The land would move Goode's lobes before his join

The land puts the homolosine’s join eleven degrees higher gave the join between the sinusoidal and the Mollweide an objective at last. Scored on the land outside Antarctica, each point read from the meridian of the lobe Goode put it in, the best join is at 51.7° rather than his 40° 44′, and moving it there gains a fifth of a degree.

It ended by pointing at the larger freedom. A homolosine is not only a join; it is a layout — where each cut between lobes runs, and where each lobe’s central meridian sits — and on the land it was drawn for, the homolosine wins by its meridians had found Goode’s placement of his meridians worth 1.63° to his own map, eight times what moving the join is worth. Goode placed them by eye, in 1923, so that each continent would sit near the middle of a lobe. Whether the land’s best layout keeps his lobes, how much of what his meridians earned he left on the table, and whether the join moves again once the lobes are free were the questions one parameter could not answer.

The layout can be found exactly, and it is not Goode’s.

A layout that can be found rather than drawn #

A layout here has the parts Goode’s has. The map has one outer edge, a full meridian where the left side of the sheet meets the right, shared by both hemispheres. Each hemisphere is divided into lobes by half-meridian cuts running from the equator to its pole, and each lobe has a central meridian from which its land’s distortion is read. The spacing curve is Goode’s own composite — the sinusoidal below the join and the Mollweide above it, which joined where the parallels agree, and the meridians turn a corner took apart — joined at his latitude, so that everything measured below is the layout’s doing and nothing else’s. The score is the one the join essay used, and the one scored lobe by lobe, the homolosine loses to its own lower half first applied to a cut map: the mean angular deformation over every land cell outside Antarctica, weighted by area, each cell read from its own lobe’s meridian.

Cuts and meridians are placed on a five-degree grid, which is a grid Goode’s own layout sits on exactly. On it, the search has no guesswork in it. A lobe’s contribution to the score depends only on which columns of land it contains and where its meridian is, so summed once per column for every candidate meridian, the cost of any lobe is a difference of two running totals. The best way to divide a hemisphere into $K$ lobes is then the best way to divide some earlier stretch of it into $K - 1$, plus the best single lobe from there to the edge — a recursion that visits every arrangement of cuts and returns the optimum. Every lobe is held to at least thirty degrees wide, or the recursion would spend a lobe on an empty sliver of ocean. The aspect of the map, the one number that stretches it sideways, is refitted to each layout it finds.

Two degrees, from the lobes alone #

At Goode’s own count of two lobes in the north and four in the south, the land’s best layout deforms it by 14.40° on average. Goode’s deforms it by 16.51°. The difference, 2.1°, is larger than the 1.63° the join essay’s predecessor found his meridians to be worth against no placement at all, and ten times what moving the join is worth. Goode’s instinct about lobes was right, and his layout was the least exact part of it.

The layout the land chooses is recognisably a homolosine and recognisably not Goode’s. In the north its edge moves from the Pacific to the Atlantic, at 20°W. Its northern cut moves into the western Pacific, at 165°E, so that one northern lobe holds Eurasia and Africa’s north on a meridian at 65°E and the other holds the Americas on 100°W — the same meridian Goode gave North America. In the south the lobes shift east: Africa on 25°E, Australia on 135°E, a narrow lobe for New Zealand and the Pacific on 175°E, and South America on 60°W, again Goode’s own choice.

Two of Goode’s six meridians survive exactly. What changes is where the Old World is cut from the New: Goode split the northern land in the Atlantic and put the edge of the whole map through the Bering Strait; the land would split it in the Pacific and put the edge down the Atlantic.

The figure shows where the two degrees come from, and it is not a uniform improvement. Weighted by area, the land that gains contributes 4.40° to the mean between them, and the land that loses gives back 2.36°. The gains are almost all Asia’s, 4.05° of the mean: on Goode’s map it sits far from his northern meridian at 30°E — eastern Siberia is 150° of longitude from it — and on the land’s layout it sits under a meridian at 65°E. The losses are North Africa and the Middle East’s, 1.27°, and Europe’s, 0.87°, which sat close to Goode’s 30°E and are now thirty-five degrees further from the nearest meridian. Every other continent moves by less than four hundredths of a degree. A layout is a trade between continents, and the land’s layout makes the trade in favour of the largest of them.

