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There are two regular polygons Pn-(R1) and Pn-(R2) which correspond to the given polygonal distances {d1, d2, ..., dn. It is proved that the radii R1, R2 of the circumcircles of the regular polygons and the n distances are connected by the system: ... where m = 1, ..., n - 1. The solutions for the equilateral triangles and the squares are investigated. The two-parametric families of the solutions are given in both cases, when three distances are rational numbers. [ABSTRACT FROM AUTHOR]

We consider the central configurations of the 1 + N -body problem, where N bodies are infinitesimal and the remaining one body is dominant. For regular polygon central configurations, we prove that the masses of all the infinitesimal bodies are equal when N is odd and the masses of the alternate infinitesimal bodies must be equal when N is even. Moreover, in the case of N being even, we present the relationship of the mass parameters between two consecutive infinitesimal bodies. [ABSTRACT FROM AUTHOR]

Solving geometry problems is both difficult and interesting. Difficult because there is no general algorithm to solve more or less non-trivial problems as every single problem requires individual and creative approach. At the same time, this is a very interesting activity, because for almost every problem there are plenty of ways to solve it. In this work, we present the method of auxiliary circle divided into equal parts. This method allows finding solution algorithm for some geometry problems which are hard to solve by the method of additional constructions. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We consider the central configurations of the 1+N-body problem, where N bodies are infinitesimal and the remaining one body is dominant. For regular polygon central configurations, we prove that the masses of all the infinitesimal bodies are equal when N is odd and the masses of the alternate infinitesimal bodies must be equal when N is even. Moreover, in the case of N being even, we present the relationship of the mass parameters between two consecutive infinitesimal bodies.

Design of one-degree-of-freedom (1-DOF) mechanisms is of paramount interest. This work deals with the generation of exact regular polygonal paths by using two particular 1-DOF mechanisms. The design of two-bar and three-bar mechanisms including non-circular gears is presented and evaluated. Differences in the kinematics of both mechanisms are discussed. The three-bar mechanism allows derivable velocity and acceleration curves during the whole trajectory, including the polygon vertices, a feature that cannot be achieved with the simpler two-bar mechanism. Hence, the three-bar mechanism makes it possible to generate perfect vertices using non-circular gears. To the best of the authors' knowledge, this mechanism is the first 1-DOF mechanism with articulated links and a non-circular gear transmission that can generate exact regular polygonal paths. The proposed mechanisms can also be applied to generate two straight lines with a given angle, which is another novel contribution of this work. A prototype of the three-bar mechanism has been developed for experimental validation. • A 1-DOF mechanism that can generate exact regular polygonal paths is presented. • Solutions with two- and three-bar mechanisms are analyzed. • The mechanism can also generate two exact straight lines with a given angle. • Non-circular gears are used to generate the required non-linear transmission. [ABSTRACT FROM AUTHOR]

This paper deals with the free vibration and buckling of heavy column, considering its own self-weight. The column has a regular polygonal cross-section with a constant area. The column is applied to an external axial load as well as the self-weight. The five end conditions of the column are considered. Based on equilibrium equations of the column element, differential equations governing the vibrational and buckled mode shapes of column are derived. In solution methods, differential equations are numerically integrated by the direct integration method and eigenvalues of the natural frequency, buckling load and self-weight buckling length are calculated by the determinant search method. The numerical results of this study were in good agreement with those of the reference. Parametric study of the end condition, side number and self-weight on the natural frequency and buckling load was carried out. [ABSTRACT FROM AUTHOR]

The paper presents the results of the study of the new set of polihedra, the Concave pyramids of the fourth sort, the construction procedures for generating them and their possible application. Correspondingly to the method of generating the Concave cupolae of fourth sort, the Concave pyramids of fourth sort have the similar logic of origination, and their counterpart in regular faced convex pyramids. They are characterised by the polygonal base, deltahedral surface net, obtained by folding the planar net of unilateral triangles, the polar distribution of the unit space cells with common apex - the top of the Concave pyramid. Polihedral surface of the planar net of Concave pyramids is produced by polar distribution of unit cells, consisting of a spatial sexagon and spatial pentagon - six, or five, unilateral triangles grouped around the common vertex. In the deltahedral surface, the two neighbouring unit cells are joined by means of a unilateral triangle in the zone of the polygonal base and a spatial quadrangle with which they share common sides. The criterion of face regularity is respected, as well as the criterion of multiple axial symmetry. The sort of the Concave pyramids is determined by the number of equilateral triangle rows in thus obtained polyhedron's net. The parameters of the solids were determined constructively by geometric methods.

