Related papers: Twisted Angles for Central Configurations Formed B…

Central configurations play an important role in the dynamics of the $n$-body problem: they occur as relative equilibria and as asymptotic configurations in colliding trajectories. We illustrate how they can be found as projective fixed…

We classify the full set of convex central configurations in the Newtonian four-body problem. Particular attention is given to configurations possessing some type of symmetry or defining geometric property. Special cases considered include…

We consider the expected value for the total curvature of a random closed polygon. Numerical experiments have suggested that as the number of edges becomes large, the difference between the expected total curvature of a random closed…

A closed-form solution for the boundary of the flat state of an orthogonal cross section of contiguous surface geometry formed by the intersection of two cylinders of equal radii oriented in dual directions of rotation about their…

We analyze the problem of quadrangulating a $n$-sided patch, each side at its boundary subdivided into a given number of edges, using a single irregular vertex (or none, when $n = 4$) that breaks the otherwise fully regular lattice. We…

We consider the construction of a polygon $P$ with $n$ vertices whose turning angles at the vertices are given by a sequence $A=(\alpha_0,\ldots, \alpha_{n-1})$, $\alpha_i\in (-\pi,\pi)$, for $i\in\{0,\ldots, n-1\}$. The problem of…

We examine a class of geometric theorems on cyclic 2n-gons. We prove that if we take n disjoint pairs of sides, each pair separated by an even number of polygon sides, then there is a linear combination of the angles between those sides…

We consider $n$-body problems given by potentials of the form ${\alpha\over r^a}+{\beta\over r^b}$ with $a,b,\alpha,\beta$ constants, $0\le a<b$. To analyze the dynamics of the problem, we first prove some properties related to central…

This is a study of the construction of particular regular sub-n-gons T in regular n-gons P using a special system of chords of P. In particular, some of these sub-n-gons have areas which are integer divisors of the area of the given n-gon…

This work provides two sufficient conditions in terms of sections or projections for a convex body to be a polytope. These conditions are necessary as well.

Nontrivial twisted boundary conditions associated with extra compact dimensions produce an ambiguity in the value of the four dimensional coupling constants of the renormalizable interactions of the twisted fields' zero modes. Resolving…

Inspired by the issue of stability of molecular structures, we investigate the strict minimality of point sets with respect to configurational energies featuring two- and three-body contributions. Our main focus is on characterizing those…

For a tetrahedron, suppose that all internal angles of faces and all dihedral angles are less than a fixed constant $C$ that is smaller than $\pi$. Then, it is said to satisfy the maximum angle condition with the constant $C$. The maximum…

An equilateral pentagon is a polygon in the plane with five sides of equal length. In this paper we classify the central configurations of the $5$-body problem having the five bodies at the vertices of an equilateral pentagon with an axis…

We establish the existence of a single-parameter family of the concave kite central configurations in the 4-body problem with two pairs of equal masses. In such configurations, one pair of the masses must lie on the base of an isosceles…

A new coordinate system on the tangent space to planar configurations is introduced to simplify some calculations on central configurations and relative equilibria in the $N$-body problem with a homogeneous potential, which includes the…

The conjecture of the existence and the uniqueness of the strictly convex quadrilateral central configuration for the Newtonian 4-body problem is one of the most-talked open problems in the study of the classical n-body problems in…

We study the relationship between the masses and the geometric properties of central configurations. We prove that in the planar four-body problem, a convex central configuration is symmetric with respect to one diagonal if and only if the…

Let $S$ be a set of $n$ points in the plane, $\wp(S)$ be the set of all simple polygons crossing $S$, $\gamma_P$ be the maximum angle of polygon $P \in \wp(S)$ and $\theta =min_{P\in\wp(S)} \gamma_P$. In this paper, we prove that…

Let t_a be the Dehn twist about a circle a on an orientable surface. It is well known that for each circle b and an integer n, I(t_a^n(b),b)=|n|I(a,b)^2, where I(,) is the geometric intersection number. We prove a similar formula for…