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Concept

Self affinity

Being built from copies of itself contracted by different factors in different directions. It is the weakening of self-similarity that separates the two definitions of dimension: the pieces are rectangles rather than squares, so a grid of squares can never match their shape at any scale.

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

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 A dimension of 1.2576, from two stretching rates. The running averages of the Hénon map's two Lyapunov exponents, settling at 0.4177 and -1.6217. Kaplan and Yorke's formula turns them into a dimension of 1.2576 without counting a single box.

A dimension from the stretching rates

An attractor has no construction rule, so its dimension has to be counted — that was the rung below's argument for defining dimension by counting at all. Kaplan and Yorke's formula computes it instead, from two numbers that describe the map and never look at the set.

dynamics · Fractal dimension

Named alongside it

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

Box dimensionIterated function systemScalingAttractorCoveringDissipationGram schmidtHausdorff dimensionLyapunov exponentMeasureSelf-similaritySensitive dependence

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