Sierpiński Packings & Fractal Applications

• Sierpiński packings are defined as countable collections of disjoint, non-separating continua in the Riemann sphere with diameters shrinking to zero, forming the basis for Sierpiński carpets when regions are Jordan domains.
• The theory employs disk-packing uniformizations and packing-conformal maps using L² estimates and modulus inequalities to connect fractal geometry with complex dynamics and geometric function theory.
• Applications include uniformization of fractal carpets, computation of closed-form packing measures on self-similar sets, and analysis of packing chromatic numbers in discrete mathematics.

A Sierpiński packing is a countable collection of pairwise disjoint, non-separating continua in the Riemann sphere $\widehat{\mathbb{C}}$ (equipped with the chordal metric), whose diameters shrink to zero. The residual set, obtained by removing all peripheral continua from the sphere, is a central object in the study of fractal geometry, geometric function theory, and connections with dynamical systems and graph theory. Sierpiński packings encompass classical constructions such as Sierpiński carpets and play a fundamental role in the analysis of homogeneous fractal sets, conformal uniformization problems, and packing chromatic numbers.

1. Definitions and Structural Properties

A family $\{p_i\}_{i\in I}$ of compact subsets of $\widehat{\mathbb{C}}$ is called a Sierpiński packing if:

• $p_i \cap p_j = \varnothing$ for $i \neq j$,
• each $p_i$ is non-separating (complement in $\widehat{\mathbb{C}}$ is connected),
• $\lim_{i\to\infty} \operatorname{diam}(p_i) = 0$.

The residual set $X$ is defined as $$X = \widehat{\mathbb{C}} \setminus \bigcup_{i\in I} p_i$$. If all $\{p_i\}_{i\in I}$0 are Jordan domains, $\{p_i\}_{i\in I}$1 is called a Sierpiński carpet. A Sierpiński packing by Jordan regions (possibly degenerate) forms a canonical class within planar fractal geometry (Ntalampekos, 2022).

A closely related notion is a packing by simple closed curves $\{p_i\}_{i\in I}$2 bounding topological disks $\{p_i\}_{i\in I}$3 with pairwise disjoint interiors, as formalized in (Merenkov et al., 2014). The associated residual set is $\{p_i\}_{i\in I}$4.

2. Uniformization by Disk Packings and Packing-Conformal Maps

A central result for Sierpiński packings with square-summable diameters is the existence of disk-packing uniformizations. Specifically, for a given Sierpiński packing $\{p_i\}_{i\in I}$5 with $\{p_i\}_{i\in I}$6, there exist:

• a collection $\{p_i\}_{i\in I}$7 of pairwise disjoint closed disks (degenerate if and only if $\{p_i\}_{i\in I}$8 is degenerate),
• a round Sierpiński packing $\{p_i\}_{i\in I}$9,
• a continuous, surjective, monotone map $\widehat{\mathbb{C}}$0 mapping the interiors appropriately,
• a Borel function $\widehat{\mathbb{C}}$1

satisfying precise modulus and area inequalities; see [(Ntalampekos, 2022), Theorem 1.1]. If the packing is "cofat" (uniform Ahlfors-2 lower bound), $\widehat{\mathbb{C}}$2 can be chosen homeomorphic. The developed concept of packing-conformal and packing-quasiconformal mappings generalizes analytic conformality to the context of circle packings and discrete continua via $\widehat{\mathbb{C}}$3-gradient and area distortion inequalities.

The proof utilizes approximation by finite circle domains, equicontinuity arguments, transboundary modulus estimates, and cell-like extension theorems. The theory extends to Sierpiński carpets (all $\widehat{\mathbb{C}}$4 are Jordan regions), more general cofat continua, and Conformal Loop Ensemble (CLE) carpets, providing a circle-packing model for probability-generated random carpets.

3. Homogeneity, Curvature Distribution, and Dimension Theory

A Sierpiński packing residual set $\widehat{\mathbb{C}}$5 is termed homogeneous if for some constants $\widehat{\mathbb{C}}$6, it satisfies:

1. (Roundness) Each complementary disk $\widehat{\mathbb{C}}$7 admits concentric Euclidean disks of radii $\widehat{\mathbb{C}}$8 with $\widehat{\mathbb{C}}$9,
2. (Scale–location property) For any $p_i \cap p_j = \varnothing$0 and $p_i \cap p_j = \varnothing$1, some $p_i \cap p_j = \varnothing$2 intersects $p_i \cap p_j = \varnothing$3, and $p_i \cap p_j = \varnothing$4,
3. (Relative separation) For all $p_i \cap p_j = \varnothing$5, $p_i \cap p_j = \varnothing$6 [(Merenkov et al., 2014), §1.2].

