The water would join Goode's map at seventy-five degrees

The ocean centres its lobes on the land’s cuts, and cannot keep its own cuts short found the interrupted homolosine’s best layout for the water — where the cuts run and where each lobe’s central meridian sits, at Goode’s count of two lobes north and four south — and set it beside the land’s. Both searches kept one thing fixed: the latitude at which Goode’s composite changes from the sinusoidal to a Mollweide cap, 40° 44′, chosen by Goode because the two maps’ parallels have the same length there. The land puts the homolosine’s join eleven degrees higher had already found that the land, scored on its own, wants that join higher, and the land would move Goode’s lobes before his join that on the land’s own lobes it wants it near sixty degrees.

The ocean essay closed by asking whether the water moves the join too. Its guess was that it might move it the other way. The water’s area sits differently from the land’s: only 27 per cent of the northern water lies above Goode’s join, against 47 per cent of the northern land, while in the south 32 per cent of the water lies in the single band from 40° to 65°S — the Southern Ocean, just poleward of the join, where the Mollweide cap takes over.

The guess was wrong in direction, and the reason it was wrong is the most useful thing the measurement finds.

The water’s join is at seventy-five degrees #

The measurement is the earlier search with one more freedom. At every join latitude on a five-degree grid, from 25° to 89°, Goode’s composite is rebuilt with its Mollweide cap rescaled to meet the sinusoidal there — still an equal-area map, still continuous along every parallel — and the best layout at Goode’s count of lobes is searched again exactly, cuts and meridians on the same five-degree grid, scored by the mean angular deformation of every half-degree cell of water to 88°, weighted by area, at the map’s own best proportions.

The water’s score falls steadily as the join rises. At Goode’s own join it is 13.31°, the earlier essay’s number. At 50° it is 12.89°, at 60° 12.52°, at 70° 12.29°, and it bottoms out at 75° with 12.27°, a degree and five hundredths better than Goode’s. Beyond that it rises by hundredths: 12.27° at 80° and 12.30° at 89°, where the cap has shrunk to a sliver round the pole and the map is all but the plain sinusoidal cut into lobes.

The land, searched the same way on its own cells outside Antarctica, bottoms out at 60° with 13.83° against 14.40° at Goode’s join, and then climbs steeply — 14.36° at 70°, 14.54° at 89°. That is the land’s own layout’s earlier answer, now found jointly with the layout. The water wants its join fifteen degrees higher than the land does, and gains twice as much by moving it: 1.05° against the land’s 0.57°.

The layout moves with the join, and hardly matters #

The dashed line in the figure is the ocean’s layout found at Goode’s join, held fixed while the join moves. It sits within five hundredths of a degree of the layout searched afresh at every join. At 75° the held layout scores 12.31° and the searched one 12.27°.

That understates how much the search actually changed. At every join up to 45° the best layout is the earlier essay’s: the edge at 40°E, through Africa and Siberia, one northern cut through North America at 110°W and southern cuts at 135°E, 140°W and 60°W. From 50° up the best layout is a different one. Its edge moves to 110°E and then 115°E, through Siberia and Western Australia; its northern lobes are centred on 175°W, the North Pacific, and on 30°W, the Atlantic with Europe and Africa beside it; its southern lobes on 170°E, 110°W, 20°W and 70°E, with cuts through the open Pacific at 150°W, through South America at 65°W and through southern Africa at 25°E.

Two quite different arrangements of lobes are, for the water, within a few hundredths of a degree of each other at every join, and the join decides which of them comes first. That is a property of the ocean the earlier essay could not have seen with the join held: the water has no single best way to be cut, but a family of nearly equal ones, and the order within that family is set by details as small as where the cap begins.

The gain is in the polar water #

Where the gain comes from is visible band by band. With the layout held at the water’s own and only the join changed, the whole gain is 1.07°, and it is concentrated at both ends of the latitude range. The water poleward of 60°, north and south, gains 1.62° — more than the whole — and the bands from 40° to 55° give back 0.73°. North of 60° every band gains, and the bands from 75° to 85° — the Arctic basin, the Barents and Kara Seas, the channels of the Canadian archipelago — gain most. South of 60° the same thing happens, from the edge of the Antarctic ice to the southern Southern Ocean. Between 40° and 55° in both hemispheres the water loses, and in the south that loss falls on exactly the band the earlier essay pointed to.

