A scaling-parameter framework for perimeter and area in self-similar planar fractals

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, they are typically analyzed in isolation and classified primarily by similarity dimension. This paper develops a unified parameter-space representation for a class of deterministic self-similar planar constructions, organized by the integers N (number of self-similar pieces) and r (inverse linear scale factor), together with two derived growth ratios, α = N / r and β = N / r 2 governing perimeter and area scaling respectively. The classical condition 1<D= l o g ( N ) / l o g ( r ) < 2 for non-integer planar fractal dimension is recast in coordinates that make perimeter and area directly comparable observables. The (N, r) parameter space is partitioned into three regimes — N ≤ r, r < N < r², and N ≥ r² — corresponding to distinct asymptotic behaviors of perimeter and area. The framework is further refined by distinguishing additive constructions, which, under a stated non-overlap assumption yield positive finite asymptotic area; and from subtractive constructions, which yield zero asymptotic area despite lying in the same dimension class. Four classical examples — the Sierpinski triangle, Sierpinski carpet, Koch snowflake, and a Koch-style square construction introduced by the author — are analyzed within the framework, together with four additional constructions used predictively to demonstrate that asymptotic behavior follows directly from (N, r, construction class) without separate re-derivation. The contribution of the paper is one of formulation and synthesis rather than new mathematics: it consolidates classical results into a single diagnostic representation in which perimeter divergence, area boundedness, and asymptotic measure behavior can be inferred directly from scaling parameters and construction class.