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

Pedro Marotta

arxiv: 2605.19128 · v1 · pith:725ZGBXSnew · submitted 2026-05-18 · 🧮 math.HO · cs.CG· math.MG

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

Pedro Marotta This is my paper

Pith reviewed 2026-05-20 07:22 UTC · model grok-4.3

classification 🧮 math.HO cs.CGmath.MG

keywords self-similar fractalsperimeter scalingarea scalingKoch snowflakeSierpinski carpetscaling regimesfractal dimensionadditive constructions

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The pith

Given N and r for a self-similar construction, its perimeter and area asymptotics follow directly from three scaling regimes plus an additive or subtractive class.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a unified parameter framework that organizes self-similar planar constructions by the integers N, the number of self-similar pieces, and r, the inverse linear scale factor. It partitions the (N,r) space into three regimes that produce distinct combinations of convergent or divergent limits for perimeter and area. In the intermediate regime a further distinction between additive constructions, which bound a positive finite area, and subtractive constructions, which yield zero area, appears under a non-overlap condition. A sympathetic reader would care because the approach replaces isolated analysis of examples such as the Koch snowflake with a single diagnostic representation from which the behaviors of any new construction in the class can be read off directly.

Core claim

The paper claims that the (N,r) parameter space for self-similar planar constructions can be divided into the three regimes N ≤ r, r < N < r², and N ≥ r², each corresponding to qualitatively distinct joint asymptotic behaviors of perimeter (scaled by α = N/r) and area (scaled by β = N/r²), and that within the intermediate regime an additive versus subtractive construction-class refinement distinguishes positive finite asymptotic area from zero asymptotic area when the pieces satisfy a stated non-overlap condition. This structural non-equivalence inside the same similarity-dimension class is not visible from D = log N / log r alone.

What carries the argument

The three-regime partition of the (N,r) parameter space together with the additive versus subtractive refinement in the middle regime, which directly determines the limiting perimeter and area from the input parameters and class without separate case-by-case limit derivations.

Load-bearing premise

The non-overlap assumption for additive constructions in the intermediate regime is required to guarantee that the bounded region yields positive finite asymptotic area rather than zero.

What would settle it

An explicit area calculation for any additive construction with r < N < r² in which the pieces overlap sufficiently to drive the enclosed region to zero area would falsify the claim that the class distinction plus the non-overlap condition produces positive finite asymptotic area.

Figures

Figures reproduced from arXiv: 2605.19128 by Pedro Marotta.

**Figure 1.** Figure 1: Sierpinski triangle, iterations n = 0 through n = 5. The construction starts from a closed equilateral triangle (n = 0). At each iteration, every triangle of side length s is replaced by three triangles of side length s/2 placed at its vertices, with the central inverted triangle removed. The number of self-similar pieces is N = 3, the linear scale factor is r = 2, and the similarity dimension is D = log 3… view at source ↗

**Figure 2.** Figure 2: Sierpinski carpet, iterations n = 0 through n = 3. The construction starts from a closed unit square (n = 0). At each iteration, every square of side s is partitioned into a 3 × 3 grid of nine sub-squares of side s/3, and the central sub-square is removed. The number of retained pieces is N = 8, the linear scale factor is r = 3, and the similarity dimension is D = log 8/ log 3 ≈ 1.893. The set is subtracti… view at source ↗

**Figure 3.** Figure 3: Koch snowflake, iterations n = 0 through n = 4. The construction starts from a closed equilateral triangle of side length 1 (n = 0). At each iteration, every boundary segment of length s is divided into three equal parts and the middle part is replaced by the two other sides of an outward-pointing equilateral triangle, so that each segment is replaced by four segments of length s/3. The generator parameter… view at source ↗

**Figure 4.** Figure 4: Koch-style construction on a square, iterations n = 0 through n = 3 (author’s construction). The construction starts from a unit square of perimeter 4 (n = 0). At each iteration, every boundary segment of length s is divided into three equal parts; on the middle part, a square of side s/3 is constructed outward; the original middle part is then deleted, so each segment is replaced by five segments of lengt… view at source ↗

