Concept

Hausdorff dimension

The exponent at which a set's measure stops being infinite and becomes zero. The measure is defined by covering the set with pieces of any sizes at all and adding the s-th powers of their diameters, and the freedom in the sizes is why this dimension is at most the box one and is the notion theorems are stated about.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

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 set on this ladder 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

Named alongside it

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

Box dimension Covering Measure Scaling Self-similarity Cantor set Iterated function system Koch curve Self affinity

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