Related papers: A Simple Proof of Thue's Theorem on Circle Packing
We investigate criteria for circle packing(CP) types of disk triangulation graphs embedded into simply connected domains in $ \mathbb{C}$. In particular, by studying combinatorial curvature and the combinatorial Gauss-Bonnet theorem…
The most efficient way to pack equally sized spheres isotropically in 3D is known as the random close packed state, which provides a starting point for many approximations in physics and engineering. However, the particle size distribution…
We show that given any tiling of Euclidean space, any geometric patterns of points, we can find a patch of tiles (of arbitrarily large size) so that copies of this patch appear in the tiling nearly centered on a scaled and translated…
We determine the symmetrized topological complexity of the circle, using primarily just general topology.
The principle of virtual power is used to derive the equilibrium field equations of a porous solid saturated with a fluid, including second density-gradient effects; the intention is the elucidation and extension of the effective stress…
A Euclidean (or hyperbolic) circle packing on a closed triangulated surface with prescribed inversive distance is locally determined by its cone angles. We prove this by applying a variational principle.
If a collection of identical particles is poured into a container, different shapes will fill to different densities. But what is the shape that fills a container as close as possible to a pre-specified, desired density? We demonstrate a…
Via circle pattern techniques, random planar triangulations (with angle variables) are mapped onto Delaunay triangulations in the complex plane. The uniform measure on triangulations is mapped onto a conformally invariant spatial point…
We determine the distribution of nearest neighbour spacings between the tangencies to a fixed circle in a class of circle packings generated by reflections. We use a combination of geometric tools and the theory of automorphic forms.
Tuza conjectured that for every graph $G$, the maximum size $\nu$ of a set of edge-disjoint triangles and minimum size $\tau$ of a set of edges meeting all triangles satisfy $\tau \leq 2\nu$. We consider an edge-weighted version of this…
In this note, we present a simpler way to prove the compactness of the closed intervals in simply ordered set with order topology.
This is the fifth in a series of papers giving a proof of the Kepler conjecture, which asserts that the density of a packing of congruent spheres in three dimensions is never greater than $\pi/\sqrt{18}\approx 0.74048...$. This is the…
A geometric inequality among three triangles, originating in circle packing problems, is introduced. In order to prove it, we reduce the original formulation to the nonnegativity of a particular polynomial in four real indeterminates.…
A proof of the continuous martingale convergence theorem is provided. It relies on a classical martingale inequality and the almost sure convergence of a uniformly bounded non-negative super-martingale, after a truncation argument.
We give an overview of various counting problems for Apollonian circle packings, which turn out to be related to problems in dynamics and number theory for thin groups. This survey article is an expanded version of my lecture notes prepared…
We apply a classic result in Theoretical Computer Science to solve a sticky problem, and give a neat and slick quick proof of half of it
This paper introduces path triangulation of points in a bounded, simply connected surface region, replacing ordinary triangles in a Delaunay triangulation with path triangles from homotopy theory. A {\bf path triangle} has a border that is…
In the context of complex algebraic varieties, the decomposition theorem for semi-small maps provides a decomposition of the direct image of the constant sheaf. In this work, we develop a decomposition theorem for branched coverings of…
We provide a simple proof for the union-closed sets conjecture, a long-standing open problem in set theory with immediate applications to graph theory, number theory, and order-theory.
In this paper we consider the problem of packing a fixed number of identical circles inside the unit circle container, where the packing is complicated by the presence of fixed size circular prohibited areas. Here the objective is to…