Scored lobe by lobe, the homolosine loses to its own lower half

One curve beats the three maps it would replace set Goode’s homolosine against every smooth equal-area curve that might replace it, and the composite lost badly. A single spacing curve with one coefficient carried 6.83 degrees less angular deformation. A five-term curve carried 9.63 less, and the composite turned out to be worse than the Mollweide it is half made of.

Its own closing section said what was wrong with the comparison. Every map in it was read as one uncut sheet, and Goode’s map is never used that way. The homolosine was designed to be cut, into two lobes in the northern hemisphere and four in the southern, so that no land is far from a central meridian. Judging it uncut is like judging a folding ruler by how straight it is when folded.

So here every map is scored the way the homolosine is actually used, with the objective the earlier comparison named but did not compute. The sphere is cut on Goode’s own lines, and every point is measured from its own lobe’s central meridian. Nothing else changes: the same eight curves, the same area weighting, each map at its own best proportions — the constant that every equal-area map is every other one shows equal area leaves free. The ranking turns almost completely over.

What cutting changes, and what it leaves alone #

On a pseudocylindrical the angular deformation at a point depends on only two things: the latitude, and the distance in longitude from the meridian the map is drawn about. That is why a single spacing curve is a whole map. It is also why cutting is easy to score. Cutting changes which distances from a meridian occur on the map, and nothing else.

Giving up continuity made the underlying point: each lobe is the same projection rotated in longitude and translated on the page, so every measure at a point is what the uncut map has at the corresponding point. Cutting adds no deformation and removes the longitudes furthest from a meridian. The only question is which maps were spending their shape on those longitudes.

Goode’s layout is lopsided, and deliberately so. The south, mostly ocean, is cut three times into four lobes, with meridians at 160° W, 60° W, 20° E and 140° E. No point there is more than 60° from its own meridian. The north is cut once, at 40° W, into a lobe for the Americas about 100° W and one for Eurasia and Africa about 30° E. That second lobe runs 150° east of its meridian, to the antimeridian, so the north keeps a strip of far longitudes the south has lost.

The scoring follows that layout exactly. Twelve half-lobes, each with its own width. Every point on each is scored at its distance from its own meridian, and the total is weighted by longitude and by the cosine of the latitude, as the uncut score was. A sheet cut into one lobe a hemisphere, 180° each side, must give back the uncut score, and it does to within 0.015°, the difference between sampling longitude at grid points and at the midpoints of equal steps. That is the control that the change of scoring changed nothing but the distances.

The ranking turns over #

Every map does better cut than uncut, which is what cutting is for. The gains are very unequal, though, and they are unequal in exactly the order that reverses the table.

The sinusoidal falls from 38.24° to 17.88°, the largest gain of any map by far. The earlier account called it the second-worst named map. Its weakness is shear that grows with distance from the central meridian, and a cut map never lets that distance grow.

Eckert IV falls only from 27.70° to 21.34°. It was the best named map uncut, and cut it is second from last, ahead of only Eckert II. Its shape was chosen to keep the far longitudes tolerable, and it pays for that everywhere else: 9.3° of deformation at the equator, where the sinusoidal has almost none. On a cut map the far longitudes never appear, and the payment buys nothing.

Goode’s composite falls from 35.39° to 18.48°. That is a large gain, but not as large as the sinusoidal’s, and it ends 0.60° behind it. The Mollweide ends at 18.59°, and Craster’s parabolic at 17.40°, the best named map on the cut sheet.

Then the result the question was asking about. The one-term curve fitted to the uncut sheet, exp(0.294 cos 2φ), which beat the composite by 6.83° uncut, reads 20.77° cut. That is 2.29° worse than the composite. The curve bought a band of excellent shape through the middle latitudes by closing its parallels fast towards the pole, and closing them fast is the Eckert IV habit. On a cut sheet that habit is paid for and not used.

So each of the earlier account’s claims fails under the scoring the homolosine is designed for. A single smooth curve does not beat the composite if it is the curve fitted uncut. The composite is not worse than the Mollweide cut: it is 0.11° better. And the named map that won uncut is nearly the worst.

The best curve for a cut sheet is almost flat #

Refitting the spacing curve on the cut score, with the same exponential cosine series, the same nested seeding and the same free proportions, gives a curve that barely moves from the sinusoidal.

The fitted coefficients are tiny: 0.039, 0.018 and 0.011 at three terms, against 0.294 for the uncut one-term curve. The curve holds 91.8 per cent of the equator’s spacing at 60°, where the uncut fit had closed to 64.3. It reaches 17.01°, and the whole improvement over the plain sinusoidal is 0.88°. The fit has nearly stopped at two terms.