The mechanism is in what Goode’s cap does to a lobe’s own central meridian. The sinusoidal keeps every central meridian free of angular deformation all the way to the pole: along it the map is locally a true scale in both directions. The Mollweide does not. Its parallels are spaced to keep area as they shorten, and on its central meridian that stretches the map north–south against east–west; rescaled to meet the sinusoidal at 40° 44′, Goode’s cap carries 43° of angular deformation on the meridian at 75° and 58° at 80°, at the proportion both maps share. Joined at 75°, the cap has only just begun there: 0° on the meridian at 75°, 16° at 80°. Away from the meridian the same holds by a smaller margin, at 75°N the higher join deforms the water sixty degrees of longitude from its meridian by 45° against Goode’s 64°.

Between the two joins the balance reverses. At 55°, sixty degrees from the meridian, Goode’s cap deforms the water by 35° and the sinusoidal it replaces by 46°: the cap was helping the water far from the meridian there, and a higher join throws that help away. So moving the join up gains everywhere poleward of the new join and loses in a band above the old one, and the question is only which there is more of.

Why the water’s join goes higher than the land’s #

The map makes the earlier guess’s error plain. That guess counted the water above Goode’s join — less of it than of the land, in the north — and supposed the water would care less about the cap. What matters is not how much water lies above the join but where it lies within its lobe, and how far poleward.

The losing water is the water just above Goode’s join and far from a lobe’s meridian, the pale wedges at the lobes’ edges between 40° and 55°. The gaining water is everything poleward of about 60°, near the meridian or not, and the faint gain across the tropics comes from the map’s proportions, which the higher join lets settle slightly differently. The land’s area is concentrated exactly in the losing band: Europe, the steppe, Canada below the Arctic, much of it far from the meridians of lobes several continents wide. The land above 70°N is a thin fringe — under 4 per cent of the northern land — and south of 60° there is none at all outside Antarctica. So the land stops moving its join at sixty degrees, where the next five degrees would cost its heartland more than they bought its fringe.

The water has a heartland of its own above sixty degrees in both hemispheres. The Arctic Ocean and its marginal seas are 7.3 per cent of the northern water above 70°; the Southern Ocean between 60° and 70°S is another band the land does not have. Both gain from the higher join at every longitude, and they outweigh the water in the losing band — the Southern Ocean’s northern half among it. The band the earlier essay picked out as the reason the water might lower the join is real and does pay, but the water south of it, nearer Antarctica, gains more than it pays.

A join for each hemisphere is worth a few hundredths #

Each hemisphere of an equal-area composite can be joined at its own latitude, since each half meets the other only along the equator, where both spacings are one. That is a freedom Goode had and did not need, and the water has reason to use it, because the Arctic and the Southern Ocean sit at different latitudes and in different shapes.

It is worth very little. Searched over every pair of joins on a ten-degree grid, with the layout searched again at the best pair, the water’s best is 80° in the north and 60° in the south, at 12.231° against 12.266° for one join at 75° — three and a half hundredths of a degree. The grid shows why. Along the northern axis, with the southern join at 60°, the score falls by more than a degree from 30° to 80°; along the southern axis, once the join is above 60°, nothing changes by more than five hundredths. The southern water’s polar gain is mostly from the band between 60° and 70°S, which every southern join of 60° or more keeps sinusoidal, and the Antarctic coast leaves little water poleward of that to decide between them.

The land’s own split, measured the same way, is 60° north and 40° south, worth a hundredth of a degree: there is almost no land south of 40° outside Antarctica to be served by the southern join at all, so the land is indifferent to it.

The jump between the two kinds of map survives #

The ocean essay’s last finding was that no single layout serves both surfaces well. Weighting the water by t and the land by 1 − t, each by its own area, the blended score’s best layout is a land layout up to t = 0.72 and jumps to an ocean layout at t = 0.74, with nothing in between, and the blended score’s own floor proved that no layout came within 2.57° of both optima at once.

With each weight free to choose its join, the jump is where it was. Every weight up to 0.72 chooses a land layout joined at 60°; from 0.74 every weight chooses an ocean layout, joined at 60° at first and at 80° as the weight on the water approaches one. The land layouts are better than before on both surfaces — at a tenth of the weight on the water, 13.94° on the land and 17.10° on the water, against 14.51° and 16.99° with Goode’s join — but they remain land layouts, and the ocean layouts remain ocean layouts.

