The ocean centres its lobes on the land's cuts, and cannot keep its own cuts short

The land would move Goode’s lobes before his join found the interrupted homolosine’s layout exactly, for the land. Scored by the mean angular deformation of the land outside Antarctica, each point read from the meridian of its own lobe, the best layout at Goode’s count of two lobes north and four south puts the map’s edge in the Atlantic and its northern cut in the western Pacific, and deforms the land by 14.40° against Goode’s 16.51°. Its cuts run through water because that is where the land is not.

An interrupted map is also drawn for the other surface. Goode published ocean-centred versions of his projection, with the continents cut and the oceans whole, and the same idea appears in every atlas that maps currents, fisheries or shipping. That map answers to a different score — the deformation of the water — and to a different cost for a cut, since a cut through open sea is the one a reader of an ocean map must jump. Whether its best layout is simply the land’s turned inside out, and whether one layout could serve both surfaces at once, were questions a search scored on the land could not ask.

The search does not care which cells it is given. Run on the water, it answers both.

The same search, given the water #

Nothing about the layout changes except what it is scored on. The map still has one outer edge, a full meridian shared by both hemispheres, and each hemisphere is still divided into lobes by straight half-meridian cuts from the equator to the pole. The spacing curve is still Goode’s composite — the sinusoidal below 40° 44′, the Mollweide above, joined where the parallels agree — so that every difference below belongs to the layout. Cuts and meridians sit on the same five-degree grid, every lobe is at least thirty degrees wide, and the recursion that visits every arrangement of cuts is the one the land’s layout came from.

What changes is the set of cells. The score is now the mean angular deformation over every half-degree cell of the raster that is not land, weighted by its area, each read from the meridian of the lobe it falls in. The Southern Ocean is included, down to the Antarctic coast, and so is the Arctic basin; as on the land, nothing poleward of 88° is scored. Water is 71 per cent of the sphere, so this is a score over most of the map rather than a score over a set of islands in it.

The cost of a cut changes too. On the land’s map a cut was charged by the land it crossed, because a coastline drawn twice and a country in two pieces is what a reader of a land map pays for one. On an ocean map the payment is the other way round: a current that leaves one lobe and reappears in another, a shipping lane broken at the sheet’s edge — the loss giving up continuity describes, charged to the water instead of the land. So a cut here is charged by the water it crosses from the equator to its pole, measured on the same raster.

Five meridians where the land put its cuts #

At Goode’s count of lobes the ocean’s best layout deforms the water by 13.31°. It is recognisably an ocean map. In the north, one lobe runs from 40°E eastward across Asia and the whole Pacific to 110°W, centred on 175°E; the other holds the Atlantic, centred on 40°W. In the south there are four: the Indian Ocean on 85°E, the western Pacific on 180°, the eastern Pacific on 105°W, and the South Atlantic on 10°W. The cuts go through Africa and Siberia (the edge, at 40°E), North America (110°W), Australia (135°E) and South America (60°W), and one runs down the open Pacific at 140°W.

The first reading — that this is the land’s layout turned inside out — is very nearly right about where the lobes sit. Five of the ocean’s six central meridians lie within twenty degrees of a cut the land’s own layout chose: its northern Pacific meridian at 175°E against the land’s northern cut at 165°E, its Atlantic meridian at 40°W against the land’s edge at 20°W, the Indian Ocean’s 85°E against a southern cut at 70°E, the western Pacific’s 180° exactly on the land’s southern cut there, and the South Atlantic’s 10°W against the edge at 20°W. The reverse holds as well: five of the land’s six meridians lie within fifteen degrees of an ocean cut, and the sixth, Eurasia’s at 65°E, within twenty-five. South America’s lobe is centred on 60°W in the land’s layout and cut at 60°W in the ocean’s.

The one exception is the telling one. The ocean’s eastern Pacific lobe, centred on 105°W, is seventy-five degrees from the nearest land cut. The land’s layout has no cut in the eastern Pacific because it has no use for one: its four southern lobes are Africa on 25°E, Australia’s two halves on 125°E and 145°E — the land’s best layout cuts Australia at 135°E rather than give a lobe to open water — and South America on 60°W, and the whole eastern Pacific sits inside South America’s lobe as empty space. The water in the same place is the largest single body of it on Earth. One lobe for the South Pacific, from Australia to South America, would be 165 degrees wide, with its edges more than eighty degrees from its meridian where the sinusoidal’s shear is worst, and the ocean’s layout spends its fourth southern lobe on splitting it.

Each layout on the other’s ground #

Scored on each other’s surface, the two optima are poor. The land’s layout deforms the water by 19.75°, six and a half degrees more than the ocean’s own; the ocean’s layout deforms the land by 26.18°, nearly twelve degrees more than the land’s. The ocean’s layout puts a cut through every inhabited continent, and a continent cut in two has each half at the far edge of a lobe, where the shear is worst. The land’s layout cuts little but Australia, and it leaves the Pacific inside a lobe built for South America.

