What a machine does with it
A screen map is a pyramid of tiles
The scheme every slippy map runs on is a coordinate system with three integers and one projection, and almost all of it is forced. A square world is what makes the quadtree work, the levels are exact powers of two, and the published resolution — 156,543 metres per pixel at zoom zero — is a distance on the ground at exactly one latitude.
The scale of a screen map is not one number
A zoom level prints one scale for the whole world, and the map is at that scale along exactly one line. At 60° north the picture labelled 1:136,495 is a 1:68,765 map — and the factor is not quite sec φ either, because the projection puts a geodetic latitude into a spherical formula.
A scale bar is right in one place
The bar in the corner of a world map is a picture of a distance, and it is a true picture along one line. On Mercator it reads 500 kilometres for a thousand at 60° north — and on an equal-area map it reads 500 one way and 2,000 the other, so the projection recommended for measuring is the one on which no single correction exists.
The square costs the poles
The cut at 85.0511287798° is the most-quoted number in web mapping and is almost never derived. It is where Mercator's northing equals half the world's width — the condition for a square — and it drops 1,901,487 square kilometres. A two-tile root would have reached 89.786° and dropped 3,558.
The pixel is a place with a size
Drawing a coordinate rounds it to a pixel, which moves it by up to half a diagonal — 16.8 metres at zoom 12. Whether two points 43 metres apart appear as two dots is not a property of the data at all: they merge for 12 per cent of the positions the tile grid could take, and the closed form predicts 12.4.
A tile is drawn without its neighbours
Independence is what makes the scheme scale: one request touches one square of the world. It also means a feature crossing nine tiles is nine features, each labelled at its own centroid — up to 1,863 kilometres from the whole one — and the displacement is bounded by the size of the feature rather than the size of a tile, so it grows as the reader zooms out.
The pyramid did not have to be Mercator
The usual defence is that a quadtree needs a square world and Mercator supplies one. So does the cylindrical equal-area with standard parallels at ±55.654° — the solution of π cos²φ₀ = 1 — and it needs no polar cut at all. What Mercator actually buys is conformality, and the price of giving it up is 13.8° of shear at 60° north.
A coordinate without its system is not a location
The same two numbers name ground 103 metres apart on one datum, 195 on another, and 7,942 kilometres away if the pair is read in the other order. The datum errors are the dangerous ones because they are plausible — and the axis swap, which everyone calls obvious, does nothing at all along a line that crosses three continents.
A degree is not a unit of length
"Within 0.01 degrees" is a condition anybody can write and no instrument can measure. On the ground it is an ellipse — 1,106 metres north–south and 558 east–west at 60° — and even on the equator it is not a circle, because the meridian's radius of curvature is smaller than the parallel's by the flattening.
Computing an area needs a surface
A shoelace over a ring of coordinates returns a number whatever the coordinates are. For one twenty-by-ten-degree cell it returns 1.75 times the true area in degrees, 3.06 in a conformal plane, and exactly the closed form in an equal-area one — and the closed form itself is 0.45 per cent out, because the sphere is a model too.
A straight segment is a claim about a plane
Two exact endpoints, joined by a straight line in the plane the file is stored in. On the ground the line is 718 kilometres from the route it claims between New York and London, and 2,961 between London and Tokyo. The departure grows as the square of the length — fitted exponent 2.001 — so a stated tolerance costs vertices as a square root.
The antimeridian is a cut in the numbers
A twenty-degree box across 180° has a bounding box of 359.4°, a planar area seventeen times too large, and a midpoint 20,015 kilometres from where it belongs — which is the antipode, exactly. Moving the cut moves the failure and never removes it, because a circle cannot be numbered by an interval.
Nearest is a question about the metric
Asked in the plane the data is stored in, a nearest-neighbour query returns a different site for 5 of 400 queries over Europe on Web Mercator — and 103 of 400 on the plate carrée, with the wrong answer up to 309 kilometres further away. Every disagreement is a contest closer than the region's own scale spread, and over a city there are none.
Reprojecting a raster invents values
Moving a picture from one projection to another moves no coordinate — the maps are exact both ways. What is lost is that a target cell's centre does not land on a source cell's centre, so a value has to be made up for it, and the making-up has an order of convergence: 1.00 for nearest, 1.98 for bilinear, 2.93 for a cubic, measured by refining the grid.
The operation decides the coordinate system
Five candidate planes over one region, scored on the three things a spatial operation depends on. The conformal conic wins shape and distance and is 11.7 per cent out on area; the equal-area member is exact on area and 38.9° out on shape. No candidate is exact in two columns, and no candidate ever will be, because one that was would be an isometry.
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.
A centroid belongs to a plane
Every renderer labels a region at its centroid, and every centroid is a shoelace over coordinates as stored — which is a statement about the plane they are in. Six planes put the middle of one 20° × 20° region up to 273 kilometres apart, the equal-area member is 66 kilometres out, and the disagreement falls as the square of the region's size.
