Related papers: On the Local-Global Principle for Integral Apollon…

We prove that a set of density one satisfies the local-global conjecture for integral Apollonian gaskets. That is, for a fixed integral, primitive Apollonian gasket, almost every (in the sense of density) admissible (passing local…

Haag, Kertzer, Rickards, and Stange disprove the Local-Global Conjecture for Apollonian circle packings. We extend their disproof to four more types of integral circle packing: the octahedral, cubic, square, and triangular packings. In each…

Four mutually tangent spheres form two gaps. In each of these, one can inscribe in a unique way four mutually tangent spheres such that each one of these spheres is tangent to exactly three of the original spheres. Repeating the process…

In a primitive integral Apollonian circle packing, the curvatures that appear must fall into one of six or eight residue classes modulo 24. The local-global conjecture states that every sufficiently large integer in one of these residue…

We introduce an orthoplicial Apollonian sphere packing, which is a sphere packing obtained by successively inverting a configuration of 8 spheres with 4-orthplicial tangency graph. We will show that there are such packings in which the…

We generalize work of Bourgain-Kontorovich and Zhang, proving an almost local-to-global property for the curvatures of certain circle packings, to a large class of Kleinian groups. Specifically, we associate in a natural way an infinite…

Apollonian circle packings arise by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. Such packings can be described in terms of the Descartes configurations they contain. It observed…

Apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further tangent circles. In Euclidean space it is possible for every circle in such a packing to have integer radius of curvature,…

Apollonian circle packings arise by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. We observe that there exist Apollonian packings which have strong integrality properties, in which…

Apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further tangent circles. It is possible for every circle in such a packing to have integer radius of curvature, and we call such a…

We use circle method prove an asymptotic local-global theorem on the heights of point orbits of thin subgroups of Bianchi groups in $\mathbb H^3$.

This paper gives $n$-dimensional analogues of the Apollonian circle packings in parts I and II. We work in the space $\sM_{\dd}^n$ of all $n$-dimensional oriented Descartes configurations parametrized in a coordinate system,…

We obtain an asymptotic formula for the number of circles of curvature at most T in any given bounded Apollonian circle packing. For an integral packing, we obtain the upper bounds for the number of circles with prime curvature as well as…

We present a generalization of Descartes' theorem for the family of polytopal sphere packings arising from uniform polytopes. The corresponding quadratic equation is expressed in terms of geometric invariants of uniform polytopes which are…

We consider Apollonian circle packings of a half Euclidean plane. We give necessary and sufficient conditions for two such packings to be related by a Euclidean similarity (that is, by translations, reflections, rotations and dilations) and…

We view space-filling circle packings as subsets of the boundary of hyperbolic space subject to symmetry conditions based on a discrete group of isometries. This allows for the application of counting methods which admit rigorous upper and…

A new locally averaged density for sphere packing in R^3 is defined by a proper combination of the local cell (Voronoi cell) and Delaunay decompositions (\S 1.2.2), using only a single layer of surrounding spheres. Local packings attaining…

The famous Kepler conjecture has a less spectacular, two-dimensional equivalent: The theorem of Thue states that the densest circle packing in the Euclidean plane has a hexagonal structure. A common proof uses Voronoi cells and analyzes…

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…

In an earlier work, we proposed a generalization for the Apollonian packing in arbitrary dimensions and showed that the resulting object in four, five, and six dimensions have properties consistent with the Apollonian circle and sphere…