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

A globe folded from flat faces

polyhedral-net is the unfolding, face-choice the projection used on each face, and solid-globe the assembled solid the net came from.

polyhedral-net is the unfolding, face-choice the projection used on each face, and solid-globe the assembled solid the net came from.

Every one of these is the same generator answering a different question, which is why they share a file, a set of colour roles and a set of assertions. Changing it changes all 3.

15 essays call it. Every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

What it draws

One picture per question the family answers. Each has a page of its own.

The sphere on an icosahedron, unfolded. A polyhedral map: the sphere projected face by face onto an icosahedron and the solid cut open along 11 of its 30 edges. The face projection here is the gnomonic, which draws every great circle as a straight line within a face, and the worst angular deformation inside a face is 6.6°. Each cut is a place where two pieces of the world that touch are drawn apart; each corner of the solid is a place where the surface has curvature and the map has an angle deficit.

polyhedral-net

The sphere on an icosahedron, unfolded 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.

solid-globe

The icosahedron's faces, drawn on the sphere One face of an icosahedron, two ways. The same spherical face mapped onto the same flat triangle by two different rules, with a grid of marks whose size is the local areal factor. The gnomonic map draws every great circle straight and stretches the corners by a factor of 1.50; the area-preserving map holds the areal factor at one to a part in a million and pays in shape, reaching 11.9° of angular deformation against the gnomonic's 8.0°. Both take the face's boundary to the face's boundary, which is what lets the pieces still fit together.

face-choice

One face of an icosahedron, two ways

Where it is called

Changing this changes every one of these figures.

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 families

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.

 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 families

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.

 One face of an icosahedron, two ways. The same spherical face mapped onto the same flat triangle by two different rules, with a grid of marks whose size is the local areal factor. The gnomonic map draws every great circle straight and stretches the corners by a factor of 1.50; the area-preserving map holds the areal factor at one to a part in a million and pays in shape, reaching 11.9° of angular deformation against the gnomonic's 8.0°. Both take the face's boundary to the face's boundary, which is what lets the pieces still fit together. The families

What a face can preserve

The obvious map onto a polyhedron's face is the gnomonic, and it draws every great circle straight while stretching the corners by a factor of 1.50. Replace it with a construction that holds the areal factor at one to a part in a million and the shape error rises from 8.0° to 11.9° — the same trade the whole sphere forces, arriving on a piece of it a twentieth the size.

 The conformal map onto a square face of a cube. The spherical face of a cube carried onto its flat face by a map that is conformal everywhere — the measured angular deformation over the drawn interior is 1.63e-6°, which is the arithmetic's own floor. The rings and spokes are circles and radii on the sphere, and they cross at right angles here because that is what conformal means. The map was solved for as 16 terms of a series rather than written down: the face's edge comes out straight to 0.33 per cent of its own half-width, and that residual — not the conformality — is what more terms buy. At each corner the map behaves like ζ^0.75, so the scale factor there is infinite. The families

A conformal map onto a face

The polyhedral ladder ended owing a conformal face map, on the grounds that it needs elliptic functions. It does not: a conformal map of the sphere is an analytic function of one conformal coordinate, so the map is a power series, choosing it is a least-squares fit — and its scale factor is infinite at the corners, which is the angle deficit arriving as a singularity.

 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. The families

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.

 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 families

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.

 The indicatrix is a limit, and here it is being taken. The axis ratio of the image of a circle of radius ρ, measured from the picture rather than computed from the derivatives, against the ratio the derivatives give. It converges, and it converges as ρ² — each halving of the radius quarters the departure. At the smallest radius drawn the two agree to 7.6e-4 relative. This is what "the indicatrix is a limit" means when it is working: there is a number for the ratio to settle on, and it settles. Measuring distortion

The indicatrix at a point that has none

Ten essays on this ladder take the derivative for granted. Rung three proved the indicatrix is a limit and every figure since has assumed the limit exists. There are points on the maps this collection draws where it does not, and they are not exotic: the horizon of a gnomonic map, and every corner of every polyhedral net.

 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. What a machine does with it

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.

The whole library