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

Euler characteristic — where it appears

The quantity V − E + F, which is 2 for every tiling of the sphere and which forces exactly twelve pentagons into any hexagonal one. Twelve pentagons in a hexagonal tiling of the sphere is not a design decision but an arithmetic consequence of it, however fine the subdivision.

Named by 9 essays across 5 fields — each of them below, with the objects they name alongside it.

The angles of a triangle, and the curvature inside it. A triangle on the sphere with sides that are great-circle arcs. Its three angles sum to 212.26°, overshooting the flat 180° by 32.26°. Integrating the curvature over the interior gives 0.56306 against an excess of 0.56306 — the same number by two routes that share no arithmetic. That is Gauss–Bonnet, and it is how a flatlander measures the curvature of a world it cannot step outside.

Measuring curvature from inside

A flatlander with a ruler can find out that its world is round, by drawing a triangle and adding the angles. Gauss–Bonnet turns that into an exact statement, and the total curvature of a closed surface turns out to be a number obtained by counting.

impossibility · Curvature 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 Ground north, on a map that has the pole on it. Every arrow points along its own meridian, towards the pole, and the pole is at the centre. Walking once anticlockwise round any loop enclosing it turns the arrow once anticlockwise as well: the index is 1, counted as a winding number with no distance anywhere in the calculation. A field like this cannot be combed flat. There is no way to choose a page direction for north at every point of the neighbourhood without the choice tearing somewhere, and the somewhere is the point in the middle.

North cannot be up everywhere

Ground north is a field of arrows on the sphere, and a field of arrows on a sphere must vanish somewhere. The failure is not measured, it is counted: the indices of the zeros sum to two, obtained here as a winding number in seven different charts with no distance anywhere in the calculation, and it is the same two that Gauss–Bonnet gets by integrating curvature.

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 The world after a map of degree 2. The graticule of the sphere, sent through the square of the stereographic coordinate and then drawn in Mollweide. Every parallel is still a closed curve and every meridian still runs pole to pole, and the whole world has been wrapped round the sphere 2 times: the 2 meridians that used to be 180° apart now lie on top of one another. Nothing has been torn. The degree measured by integrating the area the map sweeps is 2.0004.

How many times, not whether

Three rungs of this ladder answer yes or no and have no other kind of answer. The degree is the first quantity here that counts: a continuous map of the sphere to itself covers it a whole number of times, injectivity forces that number to ±1, and the number is recoverable three ways — by counting preimages, by integrating swept area, and from the rate at which the preimages coalesce.

impossibility · Topology A hexagonal tiling of the sphere, and its pentagons. 362 cells — 350 hexagons and 12 pentagons, the pentagons marked — drawn on Orthographic. The twelve are not a defect of the construction and cannot be removed by subdividing further: Euler's formula requires exactly twelve however many hexagons there are. Each pentagon here has 0.52 times the area of an average hexagon, so a count aggregated over these cells has twelve entries that mean something different from all the others.

Hexagons cannot tile the sphere

Hexagons are the best cell shape a plane offers and the sphere will not take them. Euler's formula forces exactly twelve pentagons into any such tiling — twelve at 42 cells and twelve at 642, while the hexagon count rises twenty-one-fold — and each of the twelve is measurably smaller than the hexagons around it.

applied · Cells One degenerate zero, nudged, becomes two ordinary ones. The direction of steepest ascent within twelve degrees of the north pole, for the sectoral harmonic alone and with two amounts of the tesseral added. On the left is one zero of index −2, a monkey saddle: three ways up and three ways down, and a Hessian that vanishes. On the right are two ordinary saddles of index −1 each, both of which the second-derivative test names correctly. Nothing has been added to the field but a term whose size can be made as small as anyone likes, and the classification changes at every nonzero value of it while the total does not change at all.

The second derivative cannot classify

Seven essays have used a second derivative to measure a size. The one thing a second derivative is classically used to do is say what KIND of thing is at a point, and on a sphere that use fails: the determinant test totals six where the truth is two, its failure is confined to exactly the field it is asked about, and the count that gets it right never differentiates twice.

distortion · Flexion A surface whose curvature changes sign, and integrates to nothing. a wide ring of major radius 3 and minor radius 1, shaded by its Gaussian curvature. The outer half is positively curved like a sphere, at up to 0.250; the inner half is saddle-shaped and negative, down to -0.500; and the two circles between them, drawn as lines, are exactly flat. The integral of the curvature over the whole surface is zero, which is 2π times the Euler characteristic — and every impossibility in this collection rests on that number being 2 rather than 0.

On a body with a hole, north can be up everywhere

Twelve rungs vary the body's shape and none varies its topology, which is what every impossibility here actually rests on. A torus has a nowhere-zero tangent field, a total curvature of zero rather than 4π, and a conformal world map with no cut and no singular point — and it still cannot be flattened, for the one reason that survives.

datums · Bodies The points a projection cannot move. The critical points of a stated field, found twice: once on the sphere from its own gradient, and once in Lambert cylindrical's page coordinates from the page's own numbers, with nothing shared between the two searches. seven points, 3 maxima, 2 saddles and 2 minima, and the two sets agree in position to 2.3e-7 degrees and in type at every one. The page's Jacobian is invertible wherever the map is a map, so it sends a zero gradient to a zero gradient and cannot move a critical point anywhere.

What the page cannot move

Six rungs measure what a page does to a field's readings and every one of them moves. The critical points do not: found on the sphere and found again in a projection's own page coordinates with nothing shared between the searches, they agree to 10⁻⁷ degrees and in type at every point, on every projection. What the page does move is their shape, by a factor bounded exactly by the indicatrix's axis ratio squared.

distortion · Gradient

Named alongside it

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

TopologyGauss–Bonnet theoremInvariantAngle deficitConformalityCritical pointContinuityDegeneracyGoldberg polyhedronGradientIcosahedronInterruption

All concepts