2026/05/13 by Pedro Marotta · 1 voice
Computer Science · Materials Science · Mathematics · Physics and Astronomy · #Dimension (graph theory) #Fractal #Mathematical Dynamics and Fractals #Perimeter #Planar #Quasicrystal Structures and Properties #Representation (politics) #Scaling #Sierpinski triangle #Square (algebra) #Theoretical and Computational Physics #Unit square #Zero (linguistics) #cs.CG #math.HO #math.MG
paper · pdf · doi:10.64336/001c.162173
openalex publication_date 2026/05/13 · openalex created_date 2026/05/14 · arxiv published 2026/05/18 · arxiv updated 2026/05/18 · openalex updated_date 2026/07/28
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, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>α</mml:mi> </mml:math> = <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mi>/</mml:mi> <mml:mi>r</mml:mi> </mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>β</mml:mi> </mml:math> = <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mi>/</mml:mi> <mml:msup> <mml:mi>r</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> governing perimeter and area scaling respectively. The classical condition 1<D= <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>l</mml:mi> <mml:mi>o</mml:mi> <mml:mi>g</mml:mi> <mml:mrow> <mml:mo stretchy="true" form="prefix">(</mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy="true" form="postfix">)</mml:mo> </mml:mrow> <mml:mi>/</mml:mi> <mml:mi>l</mml:mi> <mml:mi>o</mml:mi> <mml:mi>g</mml:mi> <mml:mrow> <mml:mo stretchy="true" form="prefix">(</mml:mo> <mml:mi>r</mml:mi> <mml:mo stretchy="true" form="postfix">)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> < 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.