arXiv subjects
Publications and source records attributed to Jenő Szirmai.
In this paper we study the interior angle sums of geodesic triangles in $\NIL$ geometry and prove that it can be larger, equal or less than $π$. We use for the computations the projective model of $\NIL$ introduced by E. {Molnár} in \cite{M97}.
We study the interior angle sums of translation and geodesic triangles in the universal cover of Sl2(R) geometry. We prove that the angle sum is larger then pi for translation triangles and for geodesic triangles the angle sum can be larger, equal or lessthan π.
In this paper we study the regular prism tilings and construct ball packings by geodesic balls related to the above tilings in the projective model of $\mathbf{Nil}$ geometry. Packings are generated by action of the discrete prism groups $\mathbf{pq2_{1}}$. We prove that these groups are realized by prism tilings in $\mathbf{Nil}$ space if $(p,q)=(3,6), (4,4), (6,3)$ and determine packing density formulae for geodesic ball packings generated by the above prism groups. Moreover, studying these formulae we determine the conjectured maximal dense packing arrangements and their densities and visualize them in the projective model of $\mathbf{Nil}$ geometry. We get a dense (conjectured locally densest) geodesic ball arrangement related to the parameters $(p,q)=(6,3)$ where the kissing number of the packing is $14$, similarly to the densest lattice-like $\mathbf{Nil}$ geodesic ball arrangement investigated by the second author .
The theory of the isoptic curves is widely studied in the Euclidean plane $\bE^2$ (see \cite{CMM91} and \cite{Wi} and the references given there). The analogous question was investigated by the authors in the hyperbolic $\bH^2$ and elliptic $\cE^2$ planes (see \cite{CsSz1}, \cite{CsSz2}, \cite{CsSz5}), but in the higher dimensional spaces there are only a few result in this topic. In \cite{CsSz4} we gave a natural extension of the notion of the isoptic curves to the $n$-dimensional Euclidean space $\bE^n$ $(n\ge 3)$ which are called isoptic hypersurfaces. Now we develope an algorithm to determine the isoptic surface $\mathcal{H}_{\cP}$ of a $3$-dimensional polytop $\mathcal{P}$. We will determine the isoptic surfaces for Platonic solids and for some semi-regular Archimedean polytopes and visualize them with Wolfram Mathematica.
In this paper we study congruent and non-congruent hyperball (hypersphere) packings of the truncated regular tetrahedron tilings. These are derived from the Coxeter simplex tilings $\{p,3,3\}$ $(7\le p \in \mathbb{N})$ and $\{5,3,3,3,3\}$ in $3$ and $5$-dimensional hyperbolic space. We determine the densest hyperball packing arrangements related to the above tilings. We find packing densities using congruent hyperballs and determine the smallest density upper bound of non-congruent hyperball packings generated by the above tilings.
In this paper we deal with the packings derived by horo- and hyperballs (briefly hyp-hor packings) in the $n$-dimensional hyperbolic spaces $\HYN$ ($n=2,3$) which form a new class of the classical packing problems. We construct in the $2-$ and $3-$dimensional hyperbolic spaces hyp-hor packings that are generated by complete Coxeter tilings of degree $1$ i.e. the fundamental domains of these tilings are simple frustum orthoschemes and we determine their densest packing configurations and their densities. We prove that in the hyperbolic plane ($n=2$) the density of the above hyp-hor packings arbitrarily approximate the universal upper bound of the hypercycle or horocycle packing density $\frac{3}π$ and in $\HYP$ the optimal configuration belongs to the $[7,3,6]$ Coxeter tiling with density $\approx 0.83267$. Moreover, we study the hyp-hor packings in truncated orthosche\-mes $[p,3,6]$ $(6< p < 7, ~ p\in \bR)$ whose density function is attained its maximum for a parameter which lies in the interval $[6.05,6.06]$ and the densities for parameters lying in this interval are larger that $\approx 0.85397$. That means that these locally optimal hyp-hor configurations provide larger densities that the Böröczky-Florian density upper bound $(\approx 0.85328)$ for ball and horoball packings but these hyp-hor packing configurations can not be extended to the entirety of hyperbolic space $\mathbb{H}^3$.
After having investigated the real conic sections and their isoptic curves in the hyperbolic plane $\bH^2$ we consider the problem of the isoptic curves of generalized conic sections in the extended hyperbolic plane. This topic is widely investigated in the Euclidean plane $\BE^2$ (see for example \cite{Lo}), but in the hyperbolic and elliptic planes there are few results (see \cite{CsSz1}, \cite{CsSz2} and \cite{CsSz3}). In this paper we recall the former results on isoptic curves in the hyperbolic plane geometry, and define the notion of the generalised hyperbolic angle between proper and non-proper straight lines, summarize the notions of generalized hyperbolic conic sections classified by K.~Fladt in \cite{KF1} and \cite{KF2} and gy E.~Molnár in \cite{M81}. Furthermore, we determine and visualize the generalized isoptic curves to all hyperbolic conic sections. We use for the computations the classical models which are based on the projective interpretation of the hyperbolic geometry and in this manner the isoptic curves can be visualized on the Euclidean screen of computer.
