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

TL;DR

This paper introduces a unified framework using parameters N and r to analyze and predict the perimeter and area behaviors of self-similar planar fractals, unifying classical results.

Contribution

It develops a comprehensive parameter-space representation that classifies self-similar fractals by perimeter and area growth, distinguishing different construction types within the same dimension class.

Findings

Partition of parameter space into three regimes with distinct behaviors

Differentiation between additive and subtractive constructions based on area

Predictive analysis of classical fractals using the framework

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 $\alpha = N/r$ and $\beta = N/r^2$, 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.