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

What interruption buys and what it costs

Drawn here at the parameters it defaults to, with every essay that calls it.
What interruption buys and what it costs. Mean angular deformation over the mapped world, and the total length of the tears as a fraction of the map's width, against the number of lobes. Cutting the map into eight gores drops the mean shape distortion from 38.6° to 5.5° and adds 2.2 map-widths of edge running through the middle of it. Both curves are real and only the first is usually drawn.

Mean angular deformation over the mapped world, and the total length of the tears as a fraction of the map's width, against the number of lobes. Cutting the map into eight gores drops the mean shape distortion from 38.6° to 5.5° and adds 2.2 map-widths of edge running through the middle of it. Both curves are real and only the first is usually drawn.

It is drawn by projection-figure with show: "interruption-tradeoff" — one member of a family of 16 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.

2 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.

Sinusoidal, cut into 6 lobes. six lobes, cut through the oceans so each continent stays whole. Each lobe is the same projection about its own central meridian, so each point is near a line where the shape distortion vanishes: the mean angular deformation over the mapped world falls from 38.6° uninterrupted to 17.8° here. What is given up is that the map is no longer one surface — the graticule stops at every cut because there is nothing on the other side of it. What each projection optimises

Giving up continuity

Cutting a map into lobes really does reduce the distortion, by a factor that can be measured. What is paid is that the map stops being one surface — and the size of the tear is the number that pictures of interrupted maps never carry.

 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.

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