**Figure 5.** Figure 5: (N, r) parameter-space regime plot for self-similar planar fractals. The horizontal axis is the linear scale factor r and the vertical axis is the number of self-similar pieces N. The two black curves are the boundaries N = r (solid) and N = r 2 (dashed). The plane is partitioned into three regimes: the subcritical regime N ≤ r (light grey, similarity dimension D ≤ 1, perimeter bounded under the perimeter … view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper organizes known scaling behaviors for self-similar fractals into a clean (N, r) parameter space with an additive/subtractive split, but the distinction rests on an unproven non-overlap assumption.

read the letter

The main takeaway is a synthesis that maps self-similar planar constructions onto three regimes in (N, r) space and then splits the middle regime by construction type. Given N and r plus whether the process is additive or subtractive, the asymptotic perimeter and area follow directly in the examples shown. That organizational move is the actual novelty; the underlying scaling relations for alpha = N/r and beta = N/r^2 are classical, but packaging them this way lets you read off the joint behavior without re-deriving each case from scratch. The four worked examples (Koch snowflake, Sierpinski triangle and carpet, plus the author's square variant) and the four predictive ones illustrate the point clearly and make the framework usable as a diagnostic tool. The citations stay within the standard fractal-geometry literature and the derivations are straightforward comparisons of N to r and r squared. The paper does what it sets out to do: consolidate scattered results into one representation. The soft spot is the non-overlap assumption required for positive finite area in additive constructions inside r < N < r^2. The text states the condition but does not supply a general argument that it survives iteration for arbitrary additive families satisfying only the similarity ratios. If overlaps appear in the limit, the enclosed area can still collapse to zero, which would erase the claimed distinction. That assumption is load-bearing for the strongest claim, and it needs either a proof or an explicit, checkable criterion. This is for readers who work with self-similar sets and want a compact classification scheme for teaching or quick analysis rather than new theorems. A serious referee should see it because the synthesis is coherent and the examples are concrete, even though the assumption will probably require tightening or clarification in revision.

Referee Report

1 major / 2 minor

Summary. The paper develops a unified (N, r) parameter-space framework for self-similar planar constructions, where N is the number of similar pieces and r the inverse linear scale factor. It partitions the space into three regimes (N ≤ r, r < N < r², N ≥ r²) governed by the growth ratios α = N/r and β = N/r², and further distinguishes additive versus subtractive constructions inside the intermediate regime. Under a stated non-overlap assumption for additive cases, the framework claims that the joint asymptotic behavior of perimeter and area follows directly from (N, r) and construction class, without case-by-case re-derivation. Four classical examples (Koch snowflake, Sierpinski triangle and carpet, plus an author-invented square variant) are worked out, and four additional constructions are analyzed predictively to illustrate the diagnostic use of the scheme.

Significance. If the central claims hold, the work supplies a compact diagnostic representation that consolidates known perimeter-area results for a broad class of self-similar sets and makes the distinction between additive and subtractive constructions inside the same similarity dimension visible. The parameter-free character of the regime divisions and the predictive application to new constructions are genuine strengths that could streamline classification and teaching of these examples.

major comments (1)

[§3] §3 (intermediate regime, additive case): The assertion that additive constructions with r < N < r² produce positive finite limiting area rests on an unproved non-overlap assumption. The manuscript states the condition but supplies neither a general proof that it is preserved under iteration for arbitrary additive constructions satisfying only the given (N, r) nor an explicit, checkable criterion (e.g., a measure-theoretic or geometric test) that would allow a reader to verify it for a new construction. Because this assumption is load-bearing for the claimed direct inference in the intermediate regime, its justification is required for the framework to deliver the promised generality.

minor comments (2)

[Introduction] Notation: the symbols α and β are introduced early but their explicit definitions α = N/r and β = N/r² appear only after the regime discussion; a single forward reference or boxed definition at first use would improve readability.
[§5] Examples: the four predictive constructions in the final section are described only by their (N, r) values and class; adding a brief geometric description or diagram for each would make the diagnostic outputs easier to verify independently.

