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

The scale asked for and the scale the pyramid has, at 0°

Drawn here at the parameters it defaults to, with every essay that calls it.
The scale asked for and the scale the pyramid has, at 0°. A tiling scheme exists only at integer zoom levels, a factor of two apart in resolution, so a request for any scale between them is answered by the nearest rung. The ratio runs from 0.707 to 1.405 — 1/√2 to √2 — and repeats identically at every doubling, which is four times that in area. A request for 1:10,000 is served at zoom 16, which is 1:8,531; A request for 1:25,000 is served at zoom 14, which is 1:34,124; A request for 1:50,000 is served at zoom 13, which is 1:68,247. Nothing anywhere reports it, because the map that arrives is a perfectly good map of something.

A tiling scheme exists only at integer zoom levels, a factor of two apart in resolution, so a request for any scale between them is answered by the nearest rung. The ratio runs from 0.707 to 1.405 — 1/√2 to √2 — and repeats identically at every doubling, which is four times that in area. A request for 1:10,000 is served at zoom 16, which is 1:8,531; A request for 1:25,000 is served at zoom 14, which is 1:34,124; A request for 1:50,000 is served at zoom 13, which is 1:68,247. Nothing anywhere reports it, because the map that arrives is a perfectly good map of something.

It is drawn by screen-figure with show: "zoom-ladder" — one member of a family of 12 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.

The scale asked for and the scale the pyramid has, at 0°. A tiling scheme exists only at integer zoom levels, a factor of two apart in resolution, so a request for any scale between them is answered by the nearest rung. The ratio runs from 0.707 to 1.405 — 1/√2 to √2 — and repeats identically at every doubling, which is four times that in area. A request for 1:10,000 is served at zoom 16, which is 1:8,531; A request for 1:25,000 is served at zoom 14, which is 1:34,124; A request for 1:50,000 is served at zoom 13, which is 1:68,247. Nothing anywhere reports it, because the map that arrives is a perfectly good map of something. What a machine does with it

Zoom is a ladder

A tiling scheme exists only at integer zoom levels a factor of two apart, so a request for 1:25,000 is answered with 1:34,124 — 36 per cent coarser, and 86 per cent coarser in area. The mismatch runs from 1/√2 to √2 and repeats identically at every doubling, and nothing anywhere reports it, because the map that arrives is a perfectly good map of something.

 One unit of a vector tile, in metres of ground. A vector tile's coordinates are integers on a lattice 4096 units across the tile, and the tile halves at every level, so one unit is a distance that halves too: 5.48 m at z10 and 0.086 m at z16, at 55°. It is also a different distance at every latitude, by cos φ, because the tile is in Web Mercator — the same factor that makes a grid metre a different quantity of ground at every latitude, arriving in the file format rather than in the projection. What a machine does with it

A vector tile has an integer grid

Six essays on this ladder treat a vector tile as the thing a raster tile is not: geometry, resolution-free, styled at draw time. Its coordinates are integers on a lattice 4,096 units across a tile, the tile halves at every level, and at 55° north one unit is 88 metres at zoom 6 and 21 millimetres at zoom 18.

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