Lower bound — where it appears
Named by 13 essays across 6 fields — each of them below, with the objects they name alongside it.
Total curvature and the scale rule
The impossibility has a size. A region covering a fraction of the sphere carries a fixed amount of curvature that any flat map must absorb, and for a circular region the least distortion any conformal projection can achieve is a closed form nobody can beat.
How small is flat enough
A builder works in plane coordinates and a national mapping agency does not, and the line between them is not a convention. The unavoidable error of treating a patch of the Earth as flat grows as the square of its size, and the size at any stated tolerance is a number.
The curvature of the Earth is not one number
An ellipsoid's Gaussian curvature varies by 1.35% from equator to pole, so the sphere is not developable onto the ellipsoid any more than the plane is onto the sphere. The spherical approximation is a projection with an irreducible cost, and the cost is 6,739 parts per million.
How many sheets an atlas needs
A tolerance on the scale error inverts, through Chebyshev's bound, into a sheet radius — and a covering problem turns the radius into a count. One part in a thousand costs 1,210 sheets of 403 kilometres radius, the count goes as the reciprocal of the tolerance exactly, and the projection multiplies it by anything from one to fifty-six.
The sphere is not the plane at small counts
The site's atlas arithmetic multiplies an ideal sheet count by 2π/√27, the thinnest covering density of the plane. At the four counts whose optimal covering of the sphere is a theorem the plane's number is 21 per cent high at two caps and 9, 5 and 2 per cent low at four, six and twelve — wrong in both directions, and the direction changes with the count.
What the net heuristic cannot find
Choosing a net by unfolding outwards from a face beats guessing, and the earlier measurement of that left its own limit unknown. Enumerated in full, the heuristic turns out to be exactly optimal on every solid small enough to check — rank one of 384 — and above that ceiling no sample can tell whether it still is, because eight hundred random nets never reach it.
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.
A flat picture has one direction between two places, and the Earth has two
The bearing from New York to Tokyo and the bearing back are not reverses of each other: they miss by 128 degrees. A flat picture draws one line between the two cities, so no picture containing both can have a bearing error below 64 degrees. On any set of places up to about fifty degrees across, the best flat picture sits exactly on that floor — and away from the poles, so does Mercator.
Five bearings of twelve, and only eight ways to draw them
A flat picture of four places can hold five of their twelve one-way bearings exactly, the same count as distances, and the five are found by solving equations rather than searching. The equations cannot tell a line from its reverse. Of the 192 ways to choose five, eight can be drawn with every arrow pointing forwards, all eight leave London and New York unheld, and on sixteen cities no choice made at random can be drawn at all.
Every density can be met and none is free
Four maps of the same data, all of them correct, charging between 57.6° and 104.4° of angular deformation for it. There is no such thing as the cartogram of a density — there is an infinite family, and somebody picked a member of it without saying so.
A reach set drawn from the river's outline is a bound again
Beside a bending river, a fan of bearings drawn with straight chords claims a strip of water, and the drawing stops being a lower bound. The fan's own numbers cannot repair it: what a bearing records about the river is where the bridges are, not where the bends are, and the best a rule built from them does is cut the over-claim by seven eighths while giving up twenty times as much ground. The river's outline repairs it exactly. Cutting each bearing at the first bank costs three and a half times the over-claim it removes; keeping the chords wherever a route over one bridge reaches costs one and a half. Drawn from the outline and the bridges with no fan at all, that route set misses 6.4 per cent of the reachable ground where the fan misses 35.
A layer is a meeting with the bank, not a corner turned
A reach set drawn in layers — rays from the centre, then from every place a route can leave the bank — ends in the exact answer, and how deep it must go depends on what a layer is. Counted by corners turned, the depth belongs to the polygon the bank is drawn as: five turns on a bank drawn every 400 metres, eighteen on one drawn every 50. Counted by meetings with the bank, it is three at every resolution. One meeting anywhere leaves 1.6 per cent of a ten-kilometre walk unreached, where rays from the bridges alone leave 6.4; two leave 0.47; three leave nothing. A longer walk needs more — up to six meetings at twenty kilometres across thirty-six rivers — and the bend's steepness, which prices the first layer, ranks the depth at 0.24.
The cheapest map that meets its areas
An earlier essay bracketed a cartogram's least cost between a construction charging eighty degrees and a bound valid only for symmetric densities, and recorded the gap as a shortfall. One request settles it: a density of contrast eighty whose least cost is exactly zero, met by a map written down in closed form, while the standard construction charges 43.8° for it.
Named alongside it
The objects these essays reach for when they reach for this one.
Spherical capToleranceChebyshev's boundOptimisationVerificationAngular deformationAreal factorAzimuthBarrierBearingBoundaryCartogram