Fractal geometry

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

Choose a strictly decreasing positive sequence $q_0=1,q_1,q_2,\dots$. Begin with $C_0=[0,1]$ and at step $n+1$ keep two closed end intervals of length $q_{n+1}/2^{n+1}$ from every component interval of $C_n$. This constructs a compact set $C={\bigcap }_n{C}_n$ withThe second equality is continuity from above of a measure. Every stage has positive gaps and maximum component length at most $2^{-n}$, so $C$ contains no nondegenerate interval and is a nowhere dense set. It has no isolated points: both endpoints of the stage-$n$ component containing a point remain in $C$, and at least one distinct endpoint is within $2^{-n}$. Thus its topological sparseness is compatible with positive Lebesgue measure.

Taking $q_n=(2/3{)}^n$ gives the usual Cantor set of measure zero. Taking $q_n=1/2+2^{-(n+1)}$ gives a generalized construction of measure $1/2$. This is one useful family of generalized constructions; the removed intervals need not have the same proportions at every stage.

The mass fractal dimension $d_f$ relates the particle count or mass in a self-similar aggregate to its spatial size. For primary particle radius $a$ and aggregate radius of gyration $R_g$, a common convention is $N\sim k_f(R_g/a{)}^{d_f}$, with a nonuniversal prefactor $k_f$. In three dimensions the mean occupied fraction scales as $N{a}^3/R_g^3\propto (R_g/a{)}^{d_f-3}$. A smaller $d_f$ therefore describes a more tenuous large cluster; a compact constant-density object has $d_f=3$. The scaling applies between the primary-particle and aggregate scales, not at all distances.

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.

Consider base-$b$ expansions using $m$ allowed digits with positive gaps between the corresponding first-level intervals. At depth $k$ there are $m^k$ cylinders of diameter comparable to $b^{-k}$, and separated cylinders cannot all be covered by fewer than a constant multiple of $m^k$ balls of that scale. Thus $N_{b^{-k}}(E)\asymp m^k$. Monotonicity between successive scales gives equal upper and lower Minkowski dimensions, both $\log m/\log b$.

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

For finitely many contracting Euclidean similarities, the map $\mathcal{T}$ on nonempty compact subsets of ${\mathbb{R}}^n$ is a contraction in the Hausdorff metric, with contraction constant ${\max }_i{c}_i<1$. The contraction mapping theorem gives a unique nonempty compact invariant set. If the open set condition holds, its Hausdorff dimension is the unique $s$ satisfying ${\sum }_i{c}_i^s=1$.

The Minkowski sausage replaces a segment by eight directed unit segments whose endpoint displacement is four units. A middle straight run of length two represents two collinear unit segments. After normalizing the displacement to one, the eight contracting Euclidean similarities all have ratio $1/4$. The open diamond $\{|x-1/2|+|y|<1/2\}$ is an open set condition witness: its eight image diamonds have disjoint interiors and lie within it. Hence $8(1/4{)}^s=1$, giving Hausdorff dimension $3/2$. Treating the middle run as a single similarity of ratio $1/2$ instead defines a different iterated scheme.

A contracting Euclidean similarity multiplies every Euclidean distance by a constant $c<1$; equivalently it has the form $cRx+b$ for an orthogonal matrix $R$ and translation vector $b$. Finite families of such maps generate compact sets with self-similarity.

A finite family of contracting similarities $S_1,\dots ,S_k$ satisfies the open set condition if there is a nonempty bounded open set $O$ with $S_i(O)\subseteq O$ and $S_i(O)\cap S_j(O)=\varnothing$ for $i\ne j$. This permits the closures to meet. For the invariant compact set, this separation makes its Hausdorff dimension equal to the similarity exponent $s$ defined by ${\sum }_i{c}_i^s=1$, where $c_i$ are the contraction ratios. The level-$n$ cylinder sets supply the upper dimension bound; the separated cylinder masses with weights $c_i^s$ supply the matching mass-distribution lower bound. For the 24 maps of ratio $1/5$ retaining all but the central square, $O=(0,1{)}^2$ works and $s=\log 24/\log 5$.

The Cantor set is obtained from $[0,1]$ by repeatedly deleting the open middle third of every remaining interval. Its first stages retain $[0,1/3]\cup [2/3,1]$, then four intervals of length $1/9$, then eight of length $1/27$. After level $n$, there are $2^n$ retained intervals of length $3^{-n}$. The Cantor set is the intersection of all these stages, rather than the set of endpoints visible in any finite drawing.

Equivalently,Thus its points admit a base-three expansion using only zero and two. The word “admit” handles numbers with two expansions: $1/3=0.1_3=0.02222{\dots }_3$ is retained as an endpoint. Every infinite binary choice determines a point of $C$, and differing choices give differing points, so $C$ is uncountable.

Its total retained interval length is $(2/3{)}^n\to 0$, so $C$ has zero Lebesgue measure. It is nevertheless nonempty, compact, and uncountable, and has Hausdorff dimension $\log 2/\log 3$: the covers and mass bounds are worked out in Hausdorff dimension of the Cantor set. This separates counting, ordinary length, and fine-scale geometric size in one concrete example.

The self-similarity is exact:The Cantor distribution puts half the probability into each copy at every split. It has a continuous cumulative distribution function, the Cantor function, while all its mass lies on a Lebesgue null set. This makes the construction useful in measure theory as well as fractal geometry: it provides a concrete model in which several plausible meanings of “small” disagree.

Starting from $[0,1]$, remove the central fraction $1/(2^n+2)$ of every stage-$n$ interval. The total length obeys $L_{n+1}/L_n=(2^n+1)/(2^n+2)$, which telescopes to the displayed value. The limiting intersection has Lebesgue outer measure at most $1/3$ by covering it with the stage intervals and slightly enlarging them to open intervals. This argument uses no integration.

For the middle-third Cantor set $C$ and positive integer $d$, $C^d$ is compact and totally disconnected. A connected subset projects to a point in each coordinate and is therefore itself a point. At level $n$, the product has $2^{dn}$ cubes of side $3^{-n}$. These covers give dimension at most $d\log 2/\log 3$. The product digit probability assigns each cube mass $2^{-dn}$; a ball of radius comparable to $3^{-n}$ meets only a bounded number of cubes, so its mass is at most a constant times $r^{d\log 2/\log 3}$. The cover-cost argument for Hausdorff dimension proves the matching lower bound. Thus compact totally disconnected Euclidean subsets can have arbitrarily large Hausdorff dimension when the ambient dimension is allowed to increase.

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.

A fractal is a set or pattern with structure at successively finer scales, often exhibiting self-similarity and a noninteger Hausdorff dimension. Fractal geometry studies such objects.

A plane curve generated by recursively folding or by a suitable L-system. One drawing convention uses the productions $a\mapsto aRbFR$ and $b\mapsto LFaLb$, where $F$ moves forward, $L,R$ turn by $90^{\circ }$, and $a,b$ expand but do not draw. Summaries of whole expanded blocks permit a position after a large number of forward steps to be found by descending only into the block containing that step.