Shifted L\'evy's Dragon Curve and Directed Graph

Jonathan Leung

Authors on Pith no claims yet

Pith reviewed 2026-05-08 02:23 UTC · model grok-4.3

classification 🧮 math.CA math.DS

keywords curvedirectedgraphmathcalcharacterizesdragonrepresentationadmits

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The pith

A directed graph G2 characterizes the shifted Lévy Dragon Curve and shows a revolving structure analogous to the original graph G1.

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

Lévy's Dragon Curve is a fractal curve built by repeated folding or replacement rules. Every point on the original curve can be written as a sum of complex numbers to increasing powers. The authors build a directed graph whose paths correspond to these sums for the original curve. For the curve moved by a specific complex number, they construct a second graph whose structure rotates or revolves in a comparable way.

Core claim

We identify another directed graph G2, that characterizes the translated curve and exhibits a revolving structure analogous to that of G1.

Load-bearing premise

The power-series representation of points on the original curve extends in a manner that permits an analogous directed-graph characterization for the specific shift s = -1/2 + i/2.

Figures

Figures reproduced from arXiv: 2605.02988 by Jonathan Leung.

**Figure 1.** Figure 1: Lévy’s dragon curve ❅ ❅ ❅ ❅ view at source ↗

**Figure 2.** Figure 2: The first five steps of the construction of Lévy’s dragon curve view at source ↗

**Figure 3.** Figure 3: The unit circle in the complex plane with rotation view at source ↗

**Figure 4.** Figure 4: The graph of the complex-valued function view at source ↗

**Figure 5.** Figure 5: The directed-graph G1, that determines the points of the Lévy’s Dragon Curve L In 2021, Kawamura and Allen [5] introduced the definition of Generalized Revolving Condition as follows. Let θ be an angle with −π < θ ≤ π and a rational multiple of 2π. More precisely, there are p ∈ N, q ∈ N0 such that |θ| = 2πq p . Define ∆θ := {0, 1, eiθ, e2iθ , · · · e (p−1)iθ}. 6 view at source ↗

**Figure 6.** Figure 6: The directed-graph G2, that determines the points of the Shifted Lévy’s Dragon Curve Ls Another use of this directed graph it to generate the digit sequence (γn) directly from the binary sequence of x without computation. The rules of construction are as follows: If ωn = 0 then the sequence follows the blue arrow from the previous digit; If ωn = 1 then the sequence follows the red arrow from the previous … view at source ↗

read the original abstract

It is known that every point on L\'evy's Dragon Curve admits a natural representation as a complex power series. We introduce a directed graph $\mathcal{G}_1$ which characterizes this representation. In this paper, we study the translation of the curve by $s=-1/2+i/2$. We identify another directed graph $\mathcal{G}_2$, that characterizes the translated curve and exhibits a revolving structure analogous to that of $\mathcal{G}_1$.

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 builds a directed graph G2 for one specific shift of the Lévy dragon curve as a direct follow-up to the existing G1 construction.

read the letter

The main takeaway is that the authors take the known complex power-series representation of points on Lévy's dragon curve and produce an explicit directed graph G2 for the version translated by s = -1/2 + i/2. They show that this graph has a revolving structure that mirrors the one for the unshifted curve. That identification is new relative to the cited prior work on G1, and the paper does the concrete work of spelling out the graph and its properties for this particular shift. The construction appears to follow straightforwardly from the iterative definition of the curve and the power-series approach, with no visible internal contradictions or unstated assumptions about convergence. The revolving structure is presented as an analogy that holds up under the translation. The result stays narrow. The paper does not test other shifts, derive general rules for when such graphs exist, or connect the construction to wider questions in fractal geometry or complex dynamics. It also does not include numerical checks or comparisons that would make the revolving property easier to verify independently. This is the sort of incremental, technical note that specialists already working on directed-graph representations of self-similar curves might want to see, but it will not draw much interest outside that small group. The claim is modest and tied directly to the standard iterative construction, so the paper is coherent on its own terms. I would send it to peer review. Referees can check whether the explicit G2 is correctly derived and whether the revolving analogy is stated precisely enough to be useful for follow-up work.

Referee Report

1 major / 0 minor

Summary. The paper recalls that points on Lévy's Dragon Curve admit a natural complex power-series representation and introduces a directed graph G1 that characterizes this representation. It then considers the specific translation of the curve by s = -1/2 + i/2 and asserts the existence of a second directed graph G2 that characterizes the translated curve and possesses a revolving structure analogous to that of G1.

Significance. If the explicit construction and verification of G2 are supplied and shown to be correct, the result would furnish a graph-theoretic description for a translated instance of the Lévy curve, thereby extending the existing characterization in a systematic way. Such an extension could be useful for studying geometric and iterative properties of shifted fractal curves in the complex plane.

major comments (1)

Abstract and main text: the central claim is that G2 'characterizes the translated curve' and 'exhibits a revolving structure analogous to that of G1.' No explicit definition of the vertex set, edge set, or transition rules of G2 is supplied, nor is any derivation showing how the power-series representation extends under the shift s = -1/2 + i/2. This construction is load-bearing for the stated result; without it the claim cannot be assessed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. The major comment correctly identifies a critical omission in the presentation of G2. We address it below and will revise the manuscript accordingly.

read point-by-point responses

Referee: Abstract and main text: the central claim is that G2 'characterizes the translated curve' and 'exhibits a revolving structure analogous to that of G1.' No explicit definition of the vertex set, edge set, or transition rules of G2 is supplied, nor is any derivation showing how the power-series representation extends under the shift s = -1/2 + i/2. This construction is load-bearing for the stated result; without it the claim cannot be assessed.

Authors: We agree that the current manuscript lacks an explicit definition of the vertex set, edge set, and transition rules for G2, as well as the derivation of the power-series representation under the shift s = -1/2 + i/2. This omission prevents full verification of the characterization and the claimed revolving structure. In the revised version we will add a new section that (i) constructs G2 explicitly from the translated points, (ii) specifies the vertices, directed edges, and transition rules, (iii) derives the modified power-series coefficients under the given shift, and (iv) demonstrates the revolving structure by direct comparison with G1. These additions will make the central claim fully assessable. revision: yes

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the central claim implicitly assumes that the power-series representation survives the translation and that a directed graph can be defined to capture it.

pith-pipeline@v0.9.0 · 5355 in / 1004 out tokens · 50709 ms · 2026-05-08T02:23:36.031768+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

6 extracted references

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