相关论文: Apollonian Circle Packings: Number Theory
In the classic circle packing problem, one asks whether a given set of circles can be packed into a given container. Packing problems like this have been shown to be $\mathsf{NP}$-hard. In this paper, we present new sufficient conditions…
We study some properties of a triad of circles associated with a triangle. Each circle is inside the triangle, tangent to two sides of the triangle, and externally tangent to the circle on the third side as diameter. In particular, we find…
In Euclidean geometry the circle of Apollonious is the locus of points in the plane from which two collinear adjacent segments are perceived as having the same length. In Hyperbolic geometry, the analog of this locus is an algebraic curve…
For a circle packing P on the sphere invariant under a geometrically finite Kleinian group, we compute the asymptotic of the number of circles in P of spherical curvature at most $T$ which are contained in any given region.
We have discovered the existence of polydisperse High Internal-Phase-Ratio Emulsions (HIPE) in which the internal-phase droplets, present at 95% volume fraction, remain spherical and organize themselves in the available space according to…
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…
We introduce a new class of fractal circle packings in the plane, generalizing the polyhedral packings defined by Kontorovich and Nakamura. The existence and uniqueness of these packings are guaranteed by infinite versions of the…
A family of fractal arrangements of circles is introduced for each imaginary quadratic field $K$. Collectively, these arrangements contain (up to an affine transformation) every set of circles in the extended complex plane with integral…
This article is a gentle introduction to the mathematical area known as circle packing, the study of the kinds of patterns that can be formed by configurations of non-overlapping circles. The first half of the article is an exposition of…
The configuration space of tricycles (triples of disks in contact) is shown to coincide with the complex plane resulting as a projective space costructed from the tangency and Pauli spinors. Remarkably, the fractal of the depth functions…
We study the problem of high-dimensional multiple packing in Euclidean space. Multiple packing is a natural generalization of sphere packing and is defined as follows. Let $ N>0 $ and $ L\in\mathbb{Z}_{\ge2} $. A multiple packing is a set…
The article presents the mathematical sequences describing circle packing densities in four different geometric configurations involving a hexagonal lattice based equal circle packing in the Euclidian plane. The calculated sequences take…
Circle packings with specified patterns of tangencies form a discrete counterpart of analytic functions. In this paper we study univalent packings (with a combinatorial closed disk as tangent graph) which are embedded in (or fill) a…
Recently, Connelly and Gortler gave a novel proof of the circle packing theorem for tangency packings by introducing a hybrid combinatorial-geometric operation, flip-and-flow, that allows two tangency packings whose contact graphs differ by…
Apollonian gaskets are formed by repeatedly filling the gaps between three mutually tangent circles with further tangent circles. In this paper we give explicit formulas for the the limiting pair correlation and the limiting nearest…
We revisit the scaling properties of growing spheres randomly seeded in d=2,3 and 4 dimensions using a mean-field approach. We model the insertion probability without assuming a priori a functional form for the radius distribution. The…
Circle packings are arrangement of circles satisfying specified tangency requirements. Many problems about packing of circles and spheres occur in nature particularly in material design and protein structure. Surprisingly, little is known…
In this paper, we propose a class of elementary plane geometry problems closely related to the title of this paper. Here, a circle is the 1-dimensional curve bounding a disk. For any nonnegative integer, a circle is called $n$-enclosing if…
We provide upper bounds for the sum of the multiplicities of the non-constant irreducible factors that appear in the canonical decomposition of a polynomial $f(X)\in\mathbb{Z}[X]$, in case all the roots of $f$ lie inside an Apollonius…
Applying circle inversion on a square grid filled with circles, we obtain a configuration that we call a fabric of kissing circles. The configuration and its components, which are two orthogonal frames and two orthogonal families of chains,…