Related papers: On compact packings of the plane with circles of t…
The study of rational point sets on circles over the Euclidean plane is discussed in a more general framework, i.e. we generalize the notion rational and consider these circular point sets over arbitrary fields. We also determine the…
Let R be a family of n axis-parallel rectangles with packing number p-1, meaning that among any p of the rectangles, there are two with a non-empty intersection. We show that the union complexity of R is at most O(n+p^2), and that the…
An open problem of Manuel Abellanas asks whether every set of disjoint closed unit disks in the plane can be connected by a conveyor belt, which means a tight simple closed curve that touches the boundary of each disk, possibly multiple…
A convex polygon $A$ is related to a convex $m$-gon $K= \bigcap_{i=1}^m k_i^+$, where $k_1^+,..., k_m^+$ are the $m$ halfplanes whose intersection is equal to $K$, if $A$ is the intersection of halfplanes $a_1^+,...,a_l$, each of which is a…
Polyhedral circle packings are generalizations of the Apollonian packing. We develop the theory of the Apollonian group, Descartes quadratic form, and related objects for all polyhedral packings. We use these tools to determine the domain…
We consider discretization of the 'geometric cover problem' in the plane: Given a set $P$ of $n$ points in the plane and a compact planar object $T_0$, find a minimum cardinality collection of planar translates of $T_0$ such that the union…
We construct a compact metric space that has any other compact metric space as a tangent, with respect to the Gromov-Hausdorff distance, at all points. Furthermore, we give examples of compact sets in the Euclidean unit cube, that have…
It has been shown that univalent circle packings filling in the complex plane $\bold C$ are unique up to similarities of $\bold C$. Here we prove that bounded degree branched circle packings properly covering $\bold C$ are uniquely…
We investigate in this paper the relation between Apollonian $d$-ball packings and stacked $(d+1)$-polytopes for dimension $d\ge 3$. For $d=3$, the relation is fully described: we prove that the $1$-skeleton of a stacked $4$-polytope is the…
The paper reports a generalized expression for filling the congruent circles (of radius r) in a circle (of radius R). First, a generalized expression for the biggest circle (r) inscribed in the nth part of the bigger circle (R) was…
A celebrated result of Beck shows that for any set of $N$ points on $\mathbb{S}^d$ there always exists a spherical cap $B \subset \mathbb{S}^d$ such that number of points in the cap deviates from the expected value $\sigma(B) \cdot N$ by at…
The Orbit Problem consists of determining, given a matrix $A\in \mathbb{R}^{d\times d}$ and vectors $x,y\in \mathbb{R}^d$, whether there exists $n\in \mathbb{N}$ such that $A^n=y$. This problem was shown to be decidable in a seminal work of…
In this paper we prove a theorem that provides an upper bound for the density of packings of congruent copies of a given convex body in $\mathbb{R}^n$; this theorem is a generalization of the linear programming bound for sphere packings. We…
Let $P$ be a set of $n$ points in general position on the plane. A set of closed convex polygons with vertices in $P$, and with pairwise disjoint interiors is called a convex decomposition of $P$ if their union is the convex hull of $P$,…
A set S of 2n+1 points in the plane is said to be in general position if no three points of S are collinear and no four are concyclic. A circle is called halving with respect to S if it has three points of S on its circumference, n-1 points…
Let $Q$ be a finite set of points in the plane. For any set $P$ of points in the plane, $S_{Q}(P)$ denotes the number of similar copies of $Q$ contained in $P$. For a fixed $n$, Erd\H{o}s and Purdy asked to determine the maximum possible…
This paper derives numerical bounds for and implements the splitting circle method for finding roots of a univariate polynomial in the presence of fixed precision.
We show that the nonzero roots of the torsion polynomials associated to the infinite cyclic covers of a given compact, connected, orientable 3-manifold M are contained in a compact part of the complex plane a priori determined by M. This…
We extend two results known for aspherical 3-manifolds to $PD_3$-pairs $(P,\partial{P})$ with aspherical ambient space $P$. Every such $PD_3$-pair may be assembled by attaching 1-handles to $PD_3$-pairs with aspherical; ambient space and…
How can we arrange $n$ lines through the origin in three-dimensional Euclidean space in a way that maximizes the minimum interior angle between pairs of lines? Conway, Hardin and Sloane (1996) produced line packings for $n \leq 55$ that…