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The Hessian of the renormalized volume of geometrically finite hyperbolic $3$-manifolds without rank-$1$ cusps, computed at the hyperbolic metric $g$ with totally geodesic boundary of the convex core, is shown to be a strictly positive…
In this paper we study the locally optimal geodesic ball packings with equal balls to the $\mathbf{S}^2\!\times\!\mathbf{R}$ space groups having rotation point groups and their generators are screw motions. We determine and visualize the…
The rich theory of Coxeter groups is used to provide an algebraic construction of finite volume hyperbolic n-manifolds. Combinatorial properties of finite images of these groups can be used to compute the volumes of the resulting manifolds.…
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…
The average distance of the equal hard spheres is introduced to evaluate the density of a given arrangement. The absolute smallest value is two radii because the spheres can not be closer to each other than their diameter. The absolute…
We present the first study of disordered jammed hard-sphere packings in four-, five- and six-dimensional Euclidean spaces. Using a collision-driven packing generation algorithm, we obtain the first estimates for the packing fractions of the…
We intend to continue our previous papers (\cite{MSz17} and \cite{MSz18}, as indicated there) on dense ball packing hyperbolic space $\HYP$ by equal balls, but here with centres belonging to different orbits of the fundamental group $Cw(2z,…
The densest local packings of N three-dimensional identical nonoverlapping spheres within a radius Rmin(N) of a fixed central sphere of the same size are obtained for selected values of N up to N = 1054. In the predecessor to this paper…
In this paper we obtain new upper bounds on volumes of right-angled polyhedra in hyperbolic space $\mathbb{H}^3$ in three different cases: for ideal polyhedra with all vertices on the ideal hyperbolic boundary, for compact polytopes with…
This work introduces the Hookean-Voronoi energy, a minimal model for the packing of soft, deformable balls. This is motivated by recent studies of quasi-periodic equilibria arising from dense packings of diblock and star polymers.…
We show that some hyperbolic 3-manifolds which are tessellated by copies of the regular ideal hyperbolic tetrahedron embed geodesically in a complete, finite volume, hyperbolic 4-manifold. This allows us to prove that the complement of the…
We introduce a notion of essential hyperbolic Coxeter polytope as a polytope which fits some minimality conditions. The problem of classification of hyperbolic reflection groups can be easily reduced to classification of essential Coxeter…
In this paper, we find lower bounds for volumes of hyperbolic 3-manifolds with various topological conditions. Let V_3 = 1.01494 denote the volume of a regular ideal simplex in hyperbolic 3-space. As a special case of the main theorem, if a…
In this article we obtain linear programming bounds for the maximal sphere packing density of commutative spaces. A special case of our results solves a conjecture by Cohn and Zhao on linear programming bounds for sphere packings in…
In this paper, we obtain the complete classification for compact hyperbolic Coxeter four-dimensional polytopes with eight facets.
We generate non-lattice packings of spheres in up to 22 dimensions using the geometrical constraint satisfaction algorithm RRR. Our aggregated data suggest that it is easy to double the density of Ball's lower bound, and more tentatively,…
Let M be a geometrically finite hyperbolic 3-manifold whose limit set is a round Sierpi\'nski gasket, i.e. M is geometrically finite and acylindrical with a compact, totally geodesic convex core boundary. In this paper, we classify orbit…
A divide is the image of a proper and generic immersion of a compact $1$-manifold into the $2$-disk. Due to A'Campo's theory, each divide is associated with a link in the 3-sphere. In this paper, we reveal a hidden hyperbolic structure in…
We study of the relation between the geometry of sets in complex hyperbolic space and Hilbert spaces with complete Pick kernels. We focus on the geometry associated with assembling sets into larger sets and of assembling Hilbert spaces into…
In this article we explore a symplectic packing problem where the targets and domains are $2n$-dimensional symplectic manifolds. We work in the context where the manifolds have first homology group equal to $\mathbb{Z}^n$, and we require…