Concept

Fractal dimension — where it appears

The exponent relating a curve's measured length to the ruler used: a boundary of dimension D has a length growing as the ruler's length to the power 1 − D. It is why a coastline has no length, and why any statistic with a perimeter in it has no limit either.

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

$The score does not settle at any resolution. The compactness of one stated boundary — a circle with cosine ripples at eight geometrically spaced wavenumbers, so it has structure at every scale — read at sixteen vertices up to two thousand and forty-eight. The ground score falls from 0.980 to 0.834, and it keeps falling: the boundary's length grows without bound as it is resolved while the area it encloses converges, so the quotient has no limit. The four page curves sit within a fraction of a per cent of the ground curve and of each other, which is the comparison this rung exists to make.$

The score is not stable at any scale

One boundary, read at eight resolutions from sixteen points to two thousand and forty-eight: the compactness score falls from 0.980 to 0.834 and is still falling. Changing the projection instead moves it by 0.69 per cent. The two decisions are made by the same person on the same afternoon and only one of them is ever reported.

paths · Reach

A curve built to have dimension 1.2619. The generator replaces every segment with four of equal length at headings 0, +60.0°, −60.0° and 0. Closing the displacement fixes the length ratio at 0.33335, and four copies at that ratio give a dimension of exactly log 4 / log(2 + 2 cos θ) = 1.2619. Nothing here is measured yet: this is the construction the measurement will be checked against. Drawn at depth 5, which is 1024 segments, with the second-level shape shown faint beneath it.

A line has a length only at a scale

A curve given by a formula can be sampled as finely as the picture needs. 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.

applied · Generalise

The coast copied twelve miles out gives up the sea of every narrow bay. A stated coast 116 km long with five bays 5 km deep, 6, 10, 16, 24, 36 km wide. Dashed: the envelope of arcs, the line every point of which is twelve nautical miles from the nearest point of the coast. Solid: the coast itself moved twelve miles seaward, which dents the limit by the full 5 km over every bay. Tinted: the sea between them, 240 km² in all, 9% of a straight coast's territorial sea. Over the 6 km bay the copy lies 4.8 km inside the envelope; over the 36 km bay the two meet. The coast is stated, not surveyed.

Copied out to sea, a coast keeps its length and loses its bays

Before 1930 a territorial sea was drawn by following the coast in all its sinuosities, and the phrase fits two constructions. Moved twelve miles seaward, a stated coast of five bays gives up 240 square kilometres against the envelope of arcs, most of it over the narrowest bay; moved along its own normals, it ties a loop over every bay. On a fractal coast charted at finer detail the copy grows by a quarter at every level, to 480 kilometres, while the envelope settles at 133.0.

practice · Boundary

Named alongside it

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

Boundary Convergence Generalisation Resolution Verification Baseline Buffer Closed form Convention Divider walk Estimator Ground resolution

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