Interruption — where it appears
Named by 17 essays across 4 fields — each of them below, with the objects they name alongside it.
The globe on a solid
Cylinder, cone and plane are not the only surfaces a sphere can be laid on. Project it onto a polyhedron and the curvature goes entirely to the corners — π at each of the tetrahedron's four, π/5 at each of the dodecahedron's twenty, and always 4π in total, which is exactly the curvature of the sphere it replaced.
No map of the whole sphere is one to one
Eleven essays establish that no map preserves distance, and every step of that argument needs a distance. There is a second impossibility underneath it that needs nothing at all: a sphere is compact and has no boundary, so a continuous map of it into the page cannot also be one to one. Eighteen library projections, eighteen escapes, and not one of them free.
The cut has to go somewhere
A solid lies flat only if it is cut open, and which edges to cut is a spanning tree of the face graph — so the icosahedron has exactly 5,184,000 distinct nets, a determinant rather than an estimate. All 384 of the cube's were laid flat and tested: not one overlaps, while an irregular tetrahedron overlaps in four of its sixteen.
The antimeridian is a cut in the numbers
A twenty-degree box across 180° has a bounding box of 359.4°, a planar area seventeen times too large, and a midpoint 20,015 kilometres from where it belongs — which is the antipode, exactly. Moving the cut moves the failure and never removes it, because a circle cannot be numbered by an interval.
Two opposite places on the same spot
A projection may be continuous everywhere or one to one everywhere, and the first two rungs price both. What neither says is that the choice is not symmetric: a map that keeps continuity does not lose injectivity somewhere arbitrary. It loses it, always and at minimum, on a pair of places directly opposite each other on the Earth.
More faces, less distortion, more cutting
The five regular solids are where polyhedral mapping stops, and they stop because there are only five rather than for any reason about maps. Subdivide instead, and the family runs to any number of faces with two fitted exponents: distortion falls as the reciprocal of the count and cutting rises as its square root.
The net that loses the fewest neighbours
Every one of the cube's 384 nets cuts exactly seven edges of exactly the same length, so the quantity this collection has been pricing cutting by is a constant that cannot choose between them. Measured on the reader's side — how far apart a net puts two places that touch on the globe — the best net scores 11.30 and the worst 17.97, for identical cutting.
A net can land on top of itself
Every one of the cube's 384 unfoldings is a net, and every one of the icosahedron's five million is too. Past the regular solids that stops being true: at 180 faces, 99 of every 100 randomly chosen unfoldings have faces sitting on top of each other, so choosing a net stops being a choice and becomes a search — except that the net anybody would actually draw works every time.
What a cut buys
Cuts are usually counted and never measured. A cut is a curve on the sphere with a length in kilometres, the shape distortion it removes is a falling function of that length, and the interruption everybody prints spends a hundred thousand kilometres to reach a figure the even-lobed curve reaches with sixty.
Cuts of the same size in different places
Where a body is cut decides how well it can be mapped, by a factor of ten — established with five windows of five different sizes, so *where* and *how much* were confounded and the factor could have been entirely about extent. Held to the same surface area to a quarter of a per cent, the answer survives at a factor of 1.32, and what predicts it is the curvature the window encloses.
Scored lobe by lobe, the homolosine loses to its own lower half
Read as one uncut sheet, a single smooth curve beats Goode's composite by 9.6 degrees. But the homolosine is used cut into six lobes, so here it is scored cut, with each point measured from its own lobe's meridian, and the ranking turns over. Eckert IV, the best named map uncut, falls to second from last. The curve that beat the composite is now 2.3° worse than it. The plain sinusoidal, the composite's own lower half, wins at 17.88° against 18.48. The best smooth curve for a cut map is the sinusoidal with a few per cent of correction.
