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Concept

Iterated function system

A finite list of contracting maps, whose attractor is the unique set equal to the union of its own images under them. It is the standard way of specifying a fractal, and the dimension it forces can be read off the maps' contraction ratios without covering anything.

Named by 5 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 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 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 — which was the whole case for defining dimension by counting. Kaplan and Yorke's formula computes it instead, from two numbers that describe the map and never look at the set.

dynamics · Fractal dimension A mass split 10 times, 0.3 to the left and 0.7 to the right. A self-similar measure on the unit interval: the mass is split 10 times, 0.3 of each piece's share going left and 0.7 right, and each of the 1024 pieces is drawn as a bar as tall as its mass. The heaviest carries 0.0282 and the lightest 5.9 × 10⁻⁶.

A dimension for every rate of crowding

Spread a unit of mass over an interval by splitting it unevenly, again and again, and the result covers the whole interval while crowding almost all of its weight onto a set of smaller dimension. Every rate of crowding picks out its own set of points with its own dimension, and the whole family is read off one curve.

dynamics · Fractal dimension A dust plus itself is a line. C level 6: 64 intervals, length 0.0878; C+C: 1 interval(s) [0, 2].

Two dusts that add up to a line

The middle-thirds Cantor set has no length at all, and yet adding it to itself — every sum x + y of two of its points — fills the whole interval from 0 to 2 without a gap. The reason is a single number, the thickness, and it is exactly 1 for this set. Make the dust a little thinner and its sum with itself collapses to a dust again; make two different dusts and whether they add to a line depends on their dimensions, unless arithmetic gets in the way.

dynamics · Fractal dimension

Named alongside it

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

Box dimensionScalingSelf-similarityCantor setAttractorHausdorff dimensionMeasureSelf affinityBinomial coefficientBox countingCoveringDissipation

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