The reason is the one Eckert IV illustrates from the other side. A pseudocylindrical spends its shape in two directions. Along the central meridian, a flat spacing keeps the map conformal at the equator. Away from it, a spacing that closes towards the pole lengthens the upper parallels and holds down the shear at far longitudes. Uncut, the far longitudes are half the map and the second purpose dominates, so the best curve closes fast. Cut, the far longitudes are gone and only the first purpose is left, so the best curve is nearly flat.

In one sentence: the uncut optimum and the cut optimum are different maps, because they are answering different questions about where the reader is looking. Neither can be judged by the other’s score. The cut-optimal curve scores 34.39° uncut, worse than the composite, and the uncut-optimal curve scores 20.77° cut.

How many cuts it takes to turn the ranking #

Goode’s layout is one arrangement among many, and the reversal could be a peculiarity of it. The cleaner question is how the ranking depends on how finely the sphere is cut. The same maps can be scored on sheets cut into equal lobes, from one a hemisphere, which is uncut, to eight.

The ranking changes twice. Uncut, Eckert IV leads at 27.66° and the sinusoidal trails at 38.29°. At two lobes a hemisphere, each 180° wide, the five maps close up and the Mollweide leads at 20.33°, with the composite at 20.92° and the sinusoidal at 21.28°. At three lobes the sinusoidal takes the lead and never gives it up. At eight lobes it reads 5.57° while Eckert IV has barely moved, to 18.77°.

That last pair is the clearest statement of the mechanism. The sinusoidal’s deformation is almost entirely a function of distance from the meridian, so each extra cut removes most of what is left of it. Eckert IV’s is largely a function of latitude, built into the spacing, so no amount of cutting touches it. A map whose defects are longitudinal is cured by interruption, and a map whose defects are latitudinal is not.

Goode’s own layout sits between the two regimes: two lobes in the north, where the Mollweide would lead, and four in the south, where the sinusoidal leads comfortably. The whole-sphere score blends the two, which is why the sinusoidal wins by a small margin and not a large one, and why the Mollweide caps are not simply a mistake.

Where the caps earn their place #

The homolosine is the sinusoidal below 40.7° and a scaled Mollweide above it, joined where the parallels agree and the meridians turn a corner. Scored cut, the two maps are almost identical below the join and differ only above it. So the useful comparison is latitude by latitude: where does the Mollweide cap beat the sinusoidal it replaces?

The answer is a band. From 32° to 61° the composite is ahead, by as much as 7.4° at 50°. The winning band starts below the join because the two maps take slightly different best proportions: the composite’s is 0.982 and the sinusoidal’s 0.995. Above 61° the composite falls behind and keeps falling, to 66.7° at 80° against the sinusoidal’s 33.8°.

The reason for both halves is where each map puts its deformation. The sinusoidal keeps its parallels evenly spaced and true to length, so it is conformal all the way along its central meridian, and its shear grows only with distance from that meridian, times the sine of the latitude. Cutting keeps the distance small. The Mollweide closes its parallels up towards the pole and lengthens them to keep area, so near the pole it deforms shape even on its own central meridian, and no cut can reach a defect that sits on the meridian itself. Uncut, the sinusoidal’s shear at high latitudes and far longitudes is severe, and the Mollweide cap was put there to relieve it. Cut, the cutting has already relieved that shear, and what remains is the cap’s own cost at the pole.

This is the explanation of the headline result, and it is not a criticism of Goode’s design. The homolosine was drawn, not fitted, and it was drawn to make continents look right, not to minimise a whole-sphere mean. The continents it most needed to fix are in the band where the cap wins.

The weight the band would need #

The ledger can be read exactly. The band from 32° to 61° holds 35 per cent of the sphere’s area. Summed over it, the composite’s advantage is worth 1.44° of the whole-sphere mean. Everywhere else, mostly the polar caps and the low latitudes where the proportions differ, the sinusoidal’s advantage is worth 2.04°. The sinusoidal wins by the difference, 0.60°.

For the composite to draw level, a score would have to count that band 1.41 times as heavily as its share of the area. A score weighted by land is a natural candidate. The homolosine’s own purpose was continents, and the northern mid-latitudes are where the most land lies, from the United States and Europe to the steppe and northern China. Whether land weighting reaches a factor of 1.41 depends on the coastline data and on whether Antarctica, almost all of it poleward of 61° south, is counted. That is a measurement this comparison does not make. What it does fix is the size of the question: the homolosine beats the plain interrupted sinusoidal if and only if its reader cares about the 32–61° band about forty per cent more than area alone would say.