Associated with any such packing, the curvature distribution function is $p_i \cap p_j = \varnothing$7, and the exponent is $p_i \cap p_j = \varnothing$8.

A fundamental theorem ([(Merenkov et al., 2014), §2]) for Julia sets of hyperbolic rational maps which are Sierpiński carpets relates these notions: $p_i \cap p_j = \varnothing$9 i.e., the curvature count $i \neq j$0, with $i \neq j$1 the Hausdorff dimension of the Julia set, and $i \neq j$2 the exponent. The proof combines hyperbolicity (expansion/conformal distortion yielding approximate self-similarity), verification of homogeneity, and the equality of Minkowski and Hausdorff dimension for such residual sets.

By analogy, for standard Sierpiński $i \neq j$3-carpets, $i \neq j$4 and $i \neq j$5.

4. Computability and Closed-Form Packing Measures on Self-Similar Sets

For totally disconnected self-similar sets satisfying the Strong Separation Condition (SSC), such as the Sierpiński gasket, the packing measure at its similarity dimension admits a closed-form (Llorente et al., 2014). For the planar Sierpiński gasket $i \neq j$6 with contraction ratio $i \neq j$7, similarity dimension $i \neq j$8,

$i \neq j$9

This formula is verified by explicit geometric and probabilistic calculations based on the Moran-type density function $p_i$0 and the uniqueness of measure maximizing balls. Furthermore, an iterative algorithm is provided, with convergence rigorously shown, to compute the packing measure for arbitrary contraction ratios and more general symmetric iterated function systems.

When $p_i$1 exceeds certain thresholds (e.g., $p_i$2), the maximizing ball may lose combinatorial stability and closed forms are not generally available, motivating further research into analytic and computational approaches in packing measure theory (Llorente et al., 2014).

5. Sierpiński Packing Chromatic Numbers and Graph Products

Within discrete mathematics, the Sierpiński product $p_i$3 for graphs $p_i$4, $p_i$5, and a function $p_i$6 has a vertex set $p_i$7, with two types of edges: 'fiber' edges within $p_i$8 (copies of $p_i$9 over $\widehat{\mathbb{C}}$0) and 'connecting' edges between fibers dictated by $\widehat{\mathbb{C}}$1 (Holub et al., 9 Jul 2025).

Packing-colorings assign colors to vertices so that vertices of color $\widehat{\mathbb{C}}$2 are at distance $\widehat{\mathbb{C}}$3; $\widehat{\mathbb{C}}$4 is the smallest such number for $\widehat{\mathbb{C}}$5. For Sierpiński products, one defines: $\widehat{\mathbb{C}}$6 as the lower and upper Sierpiński packing chromatic numbers.

Explicit results include:

• For $\widehat{\mathbb{C}}$7, $\widehat{\mathbb{C}}$8 and, if $\widehat{\mathbb{C}}$9, $\lim_{i\to\infty} \operatorname{diam}(p_i) = 0$0. For stars and paths as factors, bounds are established, with minimal chromatic numbers always $\lim_{i\to\infty} \operatorname{diam}(p_i) = 0$1 and upper bounds depending on structure.
• For arbitrary $\lim_{i\to\infty} \operatorname{diam}(p_i) = 0$2, explicit lower and upper bounds in terms of independence numbers and $\lim_{i\to\infty} \operatorname{diam}(p_i) = 0$3-packing numbers are provided.
• Recognition of Sierpiński product graphs where one factor is a tree can be done in polynomial time by recursive decomposition and isomorphism checks (Holub et al., 9 Jul 2025).

6. Applications and Further Directions

Sierpiński packings bridge fractal geometry, complex dynamics, conformal geometry, and discrete mathematics. Applications include:

• Uniformization of fractal carpets, facilitating transfer between analytic and combinatorial models,
• Rigorous computation of fractal packing measures for self-similar sets,
• Analysis of random fractals such as CLE carpets in probability theory and statistical physics, enabled by packing-conformal mappings (Ntalampekos, 2022),
• Combinatorial encoding of iterative structures in graphs, with efficient recognition algorithms and chromatic invariant computation (Holub et al., 9 Jul 2025).

Open problems include generalizing explicit packing measures to higher dimensions, relaxing separation and regularity conditions, and further developing the theory of packing-conformal maps for broader classes of topological packings and random fractals (Llorente et al., 2014, Ntalampekos, 2022).