In this article, the buckling optimization of axially functionally graded (AFG) columns to maximize the buckling capacity is studied. Consideration is given to an AFG column having a tapered regular polygon cross section and variable material properties. The governing differential equation is derived based on Euler–Bernoulli beam theory with the relevant boundary conditions and is solved using the direct integration method combined with a determinant search algorithm. The computed buckling loads are compared with those presented in the literature and obtained from finite element analysis. Numerical examples for buckling load and buckled mode shape are given to highlight the effect of parameters related to the Young's modulus, cross-sectional shape, tapering and column volume. In particular, the geometry and material parameters that provide buckling optimization at constant volume of the column are analysed. [ABSTRACT FROM AUTHOR]

Let G be a quadrangulation on the projective plane P , i.e., a map of a simple graph on P such that each face is quadrilateral. For a vertex v ∈ V (G) of degree 3 with neighbors v 1 , v 3 , v 5 , a Y-rotation is to delete three edges v v 1 , v v 3 , v v 5 and add v v 2 , v v 4 , v v 6 , where the union of three faces incident to v is surrounded by a closed walk v 1 v 2 v 3 v 4 v 5 v 6 . We say that G is k -minimal if its shortest noncontractible cycle is of length k and if any face contraction yields a noncontractible cycle of length less than k. It was proved that for any k ≥ 3 , any two k -minimal quadrangulations on P are Y -equivalent , i.e., can be transformed into each other by Y-rotations (Nakamoto and Suzuki, 2012). In this paper, we find wider Y-equivalence classes of quadrangulations on P , extending a result on a geometric realization of quadrangulations on P as a rhombus tiling in an even-sided regular polygon (Hamanaka et al., 2020). [ABSTRACT FROM AUTHOR]

We characterize solutions of the mean value linear elliptic equation with constant coefficients in the complex plane in the case of regular polygon. [ABSTRACT FROM AUTHOR]

Problems that satisfy various constraints while maintaining weight balance in ships or aircraft were studied. In addition, a study was conducted to solve the problem with a mathematical method under the condition that the shape and weight of the load are the same and the m×n (m and n are all odd) mesh structures. The problem is that the existing mathematical weight balancing method is not suitable for circular structures. In this paper, we studied the load stowing problem in a circular space where objects are loaded at the vertices of N equilateral polygons. Assuming that all N conformal polygons have an even number of angles, it was proved that a loading method that always maintains weight balance regardless of the variety of number of loads. By providing the structure and loading method of the drone loading ship, we showed that our method was appropriate. [ABSTRACT FROM AUTHOR]

We consider a regular plane polygon with n vertices and an arbitrary point in the plane. Let R be the circumscribed radius of the polygon and L a distance from the point to the centroid of the polygon. Then the averages of the (2m)-th powers of distances from the point to the polygon vertices satisfy the relations Sn(2) = R² + L²; Sn(2m) = (R²> + L²)m + bm 2 ∑ k=1 (m 2k) (2k k) (R² + L²)m-2k (RL)2k, where m = 2,..., n - 1. [ABSTRACT FROM AUTHOR]

Many real 3D objects have complex geometric shapes in various types of analyses. Image of these objects is recorded in the form of a 2D map. In the analysis, a simplification of this image to basic 2D figures with defined geometry is often needed. The paper presents an analysis of the geometry of a flat image (an image of a 3D object) using regular polygons. Geometry properties (F form, C centroid, S size) were determined to describe the object. Various criteria of selection of the 'best ' regular n-sided polygon for a given 2D object (solved theoretically) were put forward. In the paper, criteria for describing a 2D object by regular n-sided polygons were defined on the basis of determining the measure of object shape elongation (e). In the 'blind ' theoretical example, it was tested whether the individual shape measures listed in the paper allow for correct identification of the shapes of given 2D objects. The practical application of measures is illustrated by two actual examples. While in the first example the shape of the Canary Islands is analysed, the second example describes the shape of Poland 's borders. Actual examples deliver different results for different measures. In effect, there is no clear objective criterion for selecting a polygon shape. The simplifications of the shape of an object presented in the paper should not be equated with the object 's generalization. Such simplifications are used in GIS to visualize geographic analyses based on the data available in the primary database, because the object will retain the character of the shape in the simplest possible geometry and neighborhood, and does not lose any of the scope and accuracy of the attributes assigned to a given object in the database. [ABSTRACT FROM AUTHOR]

