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Mamuka Meskhishvili

Publications and source records attributed to Mamuka Meskhishvili.

8 recordsLinked to original sources
arXiv2026

Concentric Circles Each Passing Through One Vertex of Each of Two Regular Polygons

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

math.GM↗
arXiv2022

Two Regular Polygons with a Shared Vertex

For two non-congruent regular polygons of the same type, the method of finding the points in the plane at the equal distances to the vertices, is established. The existence of two points with this property is proved for two polygons with a shared vertex. For one of them, it is proved that it satisfies the Bottema theorem conditions and based on this, the generalized Bottema theorem for any two regular polygons is given.

math.GM↗
arXiv2022

Two Non-Congruent Regular Polygons Having Vertices at the Same Distances from the Point

For the given regular plane polygon and an arbitrary point in the plane of the polygon, the distances from the point to the vertices of the polygon are defined. We proved that there is one more non-congruent regular polygon having the vertices at the same distances from the point. The sizes of both regular polygons are uniquely determined by these distances. In general case, geometrical construction of the second regular polygon is given. It is proved that there are two points in the plane, which separately have the same set of the distances to the vertices of two non-congruent regular polygons with a shared vertex.

math.GM↗
arXiv2020

Cyclic Averages of Regular Polygons and Platonic Solids

The concept of the cyclic averages are introduced for a regular polygon $P_n$ and a Platonic solid $T_n$. It is shown that cyclic averages of equal powers are the same for various $P_n(T_n)$, but their number is characteristic of $P_n(T_n)$. Given the definition of a circle (sphere) by the vertices of $P_n(T_n)$ and on the base of the cyclic averages are established the common metrical relations of $P_n(T_n)$.

math.GM↗
arXiv2019

New Sense of a Circle

New condition is found for the set of points in the plane, for which the locus is a circle. It is proved: the locus of points, such that the sum of the $(2m)$-th powers $S_n^{(2m)}$}of the distances to the vertexes of fixed regular $n$-sided polygon is constant, is a circle if $$ S_n^{(2m)}>nr^{2m},\ {\rm where}\ m=1,2,\dots,n-1 $$ and $r$ is the distance from the center of the regular polygon to the vertex. The radius $\ell$ satisfies: $$ S_n^{(2m)}=n\Bigg[(r^2+\ell^2)^m+\sum_{k=1}^{[\frac{m}{2}]} {m\choose 2k} (r^2+\ell^2)^{m-2k}(r\ell)^{2k} {2k\choose m}\Bigg]. $$

math.GM↗
arXiv2015

Diophantine Equations and Congruent Number Equation Solutions

By using pairs of nontrivial rational solutions of congruent number equation $$ C_N:\;\;y^2=x^3-N^2x, $$ constructed are pairs of rational right (Pythagorean) triangles with one common side and the other sides equal to the sum and difference of the squares of the same rational numbers. The parametrizations are found for following Diophantine systems: \begin{align*} (p^2\pm q^2)^2-a^2 & =\square_{1,2}\,, \\[0.2cm] c^2-(p^2\pm q^2)^2 & =\square_{1,2}\,, \\[0.2cm] a^2+(p^2\pm q^2)^2 & =\square_{1,2}\,, \\[0.2cm] (p^2\pm q^2)^2-a^2 & =(r^2\pm s^2)^2. \end{align*}

math.GM↗
arXiv2013

Perfect Cuboid and Congruent Number Equation Solutions

A perfect cuboid (PC) is a rectangular parallelepiped with rational sides $a,b,c$ whose face diagonals $d_{ab}$, $d_{bc}$, $d_{ac}$ and space (body) diagonal $d_s$ are rationals. The existence or otherwise of PC is a problem known since at least the time of Leonhard Euler. This research establishes equivalent conditions of PC by nontrivial rational solutions $(X,Y)$} and $(Z,W)$} of congruent number equation $ y^2=x^3-N^2x$, where product $XZ$ is a square. By using such pair of solutions five parametrizations of nearly-perfect cuboid (NPC) (only one face diagonal is irrational) and five equivalent conditions for PC were found. Each parametrization gives all possible NPC. For example, by using one of them -- invariant parametrization for sides and diagonals of NPC are obtained: $a=2XZN$, $b=|YW|$, $c=|X-Z|\sqrt{XZ}\,N$,$d_{bc}=|XZ-N^2|\sqrt{XZ}$, $d_{ac}=|X+Z|\sqrt{XZ}\,N$, $d_s=(XZ+N^2)\sqrt{XZ}$; and condition of the existence of PC is the rationality of $d_{ab} = \sqrt{Y^2W^2+4N^2X^2Z^2}$. Because each parametrization is complete, inverse problem is discussed. For given NPC is found corresponding congruent number equation (i.e. congruent number) and its solutions.

math.NT↗