What a cut buys

No map of the whole sphere is one to one is the first rung of this ladder, and every rung since has been about what the resulting cut does. How many times, not whether counts the places a projection folds. Four colours, and what a cut cannot do to them asks what survives one. Two charts are enough, and one is not counts the charts an atlas needs.

Counting is what all of them do, and a count is not a quantity a design can be traded against. A cut is a curve on a sphere. It has a length, in kilometres, and that length is what the map is spending.

The unit, and why nobody has used it #

An uninterrupted world map has a cut. It has to: the sphere is closed and the sheet is not, so somewhere the map stops being continuous, and on every cylindrical and pseudocylindrical projection that somewhere is the antimeridian. From pole to pole that meridian is 20,015 kilometres, half the circumference of the Earth, and it is the price of admission for putting the world on a rectangle at all.

An interruption buys more sheet edge. Four equal lobes cut four full meridians and spend 80,060 kilometres. Eight lobes spend 160,121. The unit is a length rather than a count because the cuts are not interchangeable — a half-meridian is half the price of a full one, and Goode’s interruption is built almost entirely out of half-meridians.

The quantity being bought is shape. Interruption cannot change area — an interrupted equal-area projection is still exactly equal-area, because each lobe is the base projection composed with a rotation and a translation, and neither changes any distortion measure. What it changes is how far any point can be from a central meridian, and on the sinusoidal that distance is the whole of the shape error.

The rate at which it buys #

The interesting number is not any row of that table but the slope between them, because the slope is what a designer is deciding about.

The first interruption — going from one lobe to two — removes 17.5° of mean angular deformation for 20,015 kilometres of cut, a rate of 8.72 degrees per ten thousand kilometres. The step from sixteen lobes to twenty-four removes 0.93° for 160,000 kilometres, a rate of 0.058. The exchange rate falls by a factor of a hundred and fifty across the range, smoothly, with no point at which it drops away.

The shape of the fall is worth having precisely because it is so ordinary. Doubling the lobe count roughly halves the mean angular deformation — 38.9°, 21.4°, 11.1°, 5.6°, 2.8° for one, two, four, eight and sixteen lobes — while doubling the cut, so the return per kilometre halves at each doubling too. That is the arithmetic of a quantity proportional to a lobe’s angular width, and it is the sinusoidal’s shape error being proportional to distance from a central meridian, seen from the other side. Nothing more sophisticated is happening.

That smoothness is the finding, and it is a negative one. If the curve had a knee, the number of lobes would be a calculation: spend up to the knee and stop. It has none, so the number of lobes is a judgement about how much discontinuity a reader will tolerate against how much shape distortion — and that is a judgement nobody in the subject has been able to state numerically because one of its two terms had no units.

Goode’s, priced #

Goode’s homolosine is the interrupted map that is actually printed, in atlases, since 1923. Its interruption is designed around content: the cuts run through oceans so that every continent stays whole, and the lobes are consequently very unequal — the northern hemisphere is two lobes of 140° and 220°, the southern four of between 80° and 100°.

Measured against the frontier it is expensive. It spends 100,076 kilometres and returns a mean angular deformation of 17.96°. Interpolating the even-lobed curve at the same cut length gives 9.25°, so Goode’s is 94 per cent worse for the same money. Read the other way: the even-lobed scheme reaches Goode’s own 17.96° at three lobes, spending 60,045 kilometres, which is 60 per cent of Goode’s cut.

Its worst point is further out still. Goode’s maximum angular deformation is 104.57°, against 42.23° for four equal lobes and 54.62° for three. The 220° northern lobe reaches 110° from its own central meridian, which is further than any point on an uninterrupted two-lobe map, so parts of Goode’s are worse than a map with a quarter of the cutting.

The comparison that makes the size of the gap concrete is with the two-lobe map, which is the crudest interruption there is. Two equal lobes spend 40,030 kilometres — 40 per cent of Goode’s — and return 21.4° against Goode’s 18.0°. So Goode’s whole hundred thousand kilometres of cutting buys it three and a half degrees over the simplest interruption anyone would draw, and buys it a worst point that is worse than the two-lobe map’s 75.68°. On the two numbers this collection measures maps by, Goode’s is close to the cheapest interruption available and costs two and a half times as much.

