Sierpinski Series Fractals

Updated 20 October 2025

Sierpinski Series is a family of discrete self-similar fractals generated by recursive equations and modular arithmetic constructions.
The series exhibits numerical self-similarity with modular scaling, enabling efficient tiling, algorithmic self-assembly, and error correction.
It bridges fractal geometry, analytic iterations, and algebraic identities, underpinning applications in quantum computation, spectral theory, and more.

A Sierpinski Series characterizes a diverse family of discrete self-similar fractals—including the Sierpinski triangle, Sierpinski carpet, Sierpinski n-gon, generalized Sierpinski triangles, and their algebraic and analytic analogues—defined through recursive or iterative equations, modular arithmetic reductions, or generalized series and continued fraction representations. These constructions manifest deep connections between fractal geometry, algebraic identities, spectral analysis, self-assembly dynamics, quantum computation, and functional expansions on self-similar sets. In classical and modern research, the “series” notion encompasses both the mathematical series (truncated sums, rational functions) and the sequence of fractal patterns arising from modular recurrences, group-theoretic operations, or iterated function systems.

1. Recursive and Modular Construction of Sierpinski Series Fractals

The discrete Sierpinski Series fractals are defined via recursive matrices whose entries are computed by simple recurrence relations and then reduced modulo a prime $p$ . For the Sierpinski carpet, a two-dimensional infinite matrix $M$ is defined by (with $a = b = c = 1$ , $p = 3$ ):

$M[i,j]= \begin{cases} 1 & \text{if } i=0 \text{ and } j=0\ 1 & \text{if } i=0 \text{ and } j>0\ 1 & \text{if } j=0 \text{ and } i>0\ M[i,j-1] + M[i-1,j-1] + M[i-1,j] & \text{if } i,j>0 \end{cases}$

with all entries taken modulo $3$. The fractal is the set $\{(i,j): M[i,j] \not\equiv 0 \mod 3\}$ (0901.3189).

This construction generalizes to arbitrary primes $p$ and parameters $(a, b, c)$ , producing an infinite class of numerically $p$ -self-similar fractals, each one a member of the Sierpinski Series.

2. Numerical Self-Similarity and Scaling Properties

A central feature of Sierpinski Series fractals is numerical self-similarity modulo a prime $q$ , which is expressed as:

$M[s\,p^k+i,\, t\,p^k+j] \equiv M[s,t]\cdot M[i,j]\pmod{q}$

for all $k\geq0$ and $0 \le s,t < p$ . This property means that every $p^k\times p^k$ block in the matrix is, up to a modular scalar, an exact scaled copy of the “base” block, enabling the fractal pattern to assemble recursively at all scales (0901.3189). In practical terms, this permits a uniform and finite description (tile set, algorithm, or function system) that propagates the entire fractal structure indefinitely.

3. Tile Assembly, Self-Assembly, and Computation on Sierpinski Series

Winfree’s Tile Assembly Model enables the physical or simulated self-assembly of discrete Sierpinski Series. For any matrix defined by a local recurrence, a tile assembly system exists wherein each tile type encodes local submatrix information and propagates all necessary data to compute $M[i,j]$ locally. For the Sierpinski carpet, a set of 30 tile types suffices for uniform self-assembly. The local computation, error-correction properties, and scalability are direct consequences of the modular self-similarity mechanism (0901.3189).

Statistically self-similar fractals, such as the Sierpinski triangle, can also self-assemble using tile sets governed by almost-everywhere local determinism—where ambiguity may occur only on a zeta-dimension zero set—ensuring robust construction even with sparse randomness (0904.1630).

