Self affinity
Named by 4 essays across one field — each of them below, with the objects they name alongside it.
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.
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.
The room a jagged graph takes up
The graph of a continuous function is a curve, and a curve ought to have dimension one. Build the function by raising midpoints — by an amount that shrinks more slowly than the intervals do — and its graph needs more boxes than any line, at a rate fixed by one ratio. A random path built the same way needs exactly as many.
Two numbers in one jagged record
A measured record comes with no rule, so its dimension has to be estimated from a finite stretch of samples — and two things go wrong that never trouble a graph built from a formula. The popular estimator depends on the units the record is written in, and the dimension, which describes the record up close, turns out to be independent of its memory, which describes it from far away. For a self-affine path the two are tied by D = 2 − H; for a record, they are two numbers.
Named alongside it
The objects these essays reach for when they reach for this one.
Box dimensionScalingCoveringHausdorff dimensionIterated function systemSelf-similarityAttractorCorrelationDissipationFractal dimensionFractional brownian motionGram schmidt