Fractional dimension

A dimension with a non-integer value that characterizes how measured size or detail changes with scale. It is used especially to describe fractals and other irregular structures.

Hausdorff dimension: A dimension defined by how coverings of a set behave as their sizes shrink. It formalizes dimension through scale-sensitive coverings, including for sets with intricate geometry.
Cantor set: The set formed by repeatedly removing the open middle third from each remaining interval. Its Hausdorff dimension is log 2 divided by log 3, despite containing no intervals.
Topological dimension: An integer-valued measure of the local structure of a space under continuous transformations. A fractal can have integer topological dimension but a larger fractional scaling dimension.
Fractal analysis: The study of scale-dependent patterns and geometric properties in irregular structures. It uses dimension estimates to compare the complexity of shapes and measured data.
Box-counting dimension: A dimension estimated from how the number of boxes needed to cover a set grows as box size decreases. Its log–log scaling gives a practical estimate from images and numerical data.
Koch snowflake: A finite-area curve generated by repeatedly adding smaller triangular bumps to each segment. Its boundary has dimension log 4 divided by log 3, exceeding that of an ordinary curve.
Euclidean dimension: The integer dimension of a Euclidean space, corresponding to its number of independent coordinates. Fractional dimension describes scaling that ordinary coordinate counts cannot capture.
Fractal Brownian motion: A family of self-similar random processes with scale-dependent correlations and roughness. Its parameters connect path regularity with the dimensions of sampled graphs and surfaces.
Similarity dimension: The dimension implied by the number and scale ratio of identical pieces in a self-similar construction. For ideal self-similar sets, a simple scaling equation yields the fractional value.
Sierpiński triangle: A planar fractal formed by repeatedly removing the central triangle from each triangular region. Its self-similar pieces give it dimension log 3 divided by log 2.

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1/3–2/3 conjectureRelated: Probability bounds on incomparable pairs connect to quantitative measures of order complexity.