A Scaling-Parameter Framework for Perimeter and Area in Self-Similar Planar Fractals

Abstract

The Koch snowflake is a classical example of a planar curve with infinite perimeter enclosing a finite, positive area. Although such examples are well known individually, classical treatments typically analyze each construction in isolation and classify them by similarity dimension. This paper develops a unified parameter-space representation for a class of self-similar planar constructions, organized by two integers -- the number of self-similar pieces $N$ and the inverse linear scale factor $r$ -- together with two derived growth ratios $\hat{I}\pm =N/r$ and ${\hat{I}}^2=N/r^2$, governing perimeter and area scaling respectively. The $(N,r)$ parameter space is partitioned into three regimes -- $N\le r$, $r<N<r^2$, and $N\ge r^2$ -- corresponding to qualitatively distinct asymptotic behaviors of perimeter and area jointly. Within the intermediate regime $r<N<r^2$, a construction-class refinement distinguishes additive constructions (region bounded by the iterated curve), which yield positive finite asymptotic area under a stated non-overlap assumption, from subtractive constructions (iterated set itself), which yield zero asymptotic area. This records a structural non-equivalence inside the same dimension class that is not visible from $D=\log N/\log r$ alone. Four worked examples illustrate the framework -- the Sierpinski triangle, Sierpinski carpet, Koch snowflake, and a Koch-style construction on a square invented by the author -- and four further constructions are analyzed predictively to demonstrate that diagnostic outputs follow from $(N,r,\text{construction class})$ without re-derivation. The contribution lies in formulation and synthesis: the paper consolidates several classical results into a single diagnostic representation in which, given $(N,r)$ and construction class, the asymptotic behavior of perimeter and area can be inferred directly.

17 pages, 5 figures. Author's accepted manuscript. The Python code listings included as Appendices A-C in the published version are omitted here; the code is openly available at https://github.com/pedromarotta/fractal-framework-demo