A cut is also land cut in two #

A layout scored only on distortion would put its cuts wherever distortion is least, and distortion is least when every continent sits on a meridian, which may mean cutting through the land between them. What a cut buys priced a cut by its length, and giving up continuity is the reminder of what a reader pays for one that runs across land: a coastline drawn twice, a country in two pieces, a route that leaves one lobe and reappears in another.

So each cut’s length over land is measured from the same raster, and the search can be told not to use any cut that crosses more than a stated length. Measured that way, Goode’s layout is not a layout that avoided cutting land. His northern cut at 40°W runs the length of Greenland, 1,946 kilometres of it, and his map’s edge at 180° crosses 445 kilometres of the Chukchi Peninsula. His three southern cuts cross nothing at all.

The free optimum, at 14.40°, has a worst cut of 2,669 kilometres. Told to use no cut crossing more land than Goode’s worst, the search returns the layout drawn in the first figure, at 14.48°, whose cuts cross 2,057 kilometres in all against Goode’s 2,391, and whose worst cut crosses 1,056 against his 1,946. It is better than Goode’s on both counts at once.

Pushed further, the frontier is almost flat. A layout whose worst cut crosses 890 kilometres still scores 14.50°. Below that there is none: no layout at this count keeps every cut under 750 kilometres, because every northern half-meridian from the equator to the pole crosses land somewhere — Greenland, Canada’s islands, Siberia, Scandinavia or Alaska — and the least of them cross hundreds of kilometres. The Arctic is ringed by land, and a northern cut has to go through it. What a layout can choose is where, and the land’s choice goes through less of it than Goode’s did.

How sharply the land has chosen #

An optimum found exactly on a grid can still be a broad one, and this one is broad in the places Goode cared about least and narrow in the one he fixed first. Held at the land’s layout with only the northern cut moved, and both northern meridians re-chosen for each position, the land’s score barely changes across forty degrees of the Pacific: 14.55° with the cut at 145°E, 14.49° at 155°E, 14.48° at 165°E, 14.50° at 175°E, 14.52° at 175°W, 14.54° at 165°W. Anywhere from Japan’s longitude to Hawaii’s is within seven hundredths of a degree of the best.

The land those positions cut is not flat at all. At 145°E a northern cut crosses 1,612 kilometres of Siberia; at 165°E, 1,056; at 175°W, 222; at 170°W a northern cut would cross only 56, a few islands in the Bering Sea. A layout that put its northern cut at 175°W would give up 0.045° of the land’s score and cut a fifth as much land. On this axis the choice is not about distortion at all but about what a reader would rather see cut, and the measure has almost nothing to say about it.

The edge is the opposite. Moved from 20°W to 30°W the score rises by two hundredths of a degree and the edge crosses 1,724 kilometres of Greenland instead of the 890 of Iceland and north-east Greenland it crosses at 20°W; at 10°W, 2,724 kilometres of Africa and Europe; at 0°, 14.78° and 4,670. Measured every five degrees round the globe, a full meridian from pole to pole crosses less than a thousand kilometres of land in only two places: the Atlantic at 20°W, and the four meridians from 180° to 165°W around the Bering Strait. The land’s layout puts the edge of the whole map in the first. Goode put it in the second, and ran his northern cut down the Atlantic instead; the land prefers the reverse.

So what the land has chosen sharply is the edge. Where the Pacific cut falls within the Pacific, it has hardly chosen at all.

What each further lobe is worth #

The count of lobes is the other of Goode’s choices, and the recursion answers it for every count at once. One lobe a hemisphere, the uninterrupted map on its best meridians, deforms the land by 29.11°. A second lobe in each hemisphere takes that to 15.82°, most of the whole gain an interruption can make. A third southern lobe buys another 1.30°. The fourth, which Goode has, buys four hundredths of a degree: 14.52° to 14.48°. The southern continents are three, and three lobes serve them; the fourth serves the Pacific.

More northern lobes are worth a great deal if they may go anywhere — a third takes the free optimum to 10.52°, and four north with six south to 9.18° — because a cut between Europe and Asia puts each on its own meridian. But that cut runs through a continent, and the free layout with three northern lobes has a cut crossing 4,893 kilometres of land. Held to Goode’s worst cut, a third northern lobe buys 0.70°, and four north with six south only 0.21° more. Goode’s two northern lobes are the right number for a map that cuts no continent in half, and the land’s layout makes better use of the same two.

