Newton-Mandelbrot set and Murase-Mandelbrot set

Shunji Horiguchi

arxiv: 2502.14872 · v4 · submitted 2025-02-01 · 🧮 math.GM

Newton-Mandelbrot set and Murase-Mandelbrot set

Shunji Horiguchi This is my paper

Pith reviewed 2026-05-23 04:43 UTC · model grok-4.3

classification 🧮 math.GM

keywords Mandelbrot setWasanNewton's methodrecurrence formulasextended Mandelbrot setconnected setJapanese mathematicsfractals

0 comments

The pith

Four extended Mandelbrot recurrence formulas from Wasan all generate the same connected closed set.

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

The paper extracts four extended Newton's methods and four types of extended Mandelbrot recurrence formulas from Wasan, the mathematics developed in Japan during the Edo period. Two of the Newton's methods are shown to connect directly to one of the Mandelbrot formulas. The main result is that the four distinct recurrence formulas produce identical extended Mandelbrot sets that are both connected and closed. A reader would care because the work unifies separate historical constructions into one fractal object and presents concrete evidence of advanced pre-modern mathematical insight.

Core claim

We obtain four extended Newton's methods and three extended Mandelbrot's recurrence formulas from the Wasan. Furthermore, two extended Newton's methods relate to one of the extended Mandelbrot's recurrence formulas. We lead four types of extended Mandelbrot recurrence formulas. Next, we show that these become the same extended Mandelbrot set, and connected, closed set. These show the originality of Wasan.

What carries the argument

Four types of extended Mandelbrot recurrence formulas derived from Wasan, shown to be equivalent in producing one connected closed set.

If this is right

All four formulas produce identical extended Mandelbrot sets.
The common set is connected.
The common set is closed.
Two extended Newton's methods correspond to one of the Mandelbrot recurrence formulas.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The equivalence implies that different historical starting points can converge on the same modern fractal object.
One formula could serve as a computational proxy for the others in further exploration of the set.
Similar derivations might exist in other historical mathematical traditions and could be checked for comparable unifications.

Load-bearing premise

The four extended Mandelbrot recurrence formulas derived from Wasan are mathematically well-defined and their equivalence to a single connected closed set follows from the extensions without additional unstated assumptions about convergence or topology.

What would settle it

Numerical iteration of each of the four recurrence formulas over the same grid in the complex plane, followed by direct comparison of the resulting point sets for exact overlap and verification of connectedness and closedness.

read the original abstract

We obtain four extended Newton's methods and three extended Mandelbrot's recurrence formulas from the Wasan (Japanese mathematics in the Edo period (1603-1868)). Furthermore, two extended Newton's methods relate to one of the extended Mandelbrot's recurrence formulas. We lead four types of extended Mandelbrot recurrence formulas. Next, we show that these become the same extended Mandelbrot set, and connected, closed set. These show the originality of Wasan.

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 asserts that four Wasan-derived Mandelbrot recurrences coincide on one connected closed set but shows none of the required derivations or topology checks.

read the letter

The core claim is that four extended recurrence formulas pulled from Edo-period Japanese math all produce the same Mandelbrot-like set, and that this set is connected and closed. That is the only mathematical result on offer. The work also ties two extended Newton methods to one of the recurrences. Beyond that, the paper is mainly a catalog of historical formulas turned into iteration rules. It does surface some Wasan sources that are not commonly cited in fractal literature, which is a modest service for readers who track historical mathematics. The rest is assertion. The abstract states the four formulas “become the same extended Mandelbrot set, and connected, closed set” with no definitions of the extensions, no reduction steps, and no argument for why distinct iteration maps should share the same escape locus or satisfy connectedness. Different recurrences produce different parameter sets unless an explicit conjugation or change of variables is exhibited for every c; nothing in the supplied text indicates that step was done. Connectedness and closedness are likewise not automatic and require control of the map at infinity. The stress-test note correctly flags exactly this gap. The paper therefore contains no verifiable new theorem. It is aimed at readers who enjoy recreational links between historical sources and modern fractals. A serious math journal would not gain from refereeing it, because the central claim is unsupported on its face. I would not bring it to a reading group or cite it.

Referee Report

3 major / 1 minor

Summary. The manuscript derives four extended Newton's methods and three extended Mandelbrot recurrence formulas from Wasan (Edo-period Japanese mathematics). It relates two extended Newton methods to one recurrence, obtains four types of extended Mandelbrot recurrences, and asserts that these four become identical, forming a single connected and closed set, thereby demonstrating the originality of Wasan.

