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

Fracture threshold — where it appears

The tolerance at which a near-optimal set stops being one piece. Below it a search has several genuinely different answers and above it one, and where it falls is a property of how much of a projection's variation the region can see.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

The pass between the basins, measured rather than bracketed. The height of the lowest path from one near-optimal basin to the other, for regions of growing size. It is found by sorting the score surface and joining cells in order, so it is the minimax path's own height rather than the level at which a bisection stops finding two pieces. The fitted exponent is -1.34 at a coefficient of determination of 0.986. The earlier level-set measurement on the same projection gave -1.57 and the argument predicts −1, so measuring the height directly moves the answer towards the prediction, though not onto it.

The height of the pass between two basins

One thing was left owed: the fracture threshold falls faster than the argument predicts, and the difference was supposed to be the level set's blindness to the height of the pass. Measured directly, the pass between the two deepest basins falls as the −1.34 power at R-squared 0.986 — nearer the predicted −1 than the level set's −1.57 on the same projection, and not onto it. It is not the pass that breaks the set first, and at five of six sizes the second basin is the first one's mirror twin.

choosing · Choosing The basin has three widths, and they differ by a factor of 4.6. Two sections through the near-optimal basin of the Robinson aspect over Japan, drawn at one scale. Each ellipse is the set of aspects whose score is twice the optimum's, from the objective's own second derivative at the optimum: 32.1°, 16.0°, 6.9° along the three principal directions. A single number for "the width of the basin" is the cube root of their product, 15.3°, and it is not any of them.

The basins have widths as well as depths

Measuring the height of the pass left one thing owed: shape means widths too. Measured, the basin has three of them — 32°, 16° and 7° at Japan — it gets wider rather than narrower as the region grows, and the exponent it predicts overshoots the measured one by half again.

choosing · Choosing The piece count rises and falls. The number of connected pieces of the near-optimal aspect set for Robinson over Japan, swept finely through the threshold rather than sampled once below it. It is one piece at a wide threshold, reaches 13 at 1.010, and returns to one as the set shrinks onto the single best aspect. The set first disconnects at 1.941, which is above the peak: the pieces keep multiplying after the first break. This is the sweep the earlier measurement could not afford and it costs one grid, because every threshold reads the same 4992 evaluations.

The threshold is not a percolation

The near-optimal aspect set was found breaking into twelve pieces rather than two, the transition was called a percolation, and the exponent went unmeasured. Swept finely, the piece count rises from one to thirteen and falls back to one — and refining the grid by a factor of fifteen moves the peak from ten to nineteen, at the rate the surface's own minima grow, while an uncorrelated field on the same lattice grows by a factor of twelve.

choosing · Choosing

Named alongside it

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

Aspect searchBasinExponentLevel setOptimisation landscapeShortfallAspectAnisotropyDegeneracyDiscretisationHessianLocal minimum

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