None of that is a criticism, and the essay would be dishonest if it read as one. Goode’s is optimising something else entirely — where the discontinuity falls relative to what a reader wants to look at — and the trade it is making is a good one for an atlas. What the frontier supplies is the price: keeping the continents whole costs about nine degrees of mean shape distortion, or equivalently about forty thousand kilometres of cut, and until now that cost had no number.

Where Goode’s cut actually is #

Adding up meridians gives Goode’s 60,045 kilometres: one full meridian at the antimeridian and four half-meridians at the interior cuts. That is not its cut length. Forty thousand kilometres of Goode’s cut runs along the equator, because its northern and southern lobes do not line up: at 120° west the northern half is drawn about a central meridian of 100° west and the southern half about 160° west, so the two halves of the map do not join there and the reader crossing the equator crosses a discontinuity.

That equatorial cut is well known to anybody who has looked closely at the projection’s outline and is absent from every description of it, including the ones that carefully count its six lobes. It is a consequence of the lobe layout rather than a design decision, and it is the single largest item in the scheme’s bill.

The cut and the tear are not the same length #

There is a second measurement available, and keeping the two apart matters.

The cut is where the map fails to be a smooth map of the sphere. The tear is where a reader sees a gap. They are different, and the second is always shorter: on the uninterrupted sinusoidal they agree to within half a per cent, at four lobes the tear is 89 per cent of the cut, at eight it is 81, and on Goode’s it is 54.

The gap is the projection collapsing. The sinusoidal sends each pole to a point, so all lobes share it and the final degrees of every cut open nothing. Goode’s equatorial cut is the extreme case: the map is genuinely discontinuous across the equator over forty thousand kilometres, and the page shows almost nothing, because the two halves have been positioned to meet.

So a reader cannot see what a map is spending. The visible seam understates the cut by up to a factor of two, and understates it most on the scheme that spends most.

The distinction also settles a question this anchor left open. The antimeridian is a cut in the numbers is about a discontinuity that is entirely invisible on the page — the longitude wrap, where the map is continuous and the coordinates are not. That is the same relationship one level further out: three quantities in play, the cut in the surface, the tear on the page and the break in the numbers, no two of them the same set, and only the middle one visible. A projection with no visible tear can be cut along half its own equator, and a page with no cut at all can carry numbers that jump.

The base projection decides whether any of this is worth it #

The trade is not a property of interruption. It is a property of interruption applied to a particular map, and the difference between two equal-area base projections is larger than anything else measured here.

The sinusoidal’s angular deformation is a function of longitude from its central meridian and almost nothing else, because its parallels are true to scale everywhere: the east–west scale is exactly right at every point, and all the error is in the meridians’ obliquity, which grows with longitude. Interruption removes exactly that, so the sinusoidal is the projection interruption was invented for and the one it works on.

Mollweide’s error is largely a function of latitude. Its parallels are not true to scale, its shape error at the pole is there whatever the longitude, and cutting the map into lobes leaves every one of those points exactly where it was. Sixteen lobes buy it 20 degrees and then stop buying anything.

Interruption is therefore not a general technique for improving a map. It is a technique for removing longitudinal error, and it is worth its cut only on a projection whose error is longitudinal — which is a statement nobody makes, because the two projections most often interrupted are the sinusoidal and Goode’s composite of the sinusoidal with Mollweide, and the composite’s low latitudes are where all its interrupting pays.

What was computed, and how #

The cut is measured from the interruption scheme alone and not from the projection: it is the set of lobe boundaries, each counted once even where two lobes share it, plus the equatorial segments where the northern and southern layouts disagree. That makes the number a property of the interruption, so the same scheme costs the same on any base map, which is what allows the two-base comparison above.

The tear is measured the opposite way, on the page: adjacent ground samples whose page positions are far apart relative to the median step. It needs no units and no per-projection threshold, and it necessarily returns a subset of the cut, which the gate checks.

The distortion is the ordinary regional mean this collection computes everywhere — angular deformation from the projection’s own derivatives, area-weighted over the sphere, with each lobe evaluated at its own central meridian.