Simulated Author's Rebuttal

1 responses · 1 unresolved

We thank the referee for the careful reading and constructive comments on the manuscript. The observation regarding the non-overlap assumption in the intermediate regime is well taken and we address it directly below.

read point-by-point responses

Referee: [§3] §3 (intermediate regime, additive case): The assertion that additive constructions with r < N < r² produce positive finite limiting area rests on an unproved non-overlap assumption. The manuscript states the condition but supplies neither a general proof that it is preserved under iteration for arbitrary additive constructions satisfying only the given (N, r) nor an explicit, checkable criterion (e.g., a measure-theoretic or geometric test) that would allow a reader to verify it for a new construction. Because this assumption is load-bearing for the claimed direct inference in the intermediate regime, its justification is required for the framework to deliver the promised generality.

Authors: We agree that the non-overlap assumption is load-bearing for the positive finite area claim in additive constructions within r < N < r² and that its iterative preservation was not proved in general. The manuscript introduces the assumption to guarantee that successive area increments remain disjoint, permitting the area to converge as a geometric series with ratio β = N/r² < 1. In the revised version we will add an explicit, checkable geometric criterion: the N self-similar copies placed in the first iteration must have pairwise disjoint interiors and lie strictly inside the original domain. We will prove that this initial disjointness is preserved at every subsequent iteration by the uniform scaling factor 1/r. The criterion will be verified explicitly for the four classical examples and the four predictive constructions. We will also state clearly that the framework applies under this verifiable initial condition rather than for every conceivable placement satisfying only the numerical bounds (N, r). revision: partial

standing simulated objections not resolved

A general proof that non-overlap is automatically preserved under iteration for arbitrary additive constructions satisfying only the (N, r) bounds without any supplementary initial geometric conditions.

Circularity Check

0 steps flagged

No circularity: behaviors follow directly from parameter definitions and regime comparisons

full rationale

The paper defines α = N/r and β = N/r² explicitly from the input integers N and r, then partitions the (N, r) space into the three regimes by direct inequality comparisons (N ≤ r, r < N < r², N ≥ r²). Asymptotic perimeter and area statements are presented as immediate mathematical consequences of whether α ≷ 1 and β ≷ 1 together with the additive/subtractive classification; the non-overlap condition is stated as an explicit hypothesis required for the positive-finite-area conclusion in the additive intermediate case rather than being derived from or presupposed by the framework itself. No self-citations, fitted parameters, or uniqueness theorems are invoked as load-bearing steps for the central claims, and the predictive analyses of further constructions are simply applications of the same regime rules. The derivation chain is therefore self-contained against the stated inputs and does not reduce any result to itself by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The framework introduces no new free parameters or invented entities; it relies on the standard definitions of self-similarity and the stated non-overlap condition as a domain assumption.

axioms (1)

domain assumption Non-overlap assumption for additive constructions to produce positive finite asymptotic area
Invoked to distinguish area behavior in the intermediate regime for additive versus subtractive cases.

pith-pipeline@v0.9.0 · 5871 in / 1403 out tokens · 63376 ms · 2026-05-20T07:22:32.715232+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The (N,r) parameter space is partitioned into three regimes — N≤r, r<N<r², and N≥r² — corresponding to qualitatively distinct asymptotic behaviors of perimeter and area jointly. Within the intermediate regime r<N<r², a construction-class refinement distinguishes additive constructions ... from subtractive constructions
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Under a stated non-overlap assumption ... the total added area is therefore multiplied by N/r² at each step, and the area sum reduces to a geometric series with ratio N/r²

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

8 extracted references · 8 canonical work pages

[1]

(1904).Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire.Arkiv för Matematik, Astronomi och Fysik1: 681–704

von Koch, H. (1904).Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire.Arkiv för Matematik, Astronomi och Fysik1: 681–704

work page 1904
[2]

