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 $α= N/r$ and $β= N/r^2$, 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.

翻译：科赫雪花是平面曲线无限周长包围有限正面积的经典例子。尽管这类例子单独而言广为人知，但经典处理通常逐一分析每个构造，并依据相似维数进行分类。本文为自相似平面构造的一类对象建立了统一的参数空间表示，该表示由两个整数——自相似片段数$N$和逆线性尺度因子$r$——以及两个衍生增长比$α= N/r$与$β= N/r^2$（分别控制周长与面积的缩放）构成。$(N,r)$参数空间被划分为三个区域：$N \le r$、$r < N < r^2$和$N \ge r^2$，分别对应周长和面积联合渐进行为的不同定性特征。在中间区域$r < N < r^2$内，构造类的细化区分了加法构造（由迭代曲线围成的区域，在所述非重叠假设下产生正有限渐近面积）与减法构造（迭代集合本身，产生零渐近面积）。这记录了同一维数类别内部的结构非等价性——该等价性无法仅从$D = \log N / \log r$看出。四个完整示例（谢尔宾斯基三角形、谢尔宾斯基地毯、科赫雪花，以及作者发明的基于正方形的科赫式构造）阐释了该框架，并对另外四个构造进行预测性分析，以证明诊断输出可直接从$(N, r, \text{构造类})$推出，无需重新推导。本文的贡献在于构建与综合：它将若干经典结果整合为单一诊断表示，在该表示中，给定$(N, r)$和构造类，周长和面积的渐进行为可直接推断。