In this paper we study the horoball packings related to the hyperbolic 24 cell in the extended hyperbolic space $\overline{\mathbf{H}}^4$ where we allow {\it horoballs in different types} centered at the various vertices of the 24 cell. We determine, introducing the notion of the generalized polyhedral density function, the locally densest horoball packing arrangement and its density with respect to the above regular tiling. The maximal density is $\approx 0.71645$ which is equal to the known greatest ball packing density in hyperbolic 4-space given in \cite{KSz14}.
In this paper we consider ball packings in $4$-dimensional hyperbolic space. We show that it is possible to exceed the conjectured $4$-dimensional realizable packing density upper bound due to L. Fejes Tóth (Regular Figures, 1964). We give seven examples of horoball packing configurations that yield higher densities of $0.71644896\dots$, where horoballs are centered at ideal vertices of certain Coxeter simplices, and are invariant under the actions of their respective Coxeter groups.
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 densest simply transitive geodesic ball arrangements for the above space groups, moreover we compute their optimal densities and radii. The densest packing is derived from the $\mathbf{S}^2\!\times\!\mathbf{R}$ space group $\mathbf{3qe.~I.~3}$ with packing density $\approx 0.7278$. E. Molnár has shown, that the Thurston geometries have an unified interpretation in the real projective 3-sphere $\mathcal{PS}^3$. In our work we shall use this projective model of $\mathbf{S}^2\!\times\!\mathbf{R}$ geometry.
The smallest three hyperbolic compact arithmetic 5-orbifolds can be derived from two compact Coxeter polytops which are combinatorially simplicial prisms (or complete orthoschemes of degree $d=1$) in the five dimensional hyperbolic space $\mathbf{H}^5$ (see \cite{BE} and \cite{EK}). The corresponding hyperbolic tilings are generated by reflections through their delimiting hyperplanes those involve the study of the relating densest hyperball (hypersphere) packings with congruent hyperballs. The analogous problem was discussed in \cite{Sz06-1} and \cite{Sz06-2} in the hyperbolic spaces $\mathbf{H}^n$ $(n=3,4)$. In this paper we extend this procedure to determine the optimal hyperball packings to the above 5-dimensional prism tilings. We compute their metric data and the densities of their optimal hyperball packings, moreover, we formulate a conjecture for the candidate of the densest hyperball packings in the 5-dimensional hyperbolic space $\mathbf{H}^5$.
In this paper we consider the isoptic curves on the 2-dimensional geometries of constant curvature $\bE^2,~\bH^2,~\cE^2$. The topic is widely investigated in the Euclidean plane $\bE^2$ see for example \cite{CMM91} and \cite{Wi} and the references given there, but in the hyperbolic and elliptic plane there are few results in this topic (see \cite{CsSz1} and \cite{CsSz2}). In this paper we give a review on the preliminary results of the isoptics of Euclidean and hyperbolic curves and develop a procedure to study the isoptic curves in the hyperbolic and elliptic plane geometries and apply it for some geometric objects e.g. proper conic sections. We use for the computations the classical models which are based on the projectiv interpretation of the hyperbolic and elliptic geometry and in this manner the isoptic curves can be visualized on the Euclidean screen of computer.
The goal of this paper to determine the optimal horoball packing arrangements and their densities for all four fully asymptotic Coxeter tilings (Coxeter honeycombs) in hyperbolic 3-space $\mathbb{H}^3$. Centers of horoballs are required to lie at vertices of the regular polyhedral cells constituting the tiling. We allow horoballs of different types at the various vertices. Our results are derived through a generalization of the projective methodology for hyperbolic spaces. The main result states that the known Böröczky--Florian density upper bound for "congruent horoball" packings of $\mathbb{H}^3$ remains valid for the class of fully asymptotic Coxeter tilings, even if packing conditions are relaxed by allowing for horoballs of different types under prescribed symmetry groups. The consequences of this remarkable result are discussed for various Coxeter tilings.