The assertions require six things separately. That cut length rises and distortion falls with the lobe count, or there is no trade. That the exchange rate saturates by more than threefold, or the frontier is a straight line and the essay is about nothing. That an uninterrupted map still reports a cut, since a scheme returning zero would mean the measurement is not finding the antimeridian. That Goode’s sits materially off the frontier. That only the mixed-hemisphere scheme has an equatorial cut, and that the tear never exceeds the cut.

Where the model stops #

The frontier is over one family. Equal lobes about equally spaced central meridians is one arrangement out of infinitely many, and there is no claim here that it is optimal. What can be said is that it dominates Goode’s on this objective, which is enough to price Goode’s; a better arrangement would only make the price larger.

Mean angular deformation is one objective. Goode’s does better on the criterion it was designed for — no continent crossed — and that criterion is not on the axis. A fuller treatment would put continental integrity on one axis and cut length on the other and ask for the frontier of that, which needs a coastline dataset and the reasons this collection does not have one.

And the number of lobes is bounded by legibility, not by arithmetic. Twenty-four lobes reach 1.87° and are unreadable: no route, no ocean and no comparison of areas survives twenty-four discontinuities. Nothing measured here says where the limit is, because it is a fact about readers.

What it says about the composite #

Goode’s homolosine is not the interrupted sinusoidal. It is the sinusoidal below about 40° of latitude and Mollweide above it, joined along a parallel, and the whole point of that composite is that the two projections’ weaknesses are in different places. Read against the two-base measurement, the composite is doing exactly the right thing: it puts the sinusoidal — the projection interruption pays for — where the interrupting is, and Mollweide where longitude no longer dominates.

That is a defence of the design and a sharpening of the criticism at once. The composite is well judged; the lobes are the expensive part, and they are expensive because they are unequal rather than because they are numerous. A six-lobe scheme with equal lobes would spend 120,091 kilometres and return 7.43°; Goode’s spends less, 100,076, and returns two and a half times the distortion, because half its cut is buying continental integrity and the other half is buying the equatorial join that unequal lobes force on it.

The unequal lobes are load-bearing for the content and expensive for the geometry, and those are separable. Nothing in the design requires the north and south layouts to differ by as much as they do, and the forty thousand kilometres of equatorial cut is the direct cost of the amount by which they do.

The generalisation #

The rule is that a constraint nobody has given units to gets spent without being counted.

Every essay in this anchor has treated the cut as a topological necessity — something the sphere forces on the page, to be admitted and then worked around. It is that, and it is also a budget: a quantity that can be spent in different places, in different amounts, to buy different things. The moment it has kilometres attached, questions that were matters of taste become comparisons, and one of them has an answer nobody expected: the standard interrupted world map is paying about twice the going rate.

The same shape recurs wherever a design has a term that is qualitative. More faces, less distortion, more cutting is the polyhedral version and it counts faces; the net that loses the fewest neighbours counts broken adjacencies. Both are countable rather than measurable, and both would say something sharper against a length.

Who found it, and when #

Goode published the homolosine in 1923 and described its interruption by naming the meridians it cuts along. Every subsequent description does the same. The lobe count — six — is universal; the cut length appears nowhere, and neither does the equatorial discontinuity, which is two fifths of it.

The reason is not carelessness. A cut is a qualitative fact in the way the subject has always framed it: a map either is or is not interrupted, and if it is, the question is where the cuts fall relative to the content. That framing is exactly right for an atlas designer and it has no slot for a total.

The one place the subject does think quantitatively about cutting is polyhedral projection, where the net’s edges have to be counted and the trade between face count and distortion is drawn explicitly. Even there the axis is the number of faces rather than the length of the cutting, and the two are not proportional: an icosahedral net cuts far more edge per face than a cubic one.

A cheap habit follows, for anybody choosing an interruption: add up the cut before adding up the lobes. A scheme of six lobes can cost more than a scheme of eight, a half-meridian is half a full one, and a layout that changes at the equator is buying an equatorial cut it did not ask for.

Where the ladder goes next #

Seven rungs have now measured what a sphere costs a sheet: the fold, the count of charts, the colouring, and now the length of the cut. All of them treat the sphere as the thing being flattened. The next rung changes the object — a body that is not a sphere, and specifically one whose topology is different, where the obstructions this ladder has spent seven rungs on are not weaker but simply absent.