And the gap between the two optima widens. A free join improves each surface’s best by more than it improves any compromise, because each surface wants a different join and a compromise must pick one. Measured over every layout at every join on the grid, the floor rises from 2.57° to 3.30°: no layout, at any of the joins searched, comes within 3.30° of both the land’s best and the water’s best at once. Freeing the join made an interrupted map for both surfaces a worse idea, not a better one.

What a chart of the oceans can take from this #

Goode’s join was never a choice about either surface. Joined where the parallels agree and the meridians turn a corner set out what it is: the one latitude at which the sinusoidal’s parallel and the Mollweide’s have the same length, so that the two pieces meet without a step along any parallel, at the price of a corner in every meridian but the central one. It is a fact about the two maps, and it would be the same on a planet with no land at all.

Scored on what the map is for, the join stops being fixed and starts being a parameter, and the two surfaces set it differently. For a land map the measurements on this ground now agree that the join belongs near sixty degrees. For an ocean map it belongs near seventy-five, and the cap above it is so small that the map is very nearly the plain sinusoidal cut into lobes — an interrupted sinusoidal with a polar cap a few degrees wide, rather than the homolosine as Goode drew it. A cartographer drawing the oceans with a homolosine would gain a full degree of mean angular deformation by drawing a different homolosine, with no change to the lobes and none to the cuts’ cost in water, which the earlier essay found already high and hard to lower.

The corner in each meridian moves with the join, and at seventy-five degrees it sits where only the Arctic’s water and the Antarctic coast see it. That is a cost on the land it was drawn for, the homolosine wins by its meridians priced as part of what the lobes buy, and on an ocean map it is moved to where the fewest readers look.

How the joins were checked #

At Goode’s join the joined search must be the ocean search. Run with the cap joined at 40° 44′, the new search must return the earlier essay’s ocean layout, its edge and its score to a billionth of a degree. It does: 13.312°, edge at 40°E.

A searched layout is never worse than a held one. At every join, the layout searched afresh must score no worse than the ocean’s layout from Goode’s join held fixed, beyond a thousandth of a degree. It never does.

Two equal joins are one join. The same layout scored with the northern and southern joins both at 50° must give exactly the score of one join at 50°. It does.

The finding has to be there to fail. The water’s best join must lie above the land’s and the land’s above Goode’s, and moving it must be worth more to the water than to the land. They are 75°, 60° and 40.7°, and 1.05° against 0.57°.

Where the joins stop #

One family of map. Every map here is Goode’s composite — the sinusoidal to a latitude, a rescaled Mollweide beyond it. A curve with no join did better on the land than any join, and the water’s preference for keeping the sinusoidal almost to the pole suggests a smooth curve that stays flat further and closes faster would do better on the water too. That is not searched here.

A five-degree grid of joins and a ten-degree grid of pairs. The water’s curve is flat to a hundredth between 70° and 80°, so its best join is a region rather than a point, and seventy-five is that region’s middle. The hemisphere pairs are searched on a coarser grid because each pair is a full scoring of the water; a finer grid could move the pair by a few degrees and the score by less than a hundredth.

The proportions are one number for both hemispheres. A composite joined differently north and south could also be stretched differently in each, and nothing here lets it.

Water by area, to 88°. The Arctic basin weighs heavily in the result, and a map that left out the ice-covered Arctic, or weighted water by its traffic, would put the join lower. The area weighting was a readership all along found that every such weighting is a choice of reader; the polar water’s influence here is the area weighting’s choice made visible.

Still open: whether the water wants a cap at all #

The water’s score at a join of 89° — the plain sinusoidal cut into lobes, with a cap a degree wide — is three hundredths of a degree worse than its best. The cap the water keeps, between 75° and the pole, is buying almost nothing, and its one job is to close the parallels towards a pole that the water reaches only in the Arctic basin. A map for the water might do without the cap and instead close the parallels some other way, or interrupt the Arctic separately as what a cut buys would price it.

Whether a smooth spacing curve fitted to the water, with no join at all, beats the best composite by as much as it beat the composite on the land; whether its proportions move with it; and whether the Arctic basin is better served by a lobe of its own than by any closing of the parallels, are questions a composite with one join cannot ask.