Jenő Szirmai

Jenő Szirmai
Budapest University of Technology and Economics · Department of Geometry

PhD

About

120
Publications
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883
Citations
Citations since 2017
46 Research Items
581 Citations
2017201820192020202120222023020406080100120
2017201820192020202120222023020406080100120
2017201820192020202120222023020406080100120
2017201820192020202120222023020406080100120

Publications

Publications (120)
Article
Full-text available
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)=(...
Preprint
We determine the optimal horoball packing densities for the Koszul-type Coxeter simplex tilings in $\mathbb{H}^3$. We give a family of horoball packings parameterized by the Busemann function and symmetry group that achieve the simplicial packing density upper bound $d_3(\infty) = \left( 2 \sqrt{3} \Lambda\left( \frac{\pi }{3} \right) \right)^{-1}...
Article
In this paper, we consider the ball and horoball packings belonging to 3-dimensional Coxeter tilings that are derived by simply truncated orthoschemes with parallel faces. The goal of this paper is to determine the optimal ball and horoball packing arrangements and their densities for all above Coxeter tilings in hyperbolic 3-space ℍ3. The centers...
Preprint
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 hi...
Preprint
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....
Preprint
Full-text available
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...
Article
In the present paper we study $\mathbf{S}^2\!\times\!\mathbf{R}$ and $\mathbf{H}^2\!\times\!\mathbf{R}$ geometries, which are homogeneous Thurston 3-geometries. We define and determine the generalized Apollonius surfaces and with them define the ‘surface of a geodesic triangle’. Using the above Apollonius surfaces we develop a procedure to determin...
Preprint
In this paper we consider the ball and horoball packings belonging to $3$-dimensional Coxeter tilings that are derived by simply truncated orthoschemes with parallel faces. The goal of this paper to determine the optimal ball and horoball packing arrangements and their densities for all above Coxeter tilings in hyperbolic 3-space $\mathbb{H}^3$. Th...
Article
Full-text available
In connection with our works in Molnár (On isometries of space forms. Colloquia Math Soc János Bolyai 56 (1989). Differential geometry and its applications, Eger (Hungary), North-Holland Co., Amsterdam, 1992), Molnár (Acta Math Hung 59(1–2):175–216, 1992), Molnár (Beiträge zur Algebra und Geometrie 38/2:261–288, 1997) and Molnár et al. (in: Prékopa...
Preprint
Full-text available
After the investigation of the congruent and non-congruent hyperball packings related to doubly truncated Coxeter orthoscheme tilings \cite{SzJ1}, we consider the corresponding covering problems. In \cite{MSSz} the authors gave a partial classification of supergroups of some hyperbolic space groups whose fundamental domains will be integer parts of...
Article
Full-text available
In this paper we describe and visualize the densest ball and horoball packing configurations to the simply truncated 3-dimensional hyperbolic Coxeter orthoschemes with parallel faces, using the results of [24]. These beautiful packing arrangements describe and show the very interesting structure of the mentioned orthoschemes and the corresponding C...
Preprint
Full-text available
In the present paper we study $\SXR$ and $\HXR$ geometries, which are homogeneous Thurston 3-geometries. We define and determine the generalized Apollonius surfaces and with them define the "surface of a geodesic triangle". Using the above Apollonius surfaces we develop a procedure to determine the centre and the radius of the circumscribed geodesi...
Article
Full-text available
In the present paper we study S 2 ×R and H 2 ×R geometries, which are homogeneous Thurston 3-geometries. We analyse the interior angle sums of geodesic triangles in both geometries and we prove that in S 2 ×R space it can be larger than or equal to π and in H 2 ×R space the angle sums can be less than or equal to π. This proof is a new direct appro...
Article
This is an overview on ball packings in different geometries. However, parallel to reported mathematical results we discuss various appearances of related phenomena in arts and architecture. The ball (or sphere) packing problem with equal balls, without any symmetry assumption, in a 3-dimensional space of constant curvature is a very intensively re...
Article
Full-text available
In this paper we study congruent and non-congruent hyperball (hypersphere) packings of the truncated regular octahedron and cube tilings. These are derived from the Coxeter simplex tilings $\{p,3,4\}$ $(7\le p \in \mathbb{N})$ and $\{p,4,3\}$ $(5\le p \in \mathbb{N})$ in $3$-dimensional hyperbolic space $\mathbb{H}^3$. We determine the densest hype...
Article
Full-text available
Previous discoveries of the first author (1984-88) on so-called hyperbolic football manifolds and our recent works (2016-17) on locally extremal ball packing and covering hyperbolic space $\HYP$ with congruent balls had led us to the idea that our "experience space in small size" could be of hyperbolic structure. In this paper we construct an infin...
Article
Full-text available
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 plan...
Article
Full-text available
We determine the optimal horoball packings of the asymptotic or Koszul-type Coxeter simplex tilings of hyperbolic 5-space, where the symmetries of the packings are derived from Coxeter groups. The packing density \(\varTheta = \frac{5}{7 \zeta (3)} \approx 0.5942196502\ldots \) is optimal and realized in eleven cases in a commensurability class of...
Preprint
Full-text available
In our Novi Sad conference paper (1999) we described Dehn type surgeries of the famous Gieseking (1912) hyperbolic ideal simplex manifold $\mathcal{S}$, leading to compact fundamental domain $\mathcal{S}(k)$, $k = 2, 3, \dots$ with singularity geodesics of rotation order $k$, but as later turned out with cone angle $2(k-1)/k$. We computed also the...
Preprint
Full-text available
Supergroups of some hyperbolic space groups are classified as a continuation of our former works. Fundamental domains will be integer parts of truncated tetrahedra belonging to families F1 - F4, for a while, by the notation of E. Moln\'{a}r et al. in $2006$. As an application, optimal congruent hyperball packings and coverings to the truncation bas...