Inside is a claim about the edges
Whether a point is inside a polygon is not a property of the point and the polygon. It is a property of the plane the edges were understood to be straight in, and between two readings of the same file there is a band of disagreement — 29 per cent of one triangle's area, 4.2 million square kilometres, and the file does not say which reading it means.
An address is an area
A cell identifier does not name a place, it names a region — so its precision is an area rather than a length. On the obvious lon/lat scheme that area varies by a factor of 10 at level 4 and 163 at level 8, and the factor doubles with every level: the same identifier length means less ground the further north it is used.
Hexagons cannot tile the sphere
Hexagons are the best cell shape a plane offers and the sphere will not take them. Euler's formula forces exactly twelve pentagons into any such tiling — twelve at 42 cells and twelve at 642, while the hexagon count rises twenty-one-fold — and each of the twelve is measurably smaller than the hexagons around it.
A cell system trades area for shape
A grid can hold every cell to exactly the same area or hold every cell nearly square, and the measurement says it cannot do both: the equal-area cube's areas agree to a part in a thousand and its worst cell is 1.29 times as long as it is wide, while the tangent-warped cube holds shape to 1.19 and lets area vary by 20 per cent.
A query is a disc, and a disc is not a cell
Everything a cell system does is an address lookup except the one question anybody actually asks it: find everything within five kilometres of here. That is a disc, and the number of cells it fetches is not its area divided by a cell's — at the radii a query is really made at, it is three to seventeen times that.
The address is a curve through the sphere
A database does not fetch a set of cells, it reads ranges of identifiers — so the cost of a query is how many runs its cells form, not how many cells it needs. Hilbert order wins that measurement and loses the one usually quoted for it: its neighbouring cells are further apart in identifier than row-major's, on average and at worst.
An edge has no order of convergence
On a smooth field the three resampling kernels converge at orders 1, 2 and 3 and the choice is obvious. Across a discontinuity they converge at 0.78, 0.58 and 0.60 — within a factor of 1.4 of each other, in an order that puts nearest-neighbour first, and a real raster is mostly edges.
A tilted view has no zoom level
Every essay about the screen so far assumes the map lies flat on it, which was true until about 2015. Pitch the camera sixty degrees and one frame asks for 3.6 zoom levels at once, a square tile covers ground four and a half times deeper than it is wide, and the pyramid has one integer per tile to answer with.
One edge is not an edge
The three resampling kernels were measured across a discontinuity and came out at 0.78, 0.58 and 0.60 — one straight edge at 27° to the graticule. Across thirteen edges the same kernels span 0.19 to 0.87, the ranking between them reverses, and for an edge lying along a parallel the error does not fall with refinement at all.
A real edge has a width
Thirteen edges were measured and every one of them was exactly discontinuous, which no sensor has ever produced. Convolving them with a point-spread function of one degree — a cell or two — takes the three kernels from 0.78, 0.58 and 0.60 back to 1.23, 1.97 and 3.60, and takes the edge along a parallel, which converged at −1.49, up to 1.92 for bilinear and 3.73 for cubic.
The same data on two grids
Five essays have addressed, queried and ordered cells within one scheme and nobody has moved a number between two. Doing it exactly — area-weighted, both directions — preserves the total to 2 × 10⁻¹⁶ and loses 39 per cent of the field's own standard deviation in a single round trip; six round trips leave 23 per cent of its variance. The quantity that would reveal the damage is the one that never moves.
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.
Cells that are rectangles in no coordinate
The previous rung measured what moving a field between two cell schemes costs, and did it between two schemes whose cells are longitude–latitude rectangles — which is what made every overlap a rectangle with a closed-form area. The schemes anybody actually argues about have cells that are rectangles in no coordinate, and their overlaps have to be clipped.
A line has a length only at a scale
Every measurement on this site so far has been of a curve given by a formula, sampled as finely as the picture needed. A map is not that: the geometry that reaches the page has been through an algorithm whose job is to throw most of it away. The first thing that goes is the idea that the line had a length.
Simplification does not commute with the projection
A pipeline either simplifies the geometry and then projects it, or projects it and then simplifies. Both orders are in use, neither is recorded, and given the same tolerance in ground metres they keep different vertices — 129 of them on the ground, 367 in degree space at 80°, and 459 on an equal-area page.
A tolerance is a promise about the picture
Douglas–Peucker guarantees exactly one thing: no vertex it discarded is further than ε from the line drawn in its place. It says nothing about the enclosed area, nothing about which side of the boundary a point ends up on, and nothing about whether the curve still fails to cross itself — and all three are what the geometry is usually being asked.
A polygon on a sphere has no outside
Seven essays have treated a stored ring as a boundary between inside and outside. A closed curve on a sphere divides it into two pieces and neither of them is the outside, so every polygon in every file depends on a convention that no coordinate carries — and the two conventions in common use disagree by a factor of fourteen on any ring that contains a pole.
