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Lattice-like Packings and Coverings with Congruent Translation Balls and Cylinders in Sol geometry

Judit Sajtos, Jenő Szirmai

TL;DR

This work investigates lattice-like packings and coverings in Sol geometry using congruent translation balls and translation cylinders. It introduces density notions and derives upper bounds for ball coverings via circumscribed translation spheres of translation tetrahedra, while obtaining exact densities for cylinder packings within fundamental Sol lattices; an algorithm leveraging a tetrahedral decomposition of the fundamental domain and circumscribed spheres underpins the least-dense ball coverings. The study also analyzes lattice-like cylinder packings and coverings, establishing density bounds and identifying locally optimal lattice parameters, notably for $N=3$, with reported densities around 0.7854 for densest cylinder packing and about 1.2645 for thinnest cylinder coverings. Overall, the results advance non-Euclidean packing/covering theory in Thurston geometries, link Dirichlet–Voronoi structures in Sol to lattice tilings, and provide concrete computational methods for lattice-based coverings in this rich geometric setting.

Abstract

The aim of this paper is to study lattice-like coverings with congruent translation balls and the packings and coverings with a type of translation cylinders in Sol space related to the fundamental lattices. We introduce the notions of the densities of the considered problems and give upper estimate to ball coverings using the radii and the volumes of the circumscribed translation spheres of given {\it translation tetrahedra}. Moreover we determine the exact optimal packing and covering densities of a type of cylinder packings belonging to the fundamental lattices.

Lattice-like Packings and Coverings with Congruent Translation Balls and Cylinders in Sol geometry

TL;DR

This work investigates lattice-like packings and coverings in Sol geometry using congruent translation balls and translation cylinders. It introduces density notions and derives upper bounds for ball coverings via circumscribed translation spheres of translation tetrahedra, while obtaining exact densities for cylinder packings within fundamental Sol lattices; an algorithm leveraging a tetrahedral decomposition of the fundamental domain and circumscribed spheres underpins the least-dense ball coverings. The study also analyzes lattice-like cylinder packings and coverings, establishing density bounds and identifying locally optimal lattice parameters, notably for , with reported densities around 0.7854 for densest cylinder packing and about 1.2645 for thinnest cylinder coverings. Overall, the results advance non-Euclidean packing/covering theory in Thurston geometries, link Dirichlet–Voronoi structures in Sol to lattice tilings, and provide concrete computational methods for lattice-based coverings in this rich geometric setting.

Abstract

The aim of this paper is to study lattice-like coverings with congruent translation balls and the packings and coverings with a type of translation cylinders in Sol space related to the fundamental lattices. We introduce the notions of the densities of the considered problems and give upper estimate to ball coverings using the radii and the volumes of the circumscribed translation spheres of given {\it translation tetrahedra}. Moreover we determine the exact optimal packing and covering densities of a type of cylinder packings belonging to the fundamental lattices.

Paper Structure

This paper contains 9 sections, 8 theorems, 27 equations, 5 figures.

Table of Contents

  1. Introduction
  2. On Sol geometry
  3. Translation curves
  4. Translation-like bisector surfaces, translation tetrahedra and their circumscribed spheres
  5. On $\mathbf{Sol}$ lattices
  6. On the fundamental parallelepiped
  7. The lattice-like translation ball coverings
  8. Coverings of the fundamental lattice
  9. Thinnest covering

Key Result

Theorem 2.4

Figures (5)

  • Figure 1: Tetrahedron with vertices $A_1 = (1,0,0,0), \ A_2 = (1,-0.5,0,0), A_3 = (1,0,0.5,0), A_4 = (1,0,0,0.5)$ and circumradius $r\approx 0.459231$.
  • Figure 2: A fundamental parallelepiped for $N=3$.
  • Figure 3: The plane sections of the sphere with $r=2$ parallel to the plane $[y,z]$ at $y=0$
  • Figure 4: Plane sections of spheres with different radii, parallel to the $[x,y]$ plane.
  • Figure 5: The visualization of a convex translation ball and its plane sections parallel to the $[x,y]$ base plane.

Theorems & Definitions (16)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4: Sz13-3
  • Lemma 2.5: Sz17-1
  • Lemma 2.6: Sz17-1
  • Lemma 2.7: Sz17-1
  • Definition 2.8: MSz12, Sz13-3
  • Theorem 2.9: MSz12, Sz13-3
  • Definition 2.10: MSz12, Sz13-3
  • ...and 6 more