arXiv subjects
Publications and source records attributed to Jenő Szirmai.
In this paper we deal with Nil geometry, in whose projective model the geodesic curves are helix-like and fit onto geodesic Nil cylinders of revolution with fibrum axes. In this paper we investigate relations for geodesic cylinders and thus also geodesic curves, which lead to an analogous result to Pappus's hexagon theorem and provide important information about the structure of the considered space.
This paper presents a unified theory for the power of a point with respect to generalized spheres (spheres, horospheres, and hyperspheres) in $n$-dimensional hyperbolic space $\mathbf{H}^n$. By extending the classical secant theorem, we derive a novel formula for hyperspheres and also prove that the radical surface of any two non-concentric generalized spheres is a hyperplane. These results provide tools for constructing power diagrams and studying hyperball packings.
In \cite{Sz25} we generalized the famous Menelaus' and Ceva's theorems for translation triangles in each non-constant curvature Thurston geometry. In this paper based on the described method and results, we prove that the classical Desargues's and Pappus's hexagon theorems are true not only in classical geometries with constant curvature, but also in Thurston geometries with non-constant curvature on the translation surfaces. In our work we use the unified projective models of Thurston geometries.
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.
We determine the optimal horoball packing densities for Koszul-type Coxeter simplex tilings in hyperbolic $3$-space. Using a parametrization of horoballs by the Busemann function and the symmetry of the tilings, we obtain families of packings that attain the universal simplicial density upper bound \[ d_3(\infty) \;=\; \left( 2 \sqrt{3}\,Λ\!\left(\tfracπ{3}\right) \right)^{-1} \;\approx\; 0.853276, \] where $Λ$ denotes the Lobachevsky function. These results show that extremal packing densities in $\mathbb{H}^3$ are realized by multiple explicit Coxeter tilings and are closely tied to special values of $L$-functions and hyperbolic manifold volumes.
In this paper, we study the problem of hyperball (hypersphere) packings in $n$-dimensional hyperbolic space ($n \ge 4$). We prove that to each $n$-dimensional congruent saturated hyperball packing, there is an algorithm to obtain a decomposition of $n$-dimensional hyperbolic space $\mathbb{H}^n$ into truncated simplices. We prove, using the above method and the results of the paper \cite{M94}, 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 \cite{P13} regarding the monotonization of the density function in the $4$-dimensional hyperbolic space.
After having investigated and defined the ``surface of a translation-like triangle" in each non-constant curvature Thurston geometry \cite{Cs-Sz25}, we generalize the famous Menelaus' and Ceva's theorems for translation triangles in the mentioned spaces. The described method makes it possible to transfer further classical Euclidean theorems and notions to Thurston geometries with non-constant curvature. In our work we will use the projective models of Thurston geometries described by E. Molnár in \cite{M97}.
In the present paper we deal with non-constant curvature Thurston geometries \cite{M97}, \cite{S}, \cite{Sz22-3},\cite{W06}. We define and determine the generalized trans\-lation-like Apollonius surfaces and thus also bisector surfaces as a special case. Moreover, we give a possible definition of the "surface of a translation-like triangle" in each investigated geometry. In our work we will use the projective model of Thurston geometries described by E. Molnár in \cite{M97}.
In this paper, we present a new record for the densest geodesic congruent ball packing configurations in $\mathbf{H}^2\!\times\!\mathbf{R}$ geometry, generated by screw motion groups. These groups are derived from the direct product of rotational groups on $\mathbf{H}^2$ and some translation components on the real fibre direction $\mathbf{R}$ that can be determined by the corresponding Frobenius congruences. Moreover, we developed a procedure to determine the optimal radius for the densest geodesic ball packing configurations related to the considered screw motion groups. The highest packing density, $\approx0.80529$, is achieved by a multi-transitive case given by rotational parameters $(2,20,4)$. E. Molnár demonstrated that homogeneous 3-spaces can be uniformly interpreted in the projective 3-sphere $\mathcal{PS}^3(\mathbf{V}^4, \boldsymbol{V}_4, \mathbf{R})$. We use this projective model of $\mathbf{H}^2\!\times\!\mathbf{R}$ to compute and visualize the locally optimal geodesic ball arrangements.
After having investigated several types of geodesic ball packings in $\mathbf{S}^2 \times \mathbf{R}$ space, in this paper we study the locally optimal geodesic of simply and multiply transitive ball packings with equal balls to the space groups generated by rotations in $\mathbf{H}^2 \times \mathbf{R}$ geometry. These groups can be derived by direct product of the isometries on hyperbolic plane $\mathbf{H}^2$ and the real line $\mathbf{R}$. Moreover, we develop a procedure to determine the densities of the above locally densest geodesic ball packing configurations. Additionally, we examine the monotonicity properties of the densities within infinite series of the considered space groups. E. {Molnár} showed, that the homogeneous 3-spaces have a unified interpretation in the projective 3-sphere $\mathcal{PS}^3(\mathbf{V}^4,\boldsymbol{V}_4, \mathbf{R})$. In our work, we use this projective model of $\mathbf{H}^2 \times \mathbf{R}$ to visualize the locally optimal ball arrangements.
After having investigated the geodesic triangles and their angle sums in Nil and $Sl\times\mathbb{R}$ geometries we consider the analogous problem in Sol space that is one of the eight 3-dimensional Thurston geometries. We analyse the interior angle sums of geodesic triangles and we prove that it can be larger than, less than or equal to $π$. Moreover, we determine the equations of Sol isoptic surfaces of translation-like segments and as a special case of this we examine the Sol translation-like Thales sphere, which we call Thaloid. We also discuss the behavior of this surface. In our work we will use the projective model of Sol described by E. Molnár in \cite{M97}.