(1915).Sur une courbe dont tout point est un point de ramification.Comptes Rendus de l’Académie des Sciences de Paris160: 302–305

Sierpiński, W. (1915).Sur une courbe dont tout point est un point de ramification.Comptes Rendus de l’Académie des Sciences de Paris160: 302–305

work page 1915
[3]

Falconer, K. J. (2014).Fractal Geometry: Mathematical Foundations and Applications, 3rd ed. Wiley, Chichester, UK. DOI:10.1002/9781118762875

work page doi:10.1002/9781118762875 2014
[4]

(2008).Measure, Topology, and Fractal Geometry, 2nd ed

Edgar, G. (2008).Measure, Topology, and Fractal Geometry, 2nd ed. Springer, New York, NY, USA. DOI:10.1007/978-0-387-74749-1

work page doi:10.1007/978-0-387-74749-1 2008
[5]

Hutchinson, J. E. (1981).Fractals and self-similarity.Indiana Univ. Math. J.30(5): 713–747. DOI:10.1512/iumj.1981.30.30055

work page doi:10.1512/iumj.1981.30.30055 1981
[6]

(1982).The Fractal Geometry of Nature.W

Mandelbrot, B. (1982).The Fractal Geometry of Nature.W. H. Freeman, New York, NY, USA

work page 1982
[7]

Barnsley, M. F. (1993).Fractals Everywhere, 2nd ed. Academic Press / Morgan Kaufmann, San Francisco, CA, USA

work page 1993
[8]

Witten, T. A. & Sander, L. M. (1981).Diffusion-limited aggregation, a kinetic critical phenomenon.Phys. Rev. Lett.47: 1400–1403. DOI:10.1103/PhysRevLett.47.1400. 17

work page doi:10.1103/physrevlett.47.1400 1981

[1] [1]

(1904).Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire.Arkiv för Matematik, Astronomi och Fysik1: 681–704

von Koch, H. (1904).Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire.Arkiv för Matematik, Astronomi och Fysik1: 681–704

work page 1904

[2] [2]

(1915).Sur une courbe dont tout point est un point de ramification.Comptes Rendus de l’Académie des Sciences de Paris160: 302–305

Sierpiński, W. (1915).Sur une courbe dont tout point est un point de ramification.Comptes Rendus de l’Académie des Sciences de Paris160: 302–305

work page 1915

[3] [3]

Falconer, K. J. (2014).Fractal Geometry: Mathematical Foundations and Applications, 3rd ed. Wiley, Chichester, UK. DOI:10.1002/9781118762875

work page doi:10.1002/9781118762875 2014

[4] [4]

(2008).Measure, Topology, and Fractal Geometry, 2nd ed

Edgar, G. (2008).Measure, Topology, and Fractal Geometry, 2nd ed. Springer, New York, NY, USA. DOI:10.1007/978-0-387-74749-1

work page doi:10.1007/978-0-387-74749-1 2008

[5] [5]

Hutchinson, J. E. (1981).Fractals and self-similarity.Indiana Univ. Math. J.30(5): 713–747. DOI:10.1512/iumj.1981.30.30055

work page doi:10.1512/iumj.1981.30.30055 1981

[6] [6]

(1982).The Fractal Geometry of Nature.W

Mandelbrot, B. (1982).The Fractal Geometry of Nature.W. H. Freeman, New York, NY, USA

work page 1982

[7] [7]

Barnsley, M. F. (1993).Fractals Everywhere, 2nd ed. Academic Press / Morgan Kaufmann, San Francisco, CA, USA

work page 1993

[8] [8]

Witten, T. A. & Sander, L. M. (1981).Diffusion-limited aggregation, a kinetic critical phenomenon.Phys. Rev. Lett.47: 1400–1403. DOI:10.1103/PhysRevLett.47.1400. 17

work page doi:10.1103/physrevlett.47.1400 1981