The attribute is a claim about the geometry
Fourteen essays price what a stored coordinate means and not one asks what the number stored beside it means. A rate is a quantity divided by an area, the area belongs to the geometry, and no format records which area — so a simplification that moves the outline by nothing visible moves the implied total by 1.55 per cent, an unweighted average of densities is 4.09 per cent out, and a choropleth gives a polar square kilometre fifteen times the ink of an equatorial one.
The same number of cells, in two shapes
Moving a field between two cell schemes loses 18 per cent of it per cell in one geometry and 39 in another, and the earlier measurement could not say whether that was the shape of the cells or the ratio of their sizes, because changing the schemes changed both. Holding the counts settles it: the count ratio decides most of the loss, and the shape is still worth a quarter of the field.
When the edges do not line up
Rung eight held the cell counts equal so that shape could be compared without the count ratio drowning it, and recorded a doubt: a longitude–latitude source shares its boundaries with a longitude–latitude target wherever their counts share a factor. The mechanism is real and worth a factor of two. It was not what the published number was made of.
The road is drawn two pixels wide
Seven rungs measure what a screen map does to position. Nothing on a map is a point: every mark has a width, the width is chosen in pixels, and a two-pixel road covers 9.78 kilometres of ground at the equator and 1.70 at 80° north. That is a generalisation applied at a strength varying by a factor of six across one sheet, by a stylesheet with no latitude in it.
A boundary that two features share
Three rungs simplify one curve and price what a tolerance covers. Almost no boundary in a real dataset belongs to one feature: a county's edge is the next county's edge, it is stored twice, and it is simplified twice. What opens between the two answers is a region belonging to both features or to neither, and its area is not bounded by the tolerance.
A thousand features are wrong in the same direction
The area a simplification costs is unpredictable in sign for one feature. Over a population it is not: twenty-four presentations of one shape all lose area, the mean is seventy standard errors below zero, and no amount of aggregation removes it.
How many features a scale can carry
Töpfer's radical law is quoted everywhere as a rule of thumb. It is not one: it is a theorem about a size distribution with a Pareto exponent of exactly one half, exact to 1.8 per cent for that population and out by 99.4 per cent for a lognormal one.
The answer depends on the cells it was counted in
Nine essays price the cell as a shape. The number reported out of it is priced nowhere: sliding a grid without changing its resolution moves the largest reported value by 12.6 per cent, which is more than halving the resolution costs.
The renderer runs out of numbers before the zoom does
Eleven essays price the pyramid in exact arithmetic. The pipeline that draws it carries single precision, where a world coordinate near the antimeridian quantises to two metres — at every zoom, because the number does not get smaller when the pixel does. At zoom 22 that is fifty-four pixels.
The query a fast path actually answers
Fifteen rungs price what a stored coordinate means and none asks what it is searched with. No index answers "within two hundred kilometres of here"; an index answers "inside this rectangle of stored coordinates", and the rectangle is built by somebody's arithmetic — which at 55° north silently drops a tenth of the true answer on a conformal projection and three fifths of it on an equal-area one.
A cell's children do not fit inside it
Ten rungs price one cell system at one resolution, and every one of them is used hierarchically. A hexagonal hierarchy does not nest: at the aperture-seven scheme the discrete global grids use, one fourteenth of a parent is covered by no child of its own, exactly, and each of the six ring children is eleven twelfths inside.
A label belongs to no tile
Nine rungs price the tile as a piece of geometry. A label is not geometry — it is a page object placed by collision against other page objects, and collision is a global relation while a tile is rendered alone. Cut a page into tiles and 10.3 per cent of the labels are placed differently; the buffer that closes the gap is half a tile at moderate density and a whole one when it is crowded.
Which features survive is not a sample
The rung below answers how many features a scale can carry and treats the population as a number. Which ones survive is a different question: keeping one feature in ten carries 99.99 per cent of the total length and inflates the median feature by a factor of 95, and the shape of the size distribution survives both exactly.
Two routes to one scale
A national series is cascaded — the million is derived from the quarter-million, which was derived from the fifty — and the folklore is that the errors accumulate. They do not: Douglas–Peucker and Visvalingam both cascade to the same line the direct route produces, bit for bit, because both output a sublevel set of a per-vertex number. What breaks it is anything else in the chain, and a moving average puts the product nine tolerances away.
A tolerance in map units is not a tolerance
A snapping tolerance is a number, and the number is in whatever units the file is in. Five map units on Web Mercator is 4.97 metres of ground at the equator and 0.87 at eighty degrees — so a rule that merges two features three metres apart merges them everywhere below 52.8° north and refuses everywhere above it, in one pass, over one dataset, with nothing recording where the boundary is.
The orientation is a policy
A polyhedral cell system has three free angles nobody scores. They cannot improve it: rotating the solid rotates every cell rigidly, so the distribution of cell areas is identical for every orientation there is. What they decide is who stands on the bad cells — and the eight cities measured here get a spread of cell area of 1.00 under the best turn and 1.50 under the worst.
50 essays in this field, the first 16 of them shown with their opening figure.