The join climbs again #

The join essay found the land’s best join at 51.7° on Goode’s lobes, and said that moving the lobes would move the join with them. It does, and further in the same direction. On the land’s own layout the composite joined at Goode’s latitude deforms the land by 14.48°; joined at 50°, 14.09°; at 55°, 14.01°; at 60°, 13.93°; at 65°, 14.15°. The best join sits near sixty degrees, and moving it there gains 0.55° — more than twice what moving it gained on Goode’s lobes.

The reason is where the land’s layout puts its northern meridians. With Asia under 65°E instead of far to the east of 30°E, the high northern land is closer to a meridian than it was, and the sinusoidal, which is better than the Mollweide near a meridian and worse far from it, can be carried higher before the Mollweide’s cap is worth switching to. The join and the layout are not independent choices, and the order in which they are made matters: fitted after the lobes, the join gains more than it did before them.

Together the two changes take the land from 16.51° to 13.93°, two and a half degrees, and four fifths of that is the layout.

What Goode got right, measured #

It would be easy to read this as a correction of Goode, and it is only partly one. His two northern lobes are the right number for a map that will not cut a continent. His southern cuts cross no land at all, which none of the land’s layouts manages. Two of his six meridians — 100°W for North America and 60°W for South America — are exactly where the land puts them. And his map, scored on the land, beats every uninterrupted map in the library by a wide margin, including the smooth curve one curve beats the three maps it would replace fitted: the land’s best single-lobe layout is 29.11°, almost twice his 16.51°.

What he did not have was the objective. A layout drawn so that “each continent sits near a meridian” is a layout judged continent by continent, and Asia, which is too wide for one meridian at 30°E, is the continent that judgement serves worst. The land’s layout gives Asia its own meridian and gives up some of Europe’s, because Asia is the larger. That is a choice an area-weighted mean makes, and the average was a choice of norm is the reminder that a different measure could make it the other way.

How the layouts were checked #

Goode’s layout must score as the homolosine. Put through the general layout code — edge, cuts and meridians as data — Goode’s layout must give exactly the land score the earlier essays gave his map, 16.5114°, since it is the same map. It does, to the last digit.

The recursion’s cost must be the layout’s score. The layout it returns, scored cell by cell at the same aspect, must give the mean the recursion computed, to a part in a million. It does.

More lobes contain fewer. A layout of $K$ lobes can always leave one of them where another would have been, so the best score must never rise as lobes are added. It does not, across the eight counts measured.

The finding must be there to fail. With no cut crossing more land than Goode’s worst, the land’s layout must beat his by more than half a degree. It beats it by 2.03.

Where the layout stops #

A five-degree grid. Cuts and meridians are placed on it, so every number here is an optimum on that grid; a finer grid can only do better, by an amount the flatness of the frontier suggests is small.

Straight half-meridian cuts. Goode’s cuts, and the land’s, are straight lines on the sphere from the equator to the pole. A cut allowed to bend round Greenland would cross less land for the same lobes and would change the frontier — probably towards Goode’s layout, whose worst cut is the one a bend would most improve.

The land at 1:110m, counted by area. Natural Earth’s coarsest land, outside Antarctica, weighted by area. A map drawn for population would weight Asia’s coasts, India and China far more heavily and would move every meridian towards them; a map drawn for the oceans, as the land it was drawn for points out some interrupted maps are, would invert the whole problem.

One spacing curve. The layout is searched with Goode’s composite and his join, and the join then moved on the layout found. Searching both together would do at least as well as doing them in turn, and nothing here says by how much more.

Still open: whether the ocean’s map is the land’s turned inside out #

The layout found here is the land’s, and its cuts are placed in the oceans because that is where the land is not. An interrupted map drawn for the oceans — the kind Goode also published, with its lobes centred on the Pacific, the Atlantic and the Indian Ocean and its cuts running through the continents — has the same search with the raster inverted, and the frontier would then be about cutting water rather than land.

Whether the ocean’s optimal layout is the land’s with its cuts and meridians exchanged, or something with no simple relation to it, and whether one layout could serve both measures at once within a stated loss on each, are questions a search scored on the land alone cannot ask.