Related papers: A regular $n$-gon spiral
A longstanding question of Gromov asks whether every one-ended word-hyperbolic group contains a subgroup isomorphic to the fundamental group of a closed hyperbolic surface. An infinite family of word-hyperbolic groups can be obtained by…
We give a few simple methods to geometically describe some polygon and chain-spaces in R^d. They are strong enough to give tables of m-gons and m-chains when m <= 6.
Given a subset of $\mathbb C$ containing $x,y$, one can add $x + y,\,x - y,\,xy$ or (when $y\ne0$) $x/y$ or any $z$ such that $z^2=x$. Let $p$ be a prime Fermat number. We prove that it is possible to obtain from $\{1\}$ a set containing…
Two planar sets are circularly separable if there exists a circle enclosing one of the sets and whose open interior disk does not intersect the other set. This paper studies two problems related to circular separability. A linear-time…
We study the role of asymptotic curves in supporting the spiral structure of a N-body model simulating a barred spiral galaxy. Chaotic orbits with initial conditions on the unstable asymptotic curves of the main unstable periodic orbits…
A small polygon is a polygon of unit diameter. The maximal area of a small polygon with $n=2m$ vertices is not known when $m\ge 7$. Finding the largest small $n$-gon for a given number $n\ge 3$ can be formulated as a nonconvex quadratically…
The causes of spiral structure in galaxies remain uncertain. Leaving aside the grand bisymmetric spirals with their own well-known complications, here we consider the possibility that multi-armed spiral features originate from density…
We study transient spiral structures in an isothermal, thin, galactic disc. We find no instability that can grow everywhere from infinitesimal disturbances, but spiral structure does grow in the disc due to an arbitrarily strong,…
This work discusses an approach to solving geometric construction problems in which the given figure is included in a set ordered by construction steps. The flow of information is carried through the chain, allowing the original problem to…
Let R be a ring. A construction method for flexible quadratic algebras with scalar involution over R is presented which unifies various classical constructions in the literature, in particular those to construct composition algebras.
For a fixed dimension $N$ we compute the generating function of the numbers $t_N(n)$ (respectively $\bar{t}_N(n)$) of $PGL_{N+1}(k)$-orbits of rational $n$-sets (respectively rational $n$-multisets) of the projective space $\mathb{P}^N$…
A polygon is called rational if the angle between each pair of sides is a rational multiple of $\pi.$ The main theorem we will prove is Theorem 1: For rational polygons, periodic points of the billiard flow are dense in the phase space of…
In this work we prove constructively that the complement $\R^n\setminus\pol$ of a convex polyhedron $\pol\subset\R^n$ and the complement $\R^n\setminus\Int(\pol)$ of its interior are regular images of $\R^n$. If $\pol$ is moreover bounded,…
In this paper, we systematically study non-crossing chords of simple polygons in the plane. We first introduce the reduced Euler characteristic of a family of line-segments, and subsequently investigate the structure of the diagonals and…
We start from the author's metric for a rotating and accelerating Universe and we calculate the force acting on a particle in it. This force consists of two parts: one is derived from a potential, and the other describes a Coriolis force…
The spiral structure of a spiral galaxy can be seen through different observational tracers such as the dust in the interstellar medium, the free electrons in ionized regions, the molecular gas, or the atomic hydrogen in H{\alpha} regions.…
We give an algorithm to morph planar graph drawings that achieves small grid size at the expense of allowing a constant number of bends on each edge. The input is an $n$-vertex planar graph and two planar straight-line drawings of the graph…
A novel method is proposed for moving large, pyramid construction size, stone blocks. The method is inspired by a well known introductory physics homework problem, and is implemented by tying 12 identical rods of appropriately chosen radius…
Given a convex n-gon P in the Euclidean plane, it is well known that the simplicial complex \theta(P) with vertex set given by diagonals in P and facets given by triangulations of P is the boundary complex of a polytope of dimension n-3. We…
We study combinatorial configurations with the associated point and line graphs being strongly regular. Examples not belonging to known classes such as partial geometries and their generalizations or elliptic semiplanes are constructed.…