A quadrangulation on a closed surface is k-minimal if its shortest noncontractible cycle is of length k and if any face contraction yields a noncontractible cycle of length less than k. We prove that the rhombus tilings of a regular 2k-gon bijectively correspond to the pairs of a k-minimal quadrangulations on the projective plane and its specified noncontractible k-cycle. [ABSTRACT FROM AUTHOR]

Polygonal hydraulic jump is among the subjects widely studied by scientists in recent years. Although this phenomenon was discovered nearly two decades ago, many of the probable reasons behind it remain unknown. Accordingly, the main goal of the present study is to conduct a laboratory-scale phenomenological investigation on polygonal hydraulic jumps. The results indicated that the main cause of polygonal hydraulic jumps is the presence of disturbances and instabilities in flows, systems, or environments. Given the Plateau–Rayleigh instability, in the presence of surface tension and viscosity effects, the disturbances and instabilities create stable circular jumps unstable and then turn them into polygonal jumps. A stable circular jump was created with the elimination of instabilities, and the jump, unlike what observed in previous studies, was stable at a low to high range of Reynolds number. In addition, the behavior of polygonal hydraulic jumps formed in previous studies in the presence of instabilities was investigated. In a constant flow rate, the area inside the jump was equal for all the possible jumps at an error level of less than 10%. Among the polygons with an equal number of sides, which are possibly observed in a particular flow rate, the jump changes into a regular polygon, as the surface tension naturally tends to create the minimum possible surface area for the jump. [ABSTRACT FROM AUTHOR]

This study presents the conditions of MT (2)-equivalence for two systems of vectors fx1; x2; x3g and {x1, x2, x3} in R2T, where MT (2) is the group of all isometries of the 2-dimensional taxicab space RT². Firstly a minimal complete system of MT (2)-invariants of {x1, x2, x3} is obtained. Then, using the conditions of MT (2)-equivalence, an answer is given to the open prob- lem posed in [10, p.428]. Furthermore, an algorithm is given for constructing taxicab regular polygons in terms of MT (2)-invariants. This algorithm is gen- eral and useful to construct the taxicab regular 2n-gons and gives a tool to solve special cases of the open problem posed in [2, p.32]. Besides, both the conditions of the taxicab regularity of Euclidean regular polygons and Euclidean regularity of taxicab regular polygons are given in terms of MT (2)-invariants. [ABSTRACT FROM AUTHOR]

Given a regular closed curve α in the plane, a ϕ -isoptic of α is a locus of points from which pairs of tangent lines to α span a fixed angle ϕ. If, in addition, the chord that connects the two points delimiting the visibility angle is of constant length ℓ , then α is said to be (ϕ , ℓ) -isochordal viewed. Some properties of these curves have been studied, yet their full classification is not known. We approach the problem in an inverse manner, namely that we consider a ϕ -isoptic curve c as an input and construct a curve whose ϕ -isoptic is c. We provide thus a sufficient condition that constitutes a partial solution to the inverse isoptic problem. In the process, we also study a relation of isoptics to multihedgehogs. Moreover, we formulate conditions on the behavior of the visibility lines so as their envelope is a (ϕ , ℓ) -isochordal-viewed curve with a prescribed ϕ -isoptic c. Our results are constructive and offer a tool to easily generate this type of curves. In particular, we show examples of (ϕ , ℓ) -isochordal-viewed curves whose ϕ -isoptic is not circular. Finally, we prove that these curves allow the motion of a regular polygon whose vertices lie along the (ϕ , ℓ) -isochordal-viewed curve. [ABSTRACT FROM AUTHOR]