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

One face of an icosahedron, two ways

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

It is drawn by polyhedral-figure with show: "face-choice" — 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.

8 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.

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.

 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.

 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.

 The corner against where along the seam the feature crosses. A feature crossing the seam of a cube at right angles, drawn on each face's own map and unfolded, with the crossing moved along the edge. Every curve starts at zero, because the face's mirror through the edge's own midpoint reverses the along-edge direction and forces the shear term in the Jacobian to be odd. Away from it the gnomonic — the map with no corner at all in the rung below — reaches 44.4°, three times the equal-area map's and far past the conformal one's. The edge's half-length is 35.3°, so the right-hand end is still well inside it. The families

The corner is not at the midpoint

The rung below measured the corner a feature gets crossing a polyhedral seam, and found none at all under the gnomonic face map. It crossed at the edge's own midpoint every time — the one point on the edge where a face's own mirror symmetry forces the corner to vanish. Two fifths of the way to the vertex the gnomonic gives 20.15°, the equal-area map 7.28° and the conformal map under a degree, which reverses the ordering entirely.

 One of these is a corner. The corner reported at the exact conformal seam, and the corner reported at a cube's gnomonic seam fifteen degrees along its edge, each measured over a shrinking arc. The gnomonic's is 21.4572° at every span from twenty-four degrees down to three — the same number to four decimals — because it is a corner. The conformal one halves whenever the span does, fitted exponent 0.990, because a tangent read from a finite chord of a curved image departs from the true tangent in proportion to the chord. It is not a corner; it is the instrument. The families

The exact map says the seam is smooth

The previous rung could only bound the conformal seam's corner at about two degrees, because the series it was fitted with holds its boundary condition to three parts in a thousand. An exact map exists — the stereographic projection composed with ∫dt/√(1 − t⁴) — and it settles it: the corner falls in exact proportion to the arc the tangent is read over, fitted exponent 0.99, while the gnomonic's 21.4572° is the same to four decimals at every span.

 Three face maps on a cube, over a shrinking span. The corner a feature gets crossing the seam of a cube, measured by reading the tangent over an arc and then shrinking the arc by a factor of sixteen. A chord differs from a tangent in proportion to the arc, so a perfectly smooth join reports a corner that HALVES when the span halves — a slope of one on these axes. None of the three lines has a slope of one. The fitted slopes are 0.000, 0.000, -0.018, which is a flat line in each case, and a flat line is a real corner. The three differ in size and not in kind: 20.1513°, 7.2772°, 0.7687°. The families

The span ladder, run on all five

A recorded shortfall said the exact map's span ladder was one loop away from settling the seam on the five Platonic solids. The loop was run and the answer is the other one: the series conformal face map has a corner that does not shrink with the measurement span on any of them — 4.10° on the tetrahedron, 0.77 on the cube, 0.004 on the icosahedron — and it belongs to the truncation rather than to conformality.

 Walking the crossing all the way to a cube's vertex. The corner a feature gets crossing the seam, at crossings that approach the vertex geometrically — the last is 3.5e-4 degrees from it. Every face map is singular at a vertex, so the expectation is that the corner runs away. It does not. The gnomonic settles at 53.1295° and the equal-area map at 12.9656°, and both are finite. The dashed line is the solid's angle deficit, 90.0°, which is what the surface loses at that point and is not what either map's corner reaches. The families

The corner that is the curvature

Every face map is singular at a vertex, so a corner that grows as the crossing walks towards one should run away. It does not: the gnomonic settles at 53.1295° on a cube and the equal-area map at 12.9656, both finite, both reached like the first power of the remaining gap. What does not settle is the deficit beside them — 90 degrees, fixed by Descartes before any projection is chosen.

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