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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 3 essays across 2 fields — 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 Measuring a closed curve with inlets by throwing lines at it. A curve inside a disc crossed by a sample of random lines with each crossing marked, beside the mean number of crossings and the length it implies against the true length.

A length counted by the lines that cross it

Throw straight lines at random across a curve and count how often they cross it. The average count, times π times the radius of the target, is the curve's length — for a wiggly closed curve, a spiral or a snowflake alike, with no following of the curve and no derivative anywhere. It is Crofton's formula of 1868, and it measures length the way a map-reader's ruled transparency does.

analysis · Arc length

Named alongside it

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

Box dimensionCantor setScalingSelf-similarityArc lengthAttractorCoveringExpectationHausdorff dimensionIntegral geometryIterated function systemMeasure

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