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Concept

Covering

A collection of sets whose union contains the set in question. Both definitions of dimension are about covers - box counting fixes their size and takes a count, Hausdorff's allows any sizes and takes an infimum - and the difference between the two answers is exactly what that freedom buys.

Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.

((p ∧ q) ∨ (r ∧ s)) ∨ (¬p ∧ ¬r), covered by 3 rectangles. A grid of the assignments arranged so that neighbouring squares differ in one variable.

The map that puts neighbours side by side

Reorder the rows of a truth table so that neighbouring squares differ in one letter, and finding a short formula stops being algebra and becomes the problem of covering a shape with rectangles.

logic · Truth functions Infinite below 0.6309, nought above it. The total of the s-th powers of the diameters in the natural cover of the middle-thirds Cantor set, plotted against s for 4 depths. Every curve passes through one at s = 0.6309 and they separate either side of it.

Infinite on one side and nought on the other

Box counting returns a growth rate. Hausdorff's definition returns a measure — a quantity that is infinite for every exponent below the dimension and zero for every exponent above it, and the dimension is the one place where it is neither.

dynamics · Fractal dimension A carpet whose two dimensions differ by 0.076. A self-affine carpet built by keeping 5 cells of a 4 by 2 grid and repeating 4 times. Its box dimension is 1.6610 and its Hausdorff dimension 1.5850.

A carpet with two dimensions

For every self-similar set met so far the two definitions of dimension agree, and the agreement is a theorem about sets built from copies of themselves scaled equally. Stretch one direction more than the other and the two numbers come apart, by an amount that can be computed exactly.

dynamics · Fractal dimension A graph of dimension 1.585, built by raising midpoints. The Takagi–Landsberg graph with w = 0.75 on the unit interval, drawn with 16 columns shaded by how far the graph rises and falls across each. Counting boxes at seven widths gives a slope of 1.582.

The room a jagged graph takes up

The graph of a continuous function is a curve, and a curve ought to have dimension one. Build the function by raising midpoints — by an amount that shrinks more slowly than the intervals do — and its graph needs more boxes than any line, at a rate fixed by one ratio. A random path built the same way needs exactly as many.

dynamics · Fractal dimension Where discs of radius one cover the lattice, and where they leave holes. Six lattices of algebraic integers drawn as points in the plane with a unit disc around each; for five of them the discs cover the whole plane and for the sixth, the integers of the field of the square root of minus nineteen, uncovered holes remain.

Factoring uniquely with no way to divide

Unique factorisation is proved by dividing with a small remainder, and in the Gaussian integers that works because discs of radius one cover the plane. In the integers of ℚ(√−19) the discs leave holes, no division algorithm of any kind can be made to work — and factorisation is unique anyway. The same field is why n² + n + 41 is prime forty times running.

number · Unique factorisation

Named alongside it

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

Box dimensionHausdorff dimensionScalingSelf-similarityMeasureSelf affinityAlgebraic integerCantor setDiscriminantEuclidean algorithmGaussian integersGray code

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