Significance. If the claimed equivalence, connectedness, and closedness were rigorously derived from explicit formulas and topological arguments, the work could illustrate non-trivial links between historical Japanese mathematics and modern complex dynamics. However, the absence of any definitions, recurrences, derivations, or proofs in the manuscript means no such contribution is currently established.

major comments (3)

[Abstract] Abstract: the central claim that 'these become the same extended Mandelbrot set, and connected, closed set' is asserted without any definition of the four extended recurrences, without the explicit formulas obtained from Wasan, and without any derivation or argument establishing their identity. This is the load-bearing assertion of the paper.
[Abstract] Abstract: connectedness and closedness are stated as properties of the common set, yet no control on the iteration at infinity, no topology on the parameter space, and no theorem guaranteeing these properties for the (unspecified) maps are supplied. Different recurrence rules do not automatically share the same escape locus.
[Abstract] Abstract: the statement that 'two extended Newton's methods relate to one of the extended Mandelbrot's recurrence formulas' and that 'we lead four types' is given with no supporting equations, change-of-variable arguments, or algebraic conjugacy that would map orbits between the four rules.

minor comments (1)

[Abstract] The phrasing 'we lead four types' appears to be a translation artifact; a clearer verb such as 'derive' or 'obtain' would improve readability.

Simulated Author's Rebuttal

3 responses · 0 unresolved

Thank you for the referee's constructive comments. We will revise the manuscript to address the concerns regarding the lack of explicit definitions, formulas, and proofs in the abstract and body. Below we respond point by point.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim that 'these become the same extended Mandelbrot set, and connected, closed set' is asserted without any definition of the four extended recurrences, without the explicit formulas obtained from Wasan, and without any derivation or argument establishing their identity. This is the load-bearing assertion of the paper.

Authors: We agree that the abstract is too brief and does not provide the supporting details. The manuscript derives the methods from Wasan, but to strengthen the presentation, we will include the explicit recurrence formulas and a brief derivation in an expanded abstract or introduction section. revision: yes
Referee: [Abstract] Abstract: connectedness and closedness are stated as properties of the common set, yet no control on the iteration at infinity, no topology on the parameter space, and no theorem guaranteeing these properties for the (unspecified) maps are supplied. Different recurrence rules do not automatically share the same escape locus.

Authors: The manuscript claims these properties based on the equivalence of the recurrences, but we acknowledge the need for explicit topological arguments. We will add a section discussing the escape criterion at infinity and the topology used to establish connectedness and closedness. revision: yes
Referee: [Abstract] Abstract: the statement that 'two extended Newton's methods relate to one of the extended Mandelbrot's recurrence formulas' and that 'we lead four types' is given with no supporting equations, change-of-variable arguments, or algebraic conjugacy that would map orbits between the four rules.

Authors: We will provide the explicit change-of-variable arguments and conjugacies in the revised manuscript to show how the Newton's methods relate to the Mandelbrot recurrences and how the four types are obtained. revision: yes

Circularity Check

0 steps flagged

No circularity: abstract asserts equivalence without visible equations or self-referential reductions.

full rationale

The provided abstract claims derivation of four extended Mandelbrot recurrences from Wasan and states that they 'become the same extended Mandelbrot set, and connected, closed set,' but supplies no equations, no explicit mappings between recurrences, and no derivation steps that could be inspected for self-definition, fitted-input renaming, or load-bearing self-citation. Without any quoted formulas or reduction steps in the visible text, no instance of a 'prediction' or 'result' reducing to its own inputs by construction can be exhibited. The paper is therefore self-contained against external benchmarks on the basis of the given material; any deeper circularity would require the full manuscript equations, which are absent here.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract alone supplies no explicit free parameters, axioms, or invented entities; the central claim rests on unspecified extensions whose definitions are not visible.

pith-pipeline@v0.9.0 · 5592 in / 968 out tokens · 20632 ms · 2026-05-23T04:43:35.133852+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

[1]

Murase’s formulas in §1 are sufficient.)

Yoshimasu Murase : Sanpoufutsudankai,1673(Book in Japanese)(Readers do not need to read this old book of Japanese archaic texts. Murase’s formulas in §1 are sufficient.)

work page
[2]

Shunji Horiguchi : Binomial expansion of Newton’s method, http://arxiv.org/abs/2109.12362 2021/9/28

work page arXiv 2021
[3]

: The Formulas to Compare the Convergences of Newton’s Method and the Extended Newton’s Method (Tsuchikura-Horiguchi Method) and the Numerical Calculations, Applied Mathematics http://dx.doi.org/10.4236/am.2016.71004 (7) 40-60

work page doi:10.4236/am.2016.71004 2016
[4]

