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

The sphere on an icosahedron, unfolded

Drawn here at the parameters it defaults to, with every essay that calls it.
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.

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.

It is drawn by polyhedral-figure with show: "polyhedral-net" — one member of a family of 3 figures that share a generator, so the drawing above is what that generator returns when it is asked for this one and given nothing else.

10 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.

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.

 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. The families

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.

 Every one of the cube's 384 nets, scored. All 384 spanning trees of the cube's face graph, unfolded and scored on the total separation their cuts leave: how far apart, in edge lengths, the two copies of each cut edge end up on the page. Every one of them is a valid net — no Platonic unfolding overlaps — and they range from 11.30 to 17.97, a factor of 1.59. The heuristic of unfolding outwards from a chosen face lands on the first of them, exactly. The families

What the net heuristic cannot find

Rung seven chose a net by unfolding outwards from a face and showed the choice beats guessing, and recorded that its own limit was unknown. Enumerated in full, the heuristic turns out to be exactly optimal on every solid small enough to check — rank one of 384 — and above that ceiling no sample can tell whether it still is, because eight hundred random nets never reach it.

 The corner an equal-area face map puts in a feature, and the one the gnomonic does not. A great circle crossing the seam between two faces, drawn on each face's own map and unfolded flat, with the angle between the incoming and outgoing tangents plotted against how obliquely it crosses. Under the equal-area face map every solid gives a corner: zero for a perpendicular crossing, where the two faces are symmetric about the edge, peaking near thirty degrees of obliquity and falling again as the crossing lies down along the edge. Under the gnomonic it is zero at every angle on every solid, which is the flat line on the axis. The families

The gnomonic crosses a seam without a corner

Eight rungs choose a solid, a face map and where to cut. This one is about a seam the net does not cut, where two faces stay joined and carry two different maps — and the expectation was a corner in every feature crossing one. At the midpoint of the edge — which is where every crossing here is taken, and which turns out to be the one place on it a face's own mirror forces the corner to vanish — the gnomonic gives none at any obliquity on any Platonic solid, and the equal-area face map gives one of up to 0.39 degrees.

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