Hausdorff dimension

Hausdorff dimension measures how a set’s size scales with the resolution used to cover it. It can be noninteger and applies to highly irregular sets.

Hausdorff measure: A measure defined by covering a set with sets of small diameter and taking a limiting infimum of weighted sums. Hausdorff dimension is the critical exponent at which this measure changes from infinite to zero.
Metric space: A set equipped with a distance function satisfying nonnegativity, symmetry, and the triangle inequality. Hausdorff dimension is defined for subsets of metric spaces, not only Euclidean figures.
Cantor set: The subset of the unit interval left after repeatedly removing its open middle thirds. Its Hausdorff dimension is log 2 divided by log 3, despite containing no intervals.
Minkowski dimension: A dimension derived from the rate at which the volume of neighborhoods of a set shrinks with their radius. It often estimates geometric complexity computationally, though it need not equal Hausdorff dimension.
Felix Hausdorff: A German mathematician who made foundational contributions to set theory, topology, and measure theory. Hausdorff’s work established the mathematical setting and methods associated with this dimension.
Hausdorff content: The infimum of sums of diameter powers over covers whose pieces satisfy a fixed size bound. Its scale-dependent covering sums are the finite-stage quantities behind Hausdorff measure.
Box-counting dimension: A dimension defined by the asymptotic growth of the number of fixed-size boxes needed to cover a set. It is an easier scale-counting dimension, but can differ from Hausdorff dimension.
Koch snowflake: A plane curve formed by repeatedly replacing each segment with four shorter segments in a triangular pattern. Its boundary has Hausdorff dimension log 4 divided by log 3, greater than one.
Lyapunov exponent: A quantity measuring the exponential rate at which nearby trajectories in a dynamical system separate. Lyapunov exponents help determine the dimension of invariant sets in chaotic dynamics.
Carathéodory construction: A method for building measures from an outer measure satisfying a suitable covering condition. Hausdorff measure arises by applying this general construction to powers of covering diameters.

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