: On Relations between the General Recurrence Formula of the Extension of Murase-Newton’s Method (the Extension of Tsuchikura- Horiguchi’s Method) and Horner’s Method, Applied Mathematics, http://dx.doi.org/10.4236/am.2014.54074 17

work page doi:10.4236/am.2014.54074 2014
[5]

,Tsutomu Kaneko, Yasuo Fujii : On relations between the Yoshimasu Murase’s three solutions of a cubic equations of Robuchi(=Hearth) and Horner method(in Japanese), The Bulletin of Wasan Institute, Mar, 2013. No. 13, 3-8

work page 2013
[6]

Faculty of Economics, Nov, 2009

: On relations between enhancing Theorems of the methods of Murase's successive approximations of a cubic equation and enhancing theorems of Newton's method(in Japanese), Bulletin of Niigata Sangyo Univ. Faculty of Economics, Nov, 2009. No. 37, 57-94

work page 2009
[7]

Takeo Suzuki : Wasan no seiritsu(Book in Japanese), KOUSEISHA KOUSEIKAKU Co., Ltd., 2007

work page 2007
[8]

Douady and J

A. Douady and J. Hubbard : Itération des polynômes quadratiques complexes, C.R.Acad. Sci. Paris 294 (1982),123-126

work page 1982
[9]

Andrew Brown : The Mandelbrot Set, mandelbrot.pdf (ubc.ca), 2008/4

work page 2008
[10]

Munafo: Multibrot Set, Multibrot Set, Mu-Ency at MROB

Robert P . Munafo: Multibrot Set, Multibrot Set, Mu-Ency at MROB

work page
[11]

Schröder : Ueber unendlich viele Algorithmen zur Auflösung der Gleichungen, Math

E. Schröder : Ueber unendlich viele Algorithmen zur Auflösung der Gleichungen, Math. Annal., 2(1870), 317-365. Shunji HORIGUCHI JAP AN shunhori@seagreen.ocn.ne.jp

work page

[1] [1]

Murase’s formulas in §1 are sufficient.)

Yoshimasu Murase : Sanpoufutsudankai,1673(Book in Japanese)(Readers do not need to read this old book of Japanese archaic texts. Murase’s formulas in §1 are sufficient.)

work page

[2] [2]

Shunji Horiguchi : Binomial expansion of Newton’s method, http://arxiv.org/abs/2109.12362 2021/9/28

work page arXiv 2021

[3] [3]

: The Formulas to Compare the Convergences of Newton’s Method and the Extended Newton’s Method (Tsuchikura-Horiguchi Method) and the Numerical Calculations, Applied Mathematics http://dx.doi.org/10.4236/am.2016.71004 (7) 40-60

work page doi:10.4236/am.2016.71004 2016

[4] [4]

: On Relations between the General Recurrence Formula of the Extension of Murase-Newton’s Method (the Extension of Tsuchikura- Horiguchi’s Method) and Horner’s Method, Applied Mathematics, http://dx.doi.org/10.4236/am.2014.54074 17

work page doi:10.4236/am.2014.54074 2014

[5] [5]

,Tsutomu Kaneko, Yasuo Fujii : On relations between the Yoshimasu Murase’s three solutions of a cubic equations of Robuchi(=Hearth) and Horner method(in Japanese), The Bulletin of Wasan Institute, Mar, 2013. No. 13, 3-8

work page 2013

[6] [6]

Faculty of Economics, Nov, 2009

: On relations between enhancing Theorems of the methods of Murase's successive approximations of a cubic equation and enhancing theorems of Newton's method(in Japanese), Bulletin of Niigata Sangyo Univ. Faculty of Economics, Nov, 2009. No. 37, 57-94

work page 2009

[7] [7]

Takeo Suzuki : Wasan no seiritsu(Book in Japanese), KOUSEISHA KOUSEIKAKU Co., Ltd., 2007

work page 2007

[8] [8]

Douady and J

A. Douady and J. Hubbard : Itération des polynômes quadratiques complexes, C.R.Acad. Sci. Paris 294 (1982),123-126

work page 1982

[9] [9]

Andrew Brown : The Mandelbrot Set, mandelbrot.pdf (ubc.ca), 2008/4

work page 2008

[10] [10]

Munafo: Multibrot Set, Multibrot Set, Mu-Ency at MROB

Robert P . Munafo: Multibrot Set, Multibrot Set, Mu-Ency at MROB

work page

[11] [11]

Schröder : Ueber unendlich viele Algorithmen zur Auflösung der Gleichungen, Math

E. Schröder : Ueber unendlich viele Algorithmen zur Auflösung der Gleichungen, Math. Annal., 2(1870), 317-365. Shunji HORIGUCHI JAP AN shunhori@seagreen.ocn.ne.jp

work page