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 $Cw(2z, 3 \le z \in \bN$, odd number), of our new series of {\it tube or cobweb manifolds} $Cw = \HYP/\BCw$ with $z$-rotational symmetry. As we know, $\BCw$ is a fixed-point-free isometry group, acting on $\HYP$ discontinuously with appropriate tricky fundamental domain $Cw$, so that every point has a ball-like neighbourhood in the usual factor-topology. Our every $Cw(2z)$ is minimal, i.e. does not cover regularly a smaller manifold. It can be derived by its general symmetry group $\BW(u, v, w = u)$ that is a complete Coxeter orthoscheme reflection group, extended by the half-turn $\Bh$ $(0 \leftrightarrow 3, 1 \leftrightarrow 2)$ of the complete orthoscheme $A_0A_1A_2A_3 \sim b_0b_1b_2b_3$ (Fig.~1). The vertices $A_0$ and $A_3$ are outer points of the $\HYP$, as $1/u + 1/v \le 1/2$ is required, $3 \le u = w, v$ for the above orthoscheme parameters. The situation is described first in Figure 1 of the half trunc-orthoscheme $W$ and its usual extended Coxeter diagram, moreover, by the scalar product matrix $(b^{ij}) = (\langle \Bb^i, \Bb^j \rangle)$ in formula (1.1) and its inverse $(A_{jk}) = (\langle \BA_j, \BA_k \rangle)$ in (1.3). These will describe the hyperbolic angle and distance metric of the half trunc-orthoscheme $W$, then its ball packings, densities, then those of the manifolds $Cw(2z)$. As first results we concentrate only on particular constructions by computer for probable material model realizations, atoms or molekules by equal balls, for general $W(u;v;w=u)$ as well, summarized at the end of our paper.
In this paper we define the notion of infinite or bounded fibre-like geodesic cylinder in $\widetilde{\mathbf{S}\mathbf{L}_2\mathbf{R}}$ space, develop a method to determine its volume and total surface area. We prove that the common part of the above congruent fibre-like cylinders with the base plane are Euclidean circles and determine their radii. Using the former classified infinite or bounded congruent regular prism tilings with generating groups $\mathbf{pq2_1}$ we introduce the notions of cylinder packings, coverings and their densities. Moreover, we determine the densest packing, the thinnest covering cylinder arrangements in $\widetilde{\mathbf{S}\mathbf{L}_2\mathbf{R}}$ space, their densities, their connections with the extremal hyperbolic circle arrangements and with the extremal fibre-like cylinder arrangements in $\mathbf{H}X\mathbf{R}$ space In our work we use the projective model of $\widetilde{\mathbf{S}\mathbf{L}_2\mathbf{R}}$ introduced by E. Molnár in \cite{M97}.
After investigating the $3$-dimensional case [35], we continue to address and close the problems of optimal ball and horoball packings in truncated Coxeter orthoschemes with parallel faces that exist in $n$-dimensional hyperbolic space $\overline{\mathbb{H}}^n$ up to $n=6$. In this paper, we determine the optimal ball and horoball packing configurations and their densities for the aforementioned tilings in dimensions $4 \leq n \leq 6$, where the symmetries of the packings are derived from the considered Coxeter orthoscheme groups. Moreover, for each optimal horoball packing, we determine the parameter related to the corresponding Busemann function, which provides an isometry-invariant description of different optimal horoball packing configurations.
After having investigated the geodesic and translation triangles and their angle sums in $\SOL$ and $\SLR$ geometries we consider the analogous problem in $\NIL$ space that is one of the eight 3-dimensional Thurston geometries. We analyze the interior angle sums of translation triangles in $\NIL$ geometry and we provide a new approach to prove that it can be larger than or equal to $π$. Moreover, for the first time in non-constant curvature Thurston geometries we have developed a procedure for determining the equations of $\NIL$ isoptic surfaces of translation-like segments and as a special case of this we examine the $\NIL$ translation-like Thales sphere, which we call {\it Thaloid}. In our work we will use the projective model of $\NIL$ described by E. Molnár in \cite{M97}.
Of the Thurston geometries, those with constant curvature geometries (Euclidean $\EUC$, hyperbolic $\HYP$, spherical $\SPH$) have been extensively studied, but the other five geometries, $\HXR$, $\SXR$, $\NIL$, $\SLR$, $\SOL$ have been thoroughly studied only from a differential geometry and topological point of view. However, classical concepts highlighting the beauty and underlying structure of these -- such as geodesic curves and spheres, the lattices, the geodesic triangles and their surfaces, their interior sum of angles and similar statements to those known in constant curvature geometries can be formulated. These have not been the focus of attention. In this survey, we summarize our results on this topic and pose additional open questions.
In this paper, we describe and visualize the densest ball and horoball packing configurations belonging to the simply truncated $3$-dimensional hyperbolic Coxeter orthoschemes with parallel faces. These beautiful packing arrangements describe and show the very interesting structure of the mentioned orthoschemes and the corresponding Coxeter groups. We use for the visualization the Beltrami-Cayley-Klein ball model of $3$-dimensional hyperbolic space $\boldsymbol{H}^3$ and the pictures were made by the Python software.
In this paper we deal with $\NIL$ geometry, which is one of the homogeneous Thurston 3-geometries. We define the "surface of a geodesic triangle" using generalized Apollonius surfaces. Moreover, we show that the "lines" on the surface of a geodesic triangle can be defined by the famous Menelaus' condition and prove that Ceva's theorem for geodesic triangles is true in $\NIL$ space. In our work we will use the projective model of $\NIL$ geometry described by E. Molnár in \cite{M97}.