4. Algebraic and Analytic Iterations: Sierpinski Series as Rational Functions and Continued Fractions

In number theory and complex analysis, the Sierpinski series emerges as rapidly convergent series or truncated sums associated with quadratic units over imaginary quadratic fields, Hurwitz continued fractions, and iterative approximations. For example, given a quadratic polynomial $f(X)=X^2-TX+U$ , the Sierpinski series is defined as:

$\sum_{0 \leq m \leq n} \frac{U^{2^m}}{h_0(T) h_1(T, U)\cdots h_m(T, U)}$

with $h_0:=T$ , $h_{n+1}(T,U):=h_n(T,U)^2-2U^{2^n}$ ,

and is provably equivalent to both the value of a truncated Hurwitz continued fraction and the $(n+1)$ -fold Newton approximation:

$F^{(n+1)}(0;T,U),\quad F(X) := X - \frac{f(X)}{f'(X)}$

Explicit equalities link the truncated Hurwitz continued fraction expansions of quadratic units $\alpha\in\mathbb{C}$ , the rapidly convergent Sierpinski series, and the Newton approximants, with precise error estimates established (Saito et al., 17 Oct 2025). This demonstrates that Sierpinski Series is not only a geometric or combinatorial concept, but also an analytic representation tightly connected with continued fractions and iterative root-finding in complex algebraic fields.

5. Group-Theoretic and Digital Formulations: Sierpinski Series in Matrix and Carry-Free Arithmetic

The Sierpinski triangle appears as the set of nonzero residues in Pascal’s triangle modulo $2$, and its generalization via Sierpinski matrices $S_n(x)$ constructed through Kronecker products leads to a “digital binomial theorem” (Nguyen, 2014):

Matrix entries $s_{j,k}$ are indexed by $(j,k)$ : $s_{j,k} = \begin{cases} x^{s(j-k)} & \text{if %%%%23%%%% is carry-free}\ 0 & \text{otherwise} \end{cases}$ where $s(\cdot)$ is the binary sum-of-digits function.
The matrices satisfy the group property

$S(x)S(y) = S(x+y)$

implying a digital version of the Binomial Theorem where only carry-free summands contribute, and the Sierpinski Series encodes the expansion coefficients with explicit dependence on the binary arithmetic structure.

6. Generalized Iterated Function Systems and Fractal Series

Strictly self-similar Sierpinski $n$ -gons, $n$ -flakes, polyflakes, and generalized Sierpinski triangles ( $\triangle FNN$ , $\triangle FFN$ , Pedal triangle) are constructed as attractors of iterated function systems (IFS) determined by contractive maps with scaling ratios dependent on the polygon parameters (Tzanov, 2015, Steemson et al., 2018). The series of possible fractals—indexed by the number of vertices, flip/non-flip map configuration, or scaling parameter—provides a parameterized continuum of Sierpinski-like sets. The dimension of each attractor is determined by solving a Moran–Hutchinson equation:

$\sum_{i=1}^N s_i^d = 1$

and the full range of Hausdorff dimensions, from the Cantor set ( $d=1$ ), Sierpinski triangle ( $d = \log 3/\log 2$ ), up to the Sierpinski $\infty$ -flake ( $d=2$ ), are achieved within the series. Moreover, the generalized triangles show dimensions strictly less than the classical case when certain parameters are chosen.

7. Applications, Generalizations, and Significance

Sierpinski Series finds systematic application:

In constructing self-similar and statistically self-similar discrete fractals, relevant to algorithmic self-assembly and nanotechnology (0901.3189, 0904.1630).
In spectral theory and orthogonal polynomial expansions on fractals, the series frames eigenfunction and functional expansions tailored for Sierpinski-type domains (Okoudjou et al., 2011).
In quantum computation and coding theory via digital analogues and error-correcting code construction, e.g., holographic codes and fractal subregions in AdS/CFT (Bao et al., 2022).
In number-theoretic representations and fast convergence algorithms for algebraic numbers over quadratic fields, via explicit continued fraction expansions and Newton iteration analysis (Saito et al., 17 Oct 2025).
In the broader context of tiling theory, combinatorial analysis, and graph-theoretic optimization on recursive self-similar graphs (Gravier et al., 2012).
In operator-algebraic and noncommutative geometry via the construction of spectral triples, integration theory, and fractal metric recovery (Rivera, 2017, Aiello et al., 2020).

The concept extends naturally to higher-dimensional analogues, multifractal measures, and statistical generalizations, as well as analysis involving singular measures and advanced functional inequalities (Liu et al., 2017). The Sierpinski Series unifies discrete fractal patterns, analytic expansions, and algebraic identities, centralizing the paper of self-similarity through recursive schemes and modular arithmetic across a wide spectrum of mathematics, physics, and computation.