On the land it was drawn for, the homolosine wins by its meridians
Scored over the whole sphere and cut into Goode's lobes, the homolosine loses to the plain sinusoidal; for it to win on latitude alone, a reader would have to care about the band from 32° to 61° forty per cent more than its area. The land does not do that — the band holds 1.03 times its share of the continents. Weighted by land anyway, and each point read from the meridian of the lobe it lies in, the homolosine wins: 16.51° against Craster's 17.00 and the sinusoidal's 17.34. The win is in where Goode put his meridians, which lift his map from third to first; count Antarctica as land and it falls to fourth.
The land puts the homolosine's join eleven degrees higher
Goode joined the sinusoidal to the Mollweide at 40° 44′ because that is where their parallels have the same length — a fact of geometry, not of any objective. On the land his map was drawn for, read from his own lobes, the join has an objective at last, and its best value is 51.7°: a composite joined there gains a fifth of a degree over Goode's, and every join from about 46° to 60° does nearly as well. A smooth curve fitted to the land with no join in it at all stays within three per cent of flat to 50° and then closes, gaining twice as much again. Goode found the right shape and put its knee too low; with Antarctica counted, the land wants no knee at all.
The land would move Goode's lobes before his join
Goode chose his lobes by eye: where the cuts run and where each lobe's meridian sits. Scored on the land outside Antarctica and found exactly on a five-degree grid, the land's own layout at his count of two lobes north and four south deforms it by 14.48° against his 16.51° — two degrees, more than his meridians were worth in the first place — while its cuts cross less land than his: 2,057 kilometres against 2,391, the worst of them 1,056 against the 1,946 his northern cut spends on Greenland. The land wants the map's edge in the Atlantic and its northern cut in the Pacific. His fourth southern lobe is worth four hundredths of a degree, and once the lobes are the land's the best join climbs again, to sixty degrees.
The ocean centres its lobes on the land's cuts, and cannot keep its own cuts short
Run on the water instead of the land, the same exact search gives an interrupted homolosine whose lobes sit on the Pacific, the Atlantic and the Indian Ocean — five of its six central meridians within twenty degrees of a cut the land's own layout chose — and it deforms the water by 13.31° against the 19.75° the land's layout gives it. The inversion stops at the cuts. Every meridian on Earth crosses at least 7,172 kilometres of water, so the ocean's map has no layout whose cuts stay under seven thousand, where the land's went below nine hundred. And no layout serves both: none comes within 2.57° of both optima at once, though a single wide Pacific lobe buys the water 2.8° for a tenth of a degree of land.
The water would join Goode's map at seventy-five degrees
Goode joined the sinusoidal to the Mollweide at 40° 44′, and the land, given its own lobes, moves that join up to sixty degrees. The water moves it further, to seventy-five, and gains twice as much by it — a full degree of mean angular deformation against the land's half — though less of the northern water than of the northern land lies above Goode's join. The gain is in the polar water, north and south, where Goode's Mollweide cap shears even the lobes' own meridians; the water just above his join, the northern edge of the Southern Ocean among it, pays part of it back. A separate join for each hemisphere is worth a few hundredths of a degree, and with the join free the one-layout-for-both question comes out the same way it did with the join fixed, only worse: no layout comes within 3.3 degrees of both surfaces' best at once.
A smooth curve beats the water's best join from its first term
On the land, a smooth equal-area curve with no join needed three terms before it beat Goode's composite rejoined at the land's best latitude, and on the land's own lobes it needs three again. On the water a single term beats the water's best composite, a second and third add more, and the best curve is 0.54 degrees of mean angular deformation better than the composite joined at seventy-five and 1.6 better than Goode's. It is nothing like the composite in shape: its parallels never close to less than seven eighths of the equator's spacing, so the map the water wants is the sinusoidal reshaped by a few per cent at every latitude, not the sinusoidal with a cap. The lobes do not move under it. And where a separate join for each hemisphere was worth three hundredths of a degree, a separate curve for each is worth a quarter of a degree.
Named alongside it
The objects these essays reach for when they reach for this one.
Angular deformationEqual-areaTopologySinusoidalTrade-offVerificationLobePlatonic solidPolyhedral projectionPseudocylindricalDiscontinuityGoode homolosine