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arxiv: 2605.12314 · v1 · submitted 2026-05-12 · 🧮 math-ph · math.MP

Recognition: 2 theorem links

· Lean Theorem

Quasi-Sierpinski Structure for Uniform Load Distribution

Javier Rodr\'iguez-Cuadrado , Jes\'us San Mart\'in
Authors on Pith no claims yet

Pith reviewed 2026-05-13 03:53 UTC · model grok-4.3

classification 🧮 math-ph math.MP
keywords quasi-Sierpinski structureuniform load distributionTakagi classCantor functionland reclamationseabedfractal structure
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The pith

A quasi-Sierpinski fractal structure achieves uniform load distribution on the seabed when its supports displace vertically according to any function of the Takagi class.

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

This paper presents a fractal structure similar to the Sierpinski triangle for land reclamation that can rest directly on the seabed. The key to uniform load distribution is requiring the supports to displace vertically following functions of the Takagi class. As a result, the vertical deformations of the structure follow the same class of functions while horizontal deformations relate to the Cantor function. The approach works with any combination of element areas and materials, offering designers flexibility in construction.

Core claim

To achieve uniform load distribution, the supports of the quasi-Sierpinski structure must displace vertically following any function of the Takagi class. This causes the vertical deformations to follow the Takagi class and the horizontal deformations to be related to the Cantor function. The structure can be built with an unlimited number of combinations of areas and materials.

What carries the argument

Takagi-class functions that dictate the vertical displacements of the supports to produce uniform load distribution.

Load-bearing premise

That a physical structure can be constructed whose supports exactly follow arbitrary Takagi-class functions despite real-world factors like material properties, gravity, and seabed interactions.

What would settle it

Constructing a small-scale prototype and measuring if the vertical displacements under load precisely match a chosen Takagi function while achieving uniform base pressure.

Figures

Figures reproduced from arXiv: 2605.12314 by Javier Rodr\'iguez-Cuadrado, Jes\'us San Mart\'in.

Figure 1
Figure 1. Figure 1: Quasi-Sierpinski structure of N levels, corresponding to N = 5, where β is the angle formed by the inclined members with the horizontal. • Nodes: Identification is performed based on level n, n = 1, 2, . . . , N +1, and within each level, according to the natural number t that indicates its ordinal position, from left to right, with t = 1, 2, . . . , 2 n−1 for n = 1, 2, . . . , N and t = 1, 2, . . . , 2 n−… view at source ↗
Figure 2
Figure 2. Figure 2: Identification of certain nodes, inclined members (I), horizontal [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Universal Quasi-Sierpinski structure of N levels, corresponding to N = 5, where Y is the height of the structure. given that structure Q has 2 N−1 + 1 supports, supports 1 and 2 N−1 + 1 must bear a downward vertical load of F/2 N , and supports 2, 3, . . . , 2 N−1 must bear a downward vertical load of F/2 N−1 , since they cover twice the area of the previous ones. Therefore, the reactions on each support w… view at source ↗
Figure 4
Figure 4. Figure 4: Structures Q of N′ levels (first in green, second in purple, third in orange and fourth in gray) in a structure Q of N levels, corresponding to N′ = 3 and N = 5. is 1/s and for which the virtual force is the opposite, −1/s. These members counteract each other pair by pair because they have the same length, real force, area, and Young’s modulus. Thus, on the right side of Eq. 2, only the following remains: … view at source ↗
Figure 5
Figure 5. Figure 5: System VS-SU and forces generated in the members of the third [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: System VS-NV and forces generated in the members of the struc [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: System VS-NH and forces generated in the members of the struc [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Examples of functions G  x; P H ∞  from the Takagi class. In particu￾lar, (a) Double of the Takagi function, for P H ∞ = {  1 2 k } ∞ k=0 and (b) Parabola (a smooth function), for P H ∞ = {1} ∞ k=0. In general, if the ratios follow a geometric progression with parameter r, that is, if P H ∞ = {r k} ∞ k=0, then G(x; P H ∞) is twice the Takagi-Landsberg curve with parameter 1 4r [16]. Once the relationsh… view at source ↗
Figure 9
Figure 9. Figure 9: Examples of functions J  x; P H ∞  . In particular, (a) Identity func￾tion, for P H ∞ = {1} ∞ k=0 and (b) 3 4 times the pseudo-inverse of the Cantor function, for P H ∞ = {  3 2 k } ∞ k=0. displacements. We denote by fµ : {1, 2, . . . , N + 1} × [0, 1] → R the function that returns the horizontal displacement µn,t of each node (n, t) according to its horizontal coordinate x. Under the coordinate criter… view at source ↗
Figure 10
Figure 10. Figure 10: (a) Shape of structure Q before (in gray) and after load [PITH_FULL_IMAGE:figures/full_fig_p026_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Load system in a structure Q of N levels, corresponding to N = 2, where c = cos(β), s = sin(β) and F is the value of the downward force applied on the node (1, 1). support 2 N−1 + 1 of the first one generates an upward vertical reaction of value F/2 N+1 and support 1 of the second one generates an upward vertical reaction of value F/2 N+1 (the sum of both gives the reaction of value F/2 N of the structure… view at source ↗
Figure 12
Figure 12. Figure 12: Load system in a structure Q of N + 1 levels and the inclusion of two structures Q of N levels (in purple and green), corresponding to N = 4, where c = cos(β), s = sin(β) and F is the value of the downward force applied on the node (1, 1). left, respectively. Since the sum of horizontal forces is zero throughout the structure, we conclude that support 2 N−1+1 also does not generate horizontal reaction. B … view at source ↗
read the original abstract

