Fractal geometry

Fractal geometry studies recursively structured sets and spaces, often with noninteger scaling dimension.

For a nonempty bounded subset $E$ of Euclidean space, let $N_{\delta }(E)$ be the least number of radius-$\delta$ balls covering $E$. Its lower and upper box-counting dimensions are the lower and upper limits of $\log N_{\delta }(E)/\log (1/\delta )$ as $\delta \downarrow 0$. If these limits agree, their common value is its Minkowski dimension. A cover by $N_{\delta }(E)$ balls implies $|E_{\delta }|{\lesssim }_n{\delta }^n{N}_{\delta }(E)$ for the open $\delta$-neighborhood. Conversely, a maximal separated subset gives the reverse comparison up to fixed changes of scale. This connects small-scale covering counts with neighborhood Lebesgue measure.

A self-similar object is assembled from smaller copies of itself, possibly after scaling or other transformations.

The Cantor set is obtained from $[0,1]$ by repeatedly deleting the open middle third of every remaining interval. Equivalently, it consists of the numbers that admit a base-three expansion using only the digits zero and two.

A Cantor cylinder fixes finitely many of the zero-or-two ternary digits in the standard Cantor set. These sets form finite clopen partitions of diameter at most $3^{-n}$ at level $n$, and a base for the relative topology. They permit uniform approximation of continuous functions by locally constant functions.

The Sierpinski triangle is formed recursively by retaining the three corner subtriangles after subdividing an equilateral triangle into four congruent pieces.