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

Dihedral — where it appears

The angle between two faces of a solid along the edge they share, measured inside it. A net unfolds a solid by opening every kept edge to 180°, so the dihedral is how far each fold must turn, and how far it falls short of 180° is how sharply the sphere bends there.

Named by 2 essays across one field — each of them below, with the objects they name alongside 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 gnomonic crosses a seam without a corner

Choosing a polyhedral map means choosing 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.

families · Polyhedral The seam corner falls faster than the paper and crosses it at 720 faces. For an icosahedron globe of 100 mm radius with every face divided ν ways, the gnomonic face map's worst seam corner against what a hand-folded paper net is accurate to. On twenty faces the corner is 9.19°, 12.8 times the whole budget of 0.716°. The corner falls as ν to the power -1.97 and the fold's error as ν to the -0.97, so the two lines converge; the budget stops falling once the crease, a fixed quarter-millimetre radius of paper, is larger than the fold. The corner passes under it at ν = 6, 720 faces, where it is 0.315° against 0.321°.

The seam falls faster than the paper

Divide each face of an icosahedron ν ways and two things shrink at different rates. The fold's error falls as 1/ν, because it is the face's own radius times a degree. The gnomonic seam corner falls as 1/ν², because it needs both a tilt between faces and a distance along their edge. So the gnomonic, 12.8 times the whole assembly budget on twenty faces, goes under it at 720 faces on a 100 mm globe. No globe of any size needs more than 5,120, and what it costs is paper folded out of sight: 5.8 per cent of the sheet at that crossing.

families · Polyhedral

Named alongside it

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

GnomonicPolyhedral projectionSeamContinuityConvergence rateEqual-areaError budgetExponentFaceGreat circleNetScale

All concepts