$\SLR$ geometry is one of the eight 3-dimensional Thurston geometries, it can be derived from the 3-dimensional Lie group of all $2\times 2$ real matrices with determinant one. Our aim is to describe and visualize the {\it regular infinite (torus-like) or bounded} $p$-gonal prism tilings in $\SLR$ space. For this purpose we introduce the notion of the infinite and bounded prisms, prove that there exist infinite many regular infinite $p$-gonal face-to-face prism tilings $\cT^i_p(q)$ and infinitely many regular (bounded) $p$-gonal non-face-to-face $\SLR$ prism tilings $\cT_p(q)$ for parameters $p \ge 3$ where $ \frac{2p}{p-2} < q \in \mathbb{N}$. Moreover, we develope a method to determine the data of the space filling regular infinite and bounded prism tilings. We apply the above procedure to $\cT^i_3(q)$ and $\cT_3(q)$ where $6< q \in \mathbb{N}$ and visualize them and the corresponding tilings. E. Molnár showed, that the homogeneous 3-spaces have a unified interpretation in the projective 3-space $\mathcal{P}^3(\bV^4,\BV_4, \mathbf{R})$. In our work we will use this projective model of $\SLR$ geometry and in this manner the prisms and prism tilings can be visualized on the Euclidean screen of computer.
The $S^2 \times R$ geometry can be derived by the direct product of the spherical plane $\bS^2$ and the real line $\bR$. J. Z. Farkas has classified and given the complete list of the space groups of $S^2 \times R$. The $S^2 \times R$ manifolds were classified by E. Molnár and J. Z. Farkas by similarity and diffeomorphism. In Szirmai we have studied the geodesic balls and their volumes in $S^2 \times R$ space, moreover we have introduced the notion of geodesic ball packing and its density and have determined the densest geodesic ball packing for generalized Coxeter space groups of $S^2 \times R$. In this paper we study the locally optimal ball packings to the $S^2 \times R$ space groups having Coxeter point groups and at least one of the generators is a glide reflection. We determine the densest simply transitive geodesic ball arrangements for the above space groups, moreover we compute their optimal densities and radii. The density of the densest packing is $\approx 0.80407553$, may be surprising enough in comparison with the Euclidean result $\fracπ{\sqrt{18}} \approx 0.74048$. E. Molnár has shown, that the homogeneous 3-spaces have a unified interpretation in the real projective 3-sphere $\mathcal{PS}^3(\bV^4,\BV_4, \mathbb{R})$. In our work we shall use this projective model of $S^2 \times R$ geometry.
The aim of this paper to determine the locally densest horoball packing arrangements and their densities with respect to fully asymptotic tetrahedra with at least one plane of symmetry in hyperbolic 3-space $\bar{\mathbf{H}}^3$ extended with its absolute figure, where the ideal centers of horoballs give rise to vertices of a fully asymptotic tetrahedron. We allow horoballs of different types at the various vertices. Moreover, we generalize the notion of the simplicial density function in the extended hyperbolic space $\bar{\mathbf{H}}^n, ~(n \ge 2)$, and prove that, in this sense, {\it the well known Böröczky--Florian density upper bound for "congruent horoball" packings of $\bar{\mathbf{H}}^3$ does not remain valid to the fully asymptotic tetrahedra.} The density of this locally densest packing is $\approx 0.874994$, may be surprisingly larger than the Böröczky--Florian density upper bound $\approx 0.853276$ but our local ball arrangement seems not to have extension to the whole hyperbolic space.
In \cite{Sz11} we have generalized the notion of the simplicial density function for horoballs in the extended hyperbolic space $\bar{\mathbf{H}}^n, ~(n \ge 2)$, where we have allowed {\it congruent horoballs in different types} centered at the various vertices of a totally asymptotic tetrahedron. By this new aspect, in this paper we study the locally densest horoball packing arrangements and their densities with respect to totally asymptotic regular tetrahedra in hyperbolic $n$-space $\bar{\mathbf{H}}^n$ extended with its absolute figure, where the ideal centers of horoballs give rise to vertices of a totally asymptotic regular tetrahedron. We will prove that, in this sense, {\it the well known Böröczky density upper bound for "congruent horoball" packings of $\bar{\mathbf{H}}^n$ does not remain valid for $n\ge4$,} but these locally optimal ball arrangements do not have extensions to the whole $n$-dimensional hyperbolic space. Moreover, we determine an explicit formula for the density of the above locally optimal horoball packings, allowing horoballs in different types.
The Nil geometry, which is one of the eight 3-dimensional Thurston geometries, can be derived from {W. Heisenberg}'s famous real matrix group. The aim of this paper to study {\it lattice coverings} in Nil space. We introduce the notion of the density of considered coverings and give upper and lower estimations to it, moreover we formulate a conjecture for the ball arrangement of the least dense lattice-like geodesic ball covering and give its covering density $Δ\approx 1.42900615$. The homogeneous 3-spaces have a unified interpretation in the projective 3-sphere and in our work we will use this projective model of the Nil geometry.