That is a much more useful statement than either ranking. Uncut, the question “is the homolosine good?” gets the answer no, by nearly ten degrees. Cut and scored over the sphere, it gets no, by 0.6°. Cut and scored by a reader who lives between 32° and 61°, it may well get yes. The average was a choice of norm is the general form of this: the verdict belongs to the weighting, and the weighting belongs to the purpose.

What this does to the earlier result #

It does not overturn it. The earlier comparison’s arithmetic stands, and every number it quoted is reproduced above. What changes is what the result is about.

It was a result about uncut sheets, and for uncut sheets it holds: a smooth curve with one coefficient beats the composite, and the composite is worse than the Mollweide. Anybody printing an uninterrupted equal-area world map should take Eckert IV or the fitted curve over anything with Goode’s join in it.

It is not a result about the homolosine, because the homolosine is not an uncut sheet. For the use it was designed for, the composite is within 0.6° of its own lower half, 1.1° of the best named map and 1.5° of the best smooth curve, and it wins in the band its designer cared about. The earlier essay’s caveat, that the comparison was uninterrupted and that the cutting is what makes the homolosine good, is now a measured statement. Cutting is worth 16.9° to the composite. Being a composite rather than a plain sinusoidal is worth −0.6° over the sphere and +1.4° inside the band.

What a cut buys prices a cut in the deformation it removes, and here the price differs by map: 20.4° for the sinusoidal, 6.4° for Eckert IV. A cut is not worth a fixed amount. It is worth whatever deformation the map had placed at the longitudes the cut removes, which is a property of the map. That is why a family is a function, not a list matters here. Two members of one family can respond to the same interruption by a factor of three.

The controls on these numbers #

One lobe a hemisphere must be the uncut sheet. Scoring the Mollweide on two half-lobes of 180° each must reproduce the uncut score, since the distances are the same and only the order of summing differs. It does to 0.05°, which is the sampling difference between the two grids and nothing else.

Every map must improve when cut. Cutting only removes far longitudes, so a map that scored worse cut would mean the scoring had moved points rather than dropped them. All eight improve.

The reversal must be made by the numbers and not by the reading. The curve fitted uncut must lose to the composite cut, and the sinusoidal must beat it, or the claim that the ranking turns over is not being made. Both hold: 20.77° against 18.48°, and 17.88° against 18.48°.

The refitted curve must beat everything it is compared with, or the fit is reporting where it started. It reaches 17.01° against Craster’s 17.40°, the best named map cut.

And the sinusoidal cut on Goode-style lobes must agree with the earlier measurement of the same thing. Giving up continuity scored an interrupted sinusoidal on a land-centred six-lobe layout and found 17.8°. The figure here is 17.88°, on Goode’s own layout and at the sinusoidal’s best proportions.

Where the comparison is narrower than it sounds #

The whole sphere, not the land. Every score is weighted by area alone, which counts the Southern Ocean as heavily as Europe. The homolosine was designed for land, and the band ledger gives the weight a land score would need, 1.41 times area in the 32–61° band, without computing whether any real coastline supplies it.

Goode’s layout, fixed. The cuts are the published ones. A layout optimised for each map — lobes placed to suit Eckert IV’s latitudinal defects, say — would score each map differently. The equal-lobe sweep is the nearest thing here to asking that, and it shows the answer depends strongly on the count.

Angular deformation only. Every map compared is equal-area, so area is not traded, but the cut adds something the score does not see: the gaps between lobes. Giving up continuity measures the tear itself, and a reader who cares about continuity would rank six lobes below two whatever the angles say.

The mean, not the worst. On the worst case the order is different again. The cut sinusoidal’s worst point is 104.0° and the composite’s 139.9°, and the Mollweide’s pole is why. Here the composite loses more heavily, not less.

And the fit is a fit. Three coefficients from seven starts reach 17.01°. Nothing shows that no smooth curve does better, only that this one beats every named map on the cut score, which is all the claims above need.

Still open: the land the homolosine was drawn for #

The one number this comparison could not supply is the one the homolosine’s design turns on. A land-weighted score must count the 32–61° band at 1.41 times its area for the composite to draw level with the plain interrupted sinusoidal. Whether it does depends on the land, and on two choices that decide the answer before any arithmetic: whether Antarctica is land for this purpose, and whether the land counted is where people are or where the continents are.

There is a second question under the first. If land weighting does favour the band, the best curve for a land-weighted cut sheet is not the nearly-flat curve fitted here. It would close a little above 40°, where the land is, and stay flat below. That is a composite in shape, one curve with a region of fast closure in the middle latitudes, and it is exactly what Goode built out of two named maps. Whether a smooth curve fitted to land reproduces the homolosine’s join latitude of 40.7° — which would mean Goode found by eye the optimum of an objective nobody had written down — or lands somewhere else, is the question this comparison raises and cannot settle without a coastline.