Goode’s own layout sits between them on the water, at 18.12°, which is 1.63° better than the land’s optimum there while being 2.11° worse on the land. His layout differs from the land’s in several places, and one of them is a southern lobe centred on 160°W that holds almost no land at all — the kind of lobe the land’s search, finding little there to pay for it, spends instead on cutting Australia in two.

The third pair of bars measures what such a lobe is worth, and it comes from the blended score set out below. The land’s layout with Australia rejoined in one lobe on 135°E, a lobe from 170°E to 105°W on a meridian at 155°W, and the edge moved five degrees east, deforms the land by 14.51°, a tenth of a degree more than the land’s optimum, and the water by 16.99°, 2.76° less. It also cuts less land: 3,392 kilometres against 4,670, and no single cut more than 2,002 against 2,669. Of all the numbers here, that exchange is the one a cartographer drawing a land map would want to know.

Every meridian crosses thousands of kilometres of water #

The inversion stops at the cuts. Charged in water, the ocean’s layout crosses 32,526 kilometres of it: 10,175 at its edge, which runs from the North Pole through the Barents Sea, the Indian Ocean and the Southern Ocean to Antarctica; 8,340 at the open-Pacific cut; from 4,615 to 4,726 at each of the three cuts through continents, which cross the Arctic Ocean or the Southern Ocean beyond them. The land’s layout, charged in land, crossed 4,670 kilometres and never more than 2,669 at one cut. The ocean’s search did not choose badly; held to shorter cuts it can do little better, as the frontier below shows.

The figure is the reason. Measured every five degrees round the globe, a full meridian from pole to pole crosses less than a thousand kilometres of land outside Antarctica in only two places — the Atlantic at 20°W and the Bering Strait — and the land’s layout used both. The same meridians never cross less than 7,172 kilometres of water, and that minimum is at 25°E, a meridian that runs the length of Africa and Europe and still crosses the Arctic Ocean, the Mediterranean and the whole Southern Ocean. There is no gap in the water for a cut to go through.

That is a statement about the shape of the two surfaces rather than about their areas. The land is several pieces with water between them, so a cut can be threaded through a gap. The ocean is one connected body that surrounds every piece of land and closes round both poles: in the south it is a ring, the Southern Ocean, that every southern meridian must cross between the southern tip of whatever continent it passes and the Antarctic coast; in the north it is the Arctic basin, which every northern meridian crosses on its last stretch to the pole. The second cut goes across the sea found the same distinction on a single sea with islands in it — cuts go where the pieces are not — and on the whole Earth the water is the one piece there is.

The frontier shows what can be bought. Told to keep every cut under a stated length of water, the ocean’s search gives up almost nothing until nine thousand kilometres — 13.34° against 13.31° — and then climbs steeply: 14.03° with a worst cut of 7,784 kilometres and 14.19° with 7,450. Below seven thousand there is no layout at all, since the edge alone must cross at least 7,172. The land’s frontier, measured the same way in land, reached a worst cut of 890 kilometres while giving up a tenth of a degree.

So the ocean’s map is the land’s turned inside out in where it centres its lobes, and not at all in what its cuts cost. What a cut buys priced a cut by what it removes from the distortion; on a land map that price can be paid through open sea at almost no cost to the reader, and on an ocean map it cannot be paid anywhere for less than seven thousand kilometres of broken water.

What each further lobe is worth to the water #

The count of lobes answers differently too. On the land, a fourth southern lobe was worth almost nothing — four hundredths of a degree with Goode’s worst cut as a limit, a tenth with cuts free — because the southern land is three continents and three lobes serve them. On the water the fourth southern lobe is worth 1.45°, taking the score from 14.76° to 13.31°, because the southern water is not three bodies but one, and its largest part, the Pacific, is too wide for a single lobe. At three southern lobes the ocean’s search gives the whole Pacific one lobe on 155°W and pays for it at the lobe’s edges; at four it splits the Pacific at 140°W.

Elsewhere the two counts run close. An uninterrupted sheet on its best meridian deforms the water by 31.55° and the land by 29.11° — the gap on the land it was drawn for, the homolosine wins by its meridians measured between cutting and not — and the second lobe in each hemisphere does most of the work on both, taking the water to 17.87° and the land to 15.82°. A third northern lobe is worth 1.36° to the water, splitting the North Pacific in two at 175°E, and 3.88° to the land, though for the land that cut runs through a continent. The ocean is not easier or harder to serve with lobes than the land. It asks for them in different places.

One layout for both #

The obvious compromise is a score that counts both surfaces. Weighting the water by a stated share t and the land by 1 − t, each by its own area, the blended score is still a sum over cells, so the same recursion finds its best layout exactly for every t. At t = 0 that is the land’s layout and at t = 1 the ocean’s. What happens between them is the whole of the answer.

