Cartographic Projection
every map is wrong, and by exactly how much
Concept

Interruption — where it appears

A deliberate cut in a map's surface, which buys lower distortion in the remaining pieces at the price of tearing places apart. Where the cuts go is a decision with a cost, and the cost is measured on the reader's side as the neighbours the map has separated.

Named by 10 essays across 4 fields — each of them below, with the objects they name alongside it.

The icosahedron's faces, drawn on the sphere. The edges of the icosahedron projected radially onto the sphere, which is the partition of the world a polyhedral map uses: everything inside one spherical polygon is drawn on one flat face. Each face spans 25.8° from its centre to its own boundary, and the gnomonic map onto it reaches 6.6° of angular deformation at the corners. Drawn in an orthographic projection of the embedding, which is a map with its own distortion.

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.

families · Polyhedral How each projection escapes being one to one. Eighteen projections, and the theorem allows no fourth column. A map of the whole sphere either leaves ground undrawn, or draws one place as a curve, or is cut so that one ground curve appears twice — and 11 of the eighteen do more than one of those. The three columns are a solid angle, a count of places and a page length, which is why they are three columns rather than one score: there is no rate at which a hemisphere converts into a pole.

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.

impossibility · Topology Which edges to cut is a spanning tree. The icosahedron's faces as nodes and its 30 shared edges as links. A net keeps 19 of those joins and cuts the rest, and the joins have to form a spanning tree — connected, so the net is one piece, and acyclic, so it lies flat. The heavy links are one such tree. The number of distinct nets is therefore the number of spanning trees of this graph, which Kirchhoff's theorem gives as a determinant: 5,184,000 for the icosahedron.

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.

families · Polyhedral A 20° shape across the antimeridian, in the space where the numbers live. Longitude runs across the page from −180° to 180°, which is where the failure is: the shape is one rectangle on the ground and two pieces in the numbering, and every operation that treats longitude as a real number sees the two. The bounding box comes out 359° wide instead of 20°, the planar area comes out 17 times too large because the shoelace encloses the complement, and the midpoint of a segment from one edge to the other lands 20015 kilometres away — the antipode of where it belongs. The true area, from the closed form, is 4,920,667 square kilometres.

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.

applied · Dataset The map that is continuous, and the pair it pays with. The orthographic is defined and continuous at every place on the Earth — it is written in the components of the place itself, with no longitude in it to jump. What it gives up is being one to one, and it gives it up almost everywhere: 47 per cent of the sphere shares its page point with the place directly behind it. Borsuk–Ulam guarantees at least one ANTIPODAL pair among those, and here it is exactly one — the centre and the place on the far side of the world, both at the middle of the picture, found to a residual of 1.5e-14.

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.

impossibility · Topology Past the five solids: what more faces buy, and what they cost. The icosahedron subdivided 2, 3, 4, 6, 8 ways, giving 20, 80, 180, 320, 720, 1280 faces. The worst angular deformation inside a face falls from 13.1° to 0.26°, with a fitted exponent of -0.947 against the face count — the reciprocal, as it must be, because a face's angular size goes as the inverse square root of the count and the gnomonic's deformation goes as the square of that. The total cut length rises from 11.6 to 96 sphere radii, fitted at 0.509. Both exponents together say the whole economics of the family in one line: halving the distortion costs √2 times the cutting, for ever.

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.

families · Polyhedral Two nets of the same solid, cut to the same length. Every net of a Platonic solid severs exactly the same number of edges, all of the same length, so the total length of the cut is a constant and cannot choose between them. Each line joins the two places a severed edge ends up. Left: the net that keeps them closest, 11.30 edge lengths in total. Right: the net that puts them furthest apart, 17.97 — a factor of 1.59 for the same amount of cutting.

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.

families · Polyhedral Two unfoldings of the same 80-face solid. Both are edge unfoldings of the same solid along different spanning trees of its face graph, so both preserve every distance on the surface exactly. The left one is a net. The right one is not: 3 pairs of its faces occupy the same ground, so it cannot be cut out of paper and folded up. Nothing in the unfolding procedure prevents this, and past the regular solids most trees produce it.

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.

families · Polyhedral What a cut buys. The mean angular deformation of the interrupted sinusoidal against the total length of cut the interruption spends, for lobe counts from one to twenty-four. Goode's interruption — the one actually printed — is the marked point: it spends 100 thousand kilometres and returns 18.0°, where the even-lobed curve returns 9.2° for the same length. It is not on the frontier and it was never trying to be: its cuts are placed to keep continents whole.

What a cut buys

Six rungs count cuts and none measures one. 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.

impossibility · Topology Seven cuts of the same size, in different places on one body. The body's own colatitude and longitude, with the seven windows drawn on it. Each covers the same surface area to 0.24 per cent — the longitude extent is divided by sin θ and a scale is then solved per window — so their sizes are held and only their positions differ. Each is shaded by the areal spread of the conformal map solved on it, listed beside the grid and running from 1.03 to 1.36. The darker windows are the worse ones, and they are the ones over the body's lobes.

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.

datums · Bodies

Named alongside it

The objects these essays reach for when they reach for this one.

TopologyPlatonic solidPolyhedral projectionDiscontinuitySpanning treeAngle deficitCombinatoricsContinuityFaceIcosahedronTrade-offAntimeridian

All concepts