever a pair that does not include the single most expensive island, and whether a cut joining two islands becomes worth spending once a third is there to receive a second cut, are questions two islands cannot ask.