Land reclamation methods, indispensable for the proper development of modern coastal cities, are ecologically destructive. We present a fractal structure, similar to a Sierpinski triangle, which solves this problem by resting directly on the seabed thanks to the uniform load distribution we achieve on its base. To obtain this uniform distribution, we show that the supports of the structure must displace vertically following any function of the Takagi class. This causes the vertical deformations of the structure to follow this same class and the horizontal deformations to be related to the Cantor function. The structure works with an unlimited number of combinations of areas of its elements and materials, which gives designers a high degree of constructive flexibility.

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.

Referee Report

1 major / 0 minor

Summary. The manuscript proposes a quasi-Sierpinski fractal structure for land reclamation that rests directly on the seabed by achieving uniform base load distribution. The central claim is that the supports must displace vertically according to any function in the Takagi class; this forces vertical deformations of the structure to follow the same class while horizontal deformations relate to the Cantor function. The structure is asserted to admit arbitrary combinations of element areas and materials.

Significance. If the mathematical relation between Takagi-class displacements and uniform load holds under the paper's constitutive assumptions, the construction would supply a parametric family of fractal geometries that enforce uniform seabed loading without additional foundations. This could be of interest for coastal engineering models that seek to minimize ecological disruption, provided the derivation is made explicit and the physical realizability is addressed.

major comments (1)
  1. [Abstract] Abstract: The assertion that supports 'must displace vertically following any function of the Takagi class' to obtain uniform distribution is presented without derivation, equilibrium equations, or supporting calculations. The claim therefore cannot be verified from the manuscript and appears defined directly in terms of the functions that produce the uniform outcome, raising a circularity concern for the central result.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comment on the abstract. We address the concern directly below and have revised the manuscript to improve verifiability of the central claim.

read point-by-point responses

  1. Referee: The assertion that supports 'must displace vertically following any function of the Takagi class' to obtain uniform distribution is presented without derivation, equilibrium equations, or supporting calculations. The claim therefore cannot be verified from the manuscript and appears defined directly in terms of the functions that produce the uniform outcome, raising a circularity concern for the central result.

    Authors: We agree that the original abstract was too concise and did not indicate where the supporting analysis appears. The necessity of Takagi-class vertical displacements follows from the equilibrium equations for the quasi-Sierpinski truss under the assumption of linear-elastic bars and uniform base pressure; the self-similar geometry forces the admissible displacement functions to satisfy the functional equation that defines the Takagi class. This derivation is carried out explicitly in Sections 2 and 3 of the manuscript, beginning from the global force balance and proceeding to the recursive relations imposed by the fractal construction. The class is therefore not presupposed but obtained as the solution space that enforces constant seabed reaction. To remove any appearance of circularity and to make the claim verifiable from the abstract itself, we have expanded the abstract with a one-sentence outline of the equilibrium argument and inserted explicit cross-references to the relevant sections. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives that uniform base load distribution on the quasi-Sierpinski structure requires vertical support displacements to follow any Takagi-class function, with vertical deformations matching this class and horizontal ones related to the Cantor function. The provided abstract and summary present this as a shown mathematical necessity from equilibrium considerations rather than a redefinition of inputs or a fitted parameter renamed as prediction. No explicit equations, self-citations, or ansatzes are available in the text to exhibit a reduction by construction (e.g., no fitted input called prediction or self-definitional loop). The result is framed as independent first-principles content, with physical constructibility noted as a separate application issue. This is the expected honest non-finding for a self-contained mathematical claim without load-bearing self-reference.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Review based solely on the abstract; full manuscript not available for exhaustive audit of parameters or assumptions.

axioms (1)
  • domain assumption Vertical support displacements following any Takagi-class function produce uniform load distribution on the base of the quasi-Sierpinski structure.
    This is the central unproven premise stated in the abstract that enables the uniform-distribution claim.
invented entities (1)
  • Quasi-Sierpinski structure no independent evidence
    purpose: To achieve uniform load distribution while resting directly on the seabed for land reclamation.
    A new postulated geometric configuration introduced in the abstract without reference to independent physical validation.

pith-pipeline@v0.9.0 · 5406 in / 1403 out tokens · 145686 ms · 2026-05-13T03:53:56.590174+00:00 · methodology

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Reference graph

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