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Concept

Koch curve

The curve obtained by repeatedly replacing the middle third of every segment with two sides of a triangle. It has infinite length and no area, four pieces at a third the size, and a dimension of log 4 over log 3 - which is the standard first example of a dimension that is not a whole number.

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

the Koch curve, after 5 steps. the Koch curve drawn from its own rule: replace the middle third of every segment with two sides of a triangle.

A dimension that is not a whole number

Cover a set with boxes of side ε and count how many are needed. For a line the count grows like 1/ε, for a region like 1/ε². For the Koch curve it grows like 1/ε to the power 1.26, and that exponent is as good a definition of dimension as the other two.

dynamics · Fractal dimension 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

Named alongside it

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

Box dimensionCantor setScalingSelf-similarityAttractorCoveringHausdorff dimensionIterated function systemMeasureSimilarity dimension

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