Decomposition method and upper bound density related to congruent saturated hyperball packings in hyperbolic n-space

2025/06/13 by Arnasli Yahya, Yahya, Arnasli, Jenő Szirmai +1
Mathematics · #Mathematical Dynamics and Fractals #Point processes and geometric inequalities #Functional Equations Stability Results

paper · pdf · doi:10.48550/arxiv.2506.11682

Abstract

In this paper, we study the problem of hyperball (hypersphere) packings in n-dimensional hyperbolic space (n ≥ 4). We prove that to each n-dimensional congruent saturated hyperball packing, there is an algorithm to obtain a decomposition of n-dimensional hyperbolic space ℍⁿ into truncated simplices. We prove, using the above method and the results of the paper \citeM94, that the upper bound of the density for saturated congruent hyperball packings, related to the corresponding truncated tetrahedron cells, is attained in a regular truncated simplex. In 4-dimensional hyperbolic space, we determined this upper bound density to be approximately 0.75864. Moreover, we deny A.~Przeworski's conjecture \citeP13 regarding the monotonization of the density function in the 4-dimensional hyperbolic space.

Citations

• Dense ball packings by tube manifolds as new models for hyperbolic crystallography
• Classical Notions and Problems in Thurston Geometries
• Ball packings in hyperbolic space
• Optimal ball and horoball packings generated by 3-dimensional simply truncated Coxeter orthoschemes with parallel faces
• Congruent and non-congruent hyperball packings related to doubly truncated Coxeter orthoschemes in hyperbolic 3-space
• Hyperball packings related to octahedron and cube tilings in hyperbolic\n space
• Infinite series of compact hyperbolic manifolds, as possible crystal structures
• Decomposition method related to saturated hyperball packings
• Top dense hyperbolic ball packings and coverings for complete Coxeter orthoscheme groups
• Density upper bound for congruent and non-congruent hyperball packings generated by truncated regular simplex tilings
• Packings with horo- and hyperballs generated by simple frustum orthoschemes
• Hyperball packings in hyperbolic 3-space
• The optimal hyperball packings related to the smallest compact arithmetic 5-orbifolds
• A candidate to the densest packing with equal balls in the Thurston\n geometries

Related

• Upper bound of density for packing of congruent hyperballs in hyperbolic 3-space
• Decomposition method related to saturated hyperball packings
• Density upper bound for congruent and non-congruent hyperball packings generated by truncated regular simplex tilings
• Hyperball packings related to octahedron and cube tilings in hyperbolic\n space
• Congruent and non-congruent hyperball packings related to doubly truncated Coxeter orthoschemes in hyperbolic 3-space