The blended score does not move smoothly between the two. A tenth of the weight on the water changes the land’s layout by the one lobe described above, and the water’s score falls from 19.75° to 16.99° while the land’s rises by a tenth of a degree. From there to t = 0.72 the choice barely moves: the land’s layout with a Pacific lobe, the water between 16.99° and 16.67°, the land between 14.51° and 14.96°. At t = 0.74 it jumps. The layout becomes an ocean layout, cutting continents, and the land’s score goes from 14.96° to 21.54° while the water’s falls from 16.67° to 14.17°. No weighting chooses anything in between.

A jump of that kind is what a non-convex set of options produces, and it means the weighted score cannot be used to find a compromise in the gap. It can be used to rule one out. Every layout’s blended score is at least the blended optimum, so a layout within d degrees of both the land’s best and the ocean’s must have a blended score within d of (1 − t) × 14.40° + t × 13.31° — at every t. The largest shortfall, at t = 0.72, is 2.57°. So no layout of this kind — straight cuts on the five-degree grid, two lobes north and four south — comes within 2.57° of both optima at once, whether or not the search could find it, and this is a bound rather than an estimate.

The best compromise the weighting did find is the land layout at t = 0.72, which gives up 0.55° on the land and 3.35° on the water. Between that and the bound there may be layouts the weighting cannot reach — a layout that cuts one continent and not the others, say — and nothing here measures them. What is settled is that an interrupted map for both surfaces is a choice between two kinds of map, and that the cheap part of the choice is the first tenth: a land map loses almost nothing by giving the Pacific its own lobe, as Goode’s did.

The average was a choice of norm found that every score here is a decision about what to count, and the area weighting was a readership all along that weighting by area is a decision about who reads the map. The blended score makes that decision explicit. A reader who wants a land map with honest oceans can weight the water anywhere between a tenth and seven tenths and get a land layout with a Pacific lobe every time, its two scores never moving by more than half a degree across the range.

How the layouts were checked #

The search on the land must be the land’s own search. Given the land’s cells and its cuts charged in land, the general search must return the layout the land would move Goode’s lobes before his join found, with the same edge and the same score to twelve decimal places. It does: 14.4018°, edge at 20°W.

The ocean layout’s score must be its cells’ mean. Scored cell by cell at its own proportions, the ocean’s layout must give the mean the recursion computed, to a part in a billion. It does.

More lobes contain fewer. A layout of more lobes can always reproduce one of fewer, so the ocean’s best score must never rise as lobes are added, across the seven counts measured. It does not.

The findings must be there to fail. The ocean’s layout must beat the land’s on the water by more than three degrees (it beats it by 6.44), and the blended score’s floor must rule out any layout within a degree of both optima (it rules out 2.57°).

Where the ocean’s layout stops #

The same spacing curve and join. The ocean’s layout is found with Goode’s composite joined at 40° 44′. The land puts the homolosine’s join eleven degrees higher found the land’s best join at 51.7° on Goode’s lobes and near sixty on its own. The water may move its join too, by an amount and in a direction nothing here measures.

Water by area, and every cut charged the same. The score counts a square kilometre of the Southern Ocean like a square kilometre of the Caribbean, and the cut cost counts a kilometre of the Drake Passage like a kilometre of the central Pacific. A map for shipping would weight the lanes; a map for currents would charge a cut by the flow it breaks.

Straight half-meridian cuts on a five-degree grid. A cut allowed to bend could thread a gap the straight ones cannot — round the southern tip of Africa, through the Drake Passage — though no bend removes the Southern Ocean itself, which every route from a continent to the pole must cross.

The land at 1:110m. Natural Earth’s coarsest land, in which Antarctica’s ice shelves are land and the Arctic’s sea ice is water. An ocean map that drew the coast at the grounding line instead would move the Southern Ocean’s edge and the scores near it.

One interrupted map, not two sheets. An atlas can always print both maps, and the bound above is the measured case for doing so: at Goode’s count of lobes, no single sheet of this kind comes within two and a half degrees of both.

Still open: whether the water moves the join as the land did #

The two searches used one join, Goode’s, and the land’s own search found that moving it was worth half a degree once the lobes were free. The water’s area sits differently from the land’s. Of the northern water only 27 per cent lies above Goode’s join, while in the south 32 per cent of the water lies in the band from 40° to 65°S alone — the Southern Ocean, just poleward of the join, where the Mollweide cap takes over from the sinusoidal and the ocean’s lobes are widest.

Whether the ocean’s best join sits below Goode’s where the land’s sat above it, whether a join chosen separately for each hemisphere is worth more to the water than to the land, and whether the jump in the blended score survives when each weight is allowed its own join, are questions a search with the join held fixed cannot ask.