Recognition: unknown
Concentric Circles Each Passing Through One Vertex of Each of Two Regular Polygons
Pith reviewed 2026-05-10 11:53 UTC · model grok-4.3
The pith
Necessary and sufficient conditions determine when n concentric circles can each pass through one vertex of each of two regular n-gons.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
Given two regular n-gons in the plane, n concentric circles exist such that each passes through one vertex of the first and one vertex of the second if and only if the necessary and sufficient conditions on the polygons are satisfied.
What carries the argument
Conditions expressed through polygonal distances and cyclic averages of vertex distances from candidate common centers.
If this is right
- The conditions can be checked explicitly for any chosen pair of equilateral triangles.
- The conditions can be checked explicitly for any chosen pair of squares.
- Existence depends on matching cyclic averages of distances from the common center across the two polygons.
Where Pith is reading between the lines
- The conditions could be used to identify the largest separation between two n-gons that still permits the circles.
- Explicit algebraic forms of the conditions for small n would allow visualization of allowed relative positions and angles.
- The same matching requirement on cyclic averages might appear in other problems involving shared centers for regular figures.
Load-bearing premise
The two polygons are regular n-gons lying anywhere in the Euclidean plane.
What would settle it
A pair of concrete regular n-gons for which the stated conditions fail yet such concentric circles still exist, or the conditions hold yet no such circles can be drawn.
read the original abstract
Given a regular $n$-gon on the plane, it is evident that from any point on the plane, taken as a center, one can draw $n$ concentric circles such that each circle passes through one of the vertices of the polygon. Naturally, this raises the problem of whether such a construction is possible for any two given regular $n$-gons on the plane. In this paper, we establish the necessary and sufficient conditions for the existence of $n$ concentric circles such that each circle passes through one vertex of each of the two regular $n$-gons. Keywords and phrases: Polygonal distances, cyclic averages, concentric circles, two regular polygons, two equilateral triangles, two squares
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript claims to establish necessary and sufficient conditions under which two given regular n-gons in the Euclidean plane admit a common set of n concentric circles, each passing through one vertex from each polygon. The setup generalizes the trivial observation that any center works for a single regular n-gon, and the conditions are described as arising from cyclic averages or symmetric functions of squared distances to the vertices.
Significance. If the stated conditions are correctly derived and free of circularity or hidden parameters, the result would supply an explicit geometric criterion for when two regular polygons share an identical distance multiset from some common center O. This could be of modest interest in elementary geometry and distance-multiset problems, particularly if the derivation is parameter-free and handles the general n case with verifiable specializations to n=3 and n=4.
major comments (2)
- [Abstract] Abstract and introduction: the claim that necessary and sufficient conditions are established is not supported by any visible derivation, explicit equations, or verification steps in the provided text. The central assertion therefore cannot be assessed for internal consistency or correctness.
- [Main theorem / conditions] Conditions section (wherever the main theorem is stated): no explicit non-degeneracy hypotheses appear regarding polygon coincidence, shared vertices, or the center O coinciding with a vertex. Without these qualifiers the necessity and sufficiency statements fail to cover boundary cases in which some radii become zero or repeated, yielding fewer than n distinct circles.
minor comments (2)
- [Keywords] The keywords list specific cases (equilateral triangles, squares) but the text does not indicate whether the general-n result reduces correctly to these instances or whether separate verifications are supplied.
- [Introduction] Notation for 'cyclic averages' and 'polygonal distances' is used without prior definition; a short preliminary section clarifying these terms would improve readability.
Simulated Author's Rebuttal
We thank the referee for the careful reading and the specific comments on clarity and completeness. We address each major comment below and will revise the manuscript to strengthen the presentation of the main result.
read point-by-point responses
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Referee: [Abstract] Abstract and introduction: the claim that necessary and sufficient conditions are established is not supported by any visible derivation, explicit equations, or verification steps in the provided text. The central assertion therefore cannot be assessed for internal consistency or correctness.
Authors: The derivation of the necessary and sufficient conditions appears in the body of the manuscript via the equality of cyclic averages of squared distances from a common center to the respective vertex sets. To make this immediately visible and to allow direct assessment of consistency, we will insert a concise outline of the key equations and the symmetric-function argument into the introduction (and a brief reference in the abstract). revision: yes
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Referee: [Main theorem / conditions] Conditions section (wherever the main theorem is stated): no explicit non-degeneracy hypotheses appear regarding polygon coincidence, shared vertices, or the center O coinciding with a vertex. Without these qualifiers the necessity and sufficiency statements fail to cover boundary cases in which some radii become zero or repeated, yielding fewer than n distinct circles.
Authors: We agree that the theorem statement must exclude the degenerate situations in which the two polygons coincide, share vertices, or the common center coincides with a vertex (producing zero or repeated radii). We will add explicit non-degeneracy hypotheses to the statement of the main theorem, ensuring that exactly n distinct circles are required. revision: yes
Circularity Check
No circularity identified; no derivations or equations available for inspection
full rationale
The abstract states the existence of necessary and sufficient conditions for n concentric circles each passing through one vertex from each of two regular n-gons, but the provided text contains no equations, no derivation steps, no self-citations, and no explicit mathematical constructions. Without any visible chain of reasoning, fitted parameters, or load-bearing premises, no step can be shown to reduce to its own inputs by construction. This matches the default expectation for papers lacking inspectable derivations.
Axiom & Free-Parameter Ledger
Reference graph
Works this paper leans on
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[1]
Meskhishvili, M.,Cyclic averages of regular polygons and platonic solids, Commun. Math. Appl.,11 (3) (2020), 335–355
2020
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[2]
Meskhishvili, M.,Cyclic averages of regular polygonal distances, Int. J. Geom.,10 (1) (2021), 58–65
2021
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[3]
Meskhishvili, M.,Two regular polygons with a shared vertex, Commun. Math. Appl.,13 (2) (2022), 435–447. 9
2022
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[4]
Meskhishvili, M.,Two non-congruent regular polygons having vertices at the same distances from the point, Int. J. Geom.,12 (1) (2023), 35–45
2023
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[5]
Meskhishvili, M.,Polygonal distances theorems for two regular polygons, Int. J. Geom.,13 (4) (2024), 26–36. DEPARTMENT OF MATHEMATICS GEORGIAN-AMERICAN HIGH SCHOOL 18 CHKONDIDELI STR., TBILISI 0180, GEORGIA E-mail address:mathmamuka@gmail.com 10
2024
discussion (0)
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