Preprint
Full-text available
After having investigated the packings derived by horo- and hyperballs related to simple frustum Coxeter orthoscheme tilings we consider the corresponding covering problems (briefly hyp-hor coverings) in $n$-dimensional hyperbolic spaces $\mathbb{H}^n$ ($n=2,3$). We construct in the $2-$ and $3-$dimensional hyperbolic spaces hyp-hor coverings that...
Article
Full-text available
In [17] we considered hyperball packings in 3-dimensional hyperbolic space. We developed a decomposition algorithm that for each saturated hyperball packing has provided a decomposition of ℍ ³ into truncated tetrahedra. Thus, in order to get a density upper bound for hyperball packings, it is sufficient to determine the density upper bound of hyper...
Preprint
Koszul type Coxeter Simplex tilings exist in hyperbolic space $\mathbb{H}^n$ for $2 \leq n \leq 9$, and their horoball packings have the highest known regular ball packing densities for $3 \leq n \leq 5$. In this paper we determine the optimal horoball packings of Koszul type Coxeter simplex tilings of $n$-dimensional hyperbolic space for $6 \leq n...
Preprint
Full-text available
In the present paper we study $S^2 \times R$ and $H^2 \times R$ geometries, which are homogeneous Thurston 3-geometries. We analyse the interior angle sums of geodesic triangles in both geometries and prove, that in $S^2 \times R$ space it can be larger or equal than $\pi$ and in $H^2 \times R$ space the angle sums can be less or equal than $\pi$....
Article
Full-text available
In this paper we study the problem of hyperball (hypersphere) packings in $3$-dimensional hyperbolic space. We introduce a new definition of the non-compact saturated ball packings and describe to each saturated hyperball packing, a new procedure to get a decomposition of 3-dimensional hyperbolic space $\HYP$ into truncated tetrahedra. Therefore, i...
Article
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In the present paper we study the $\SOL$ geometry that is one of the eight homogeneous Thurston 3-geomet\-ri\-es. We determine the equation of the translation-like bisector surface of any two points. We prove, that the isosceles property of a translation triangle is not equivalent to two angles of the triangle being equal and that the triangle ineq...
Article
Full-text available
In this paper we study the Nil geometry that is one of the eight homogeneous Thurston 3-geometries. We determine the equation of the translation-like bisector surface of any two points. We prove, that the isosceles property of a translation triangle is not equivalent to two angles of the triangle being equal and that the triangle inequalities do no...
Preprint
Full-text available
In \cite{Sz17-2} we proved that to each saturated congruent hyperball packing exists a decomposition of $3$-dimensional hyperbolic space $\mathbb{H}^3$ into truncated tetrahedra. Therefore, in order to get a density upper bound for hyperball packings, it is sufficient to determine the density upper bound of hyperball packings in truncated simplices...
Article
Full-text available
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 \pi.
Preprint
Full-text available
In \cite{Sz17-2} we considered hyperball packings in $3$-dimensional hyperbolic space. We developed a decomposition algorithm that for each saturated hyperball packing provides a decomposition of $\HYP$ into truncated tetrahedra. In order to get a density upper bound for hyperball packings, it is sufficient to determine the density upper bound of h...
Preprint
Full-text available
We describe the optimal horoball packings of asymptotic Koszul type Coxeter simplex tilings of $5$-dimensional hyperbolic space where the symmetries of the packings are generated by Coxeter groups. We find that the optimal horoball packing density of $\delta_{opt}=0.59421\dots$ is realized in an entire commensurability class of arithmetic Coxeter t...
Chapter
In this paper we study a family of compact 3-dimensional manifolds, i.e. space forms - more popularly, finite worlds - that are derived from famous Euclidean and non-Euclidean polyhedral tilings by the unified method of face identification, i.e. logical gluings. All these seem to have application in modern crystallography, as fullerenes and nanotub...
Article
Full-text available
In the earlier works \cite{Sz06-1}, \cite{Sz06-2}, \cite{Sz13-3} and \cite{Sz13-4} we have investigated the the densest packings and the least dense coverings by congruent hyperballs (hyperspheres) to the regular prism tilings in the $n$-dimensional hyperbolic space $\HYN$ ($n \in \mathbb{N},~n \ge 3)$. In this paper we study the problem of hyperba...
Article
Full-text available
In $n$-dimensional hyperbolic space $\mathbf{H}^n$ $(n\ge2)$ there are $3$-types of spheres (balls): the sphere, horosphere and hypersphere. If $n=2,3$ we know an universal upper bound of the ball packing densities, where each ball volume is related to the volume of the corresponding Dirichlet-Voronoi (D-V) cell. E.g. in $\mathbf{H}^3$ the densest...
Article
Full-text available
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...
Poster
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In this poster we briefly summarize some of our results related to Thurston geometries.
Article
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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 $\pi$. We use for the computations the projective model of $\NIL$ introduced by E. {Moln\'ar} in \cite{M97}.
Article
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The motivation for this talk and paper is related to the classification of the homogeneous simply connected maximal 3-geometries (the so-called Thurston geometries: E ³ , S ³ , H ³ , S ² ×R, H ² ×R, SL g2 R, Nil, and Sol) and their applications in crystallography. The first author found in (Molnár 1997) (see also the more popular (Molnár et al. 201...
Article
Full-text available
In this paper we study the Nil geometry that is one of the eight homogeneous Thurston 3-geomet\-ri\-es. We determine the equation of the translation-like bisector surface of any two points. We prove, that the isosceles property of a translation triangle is not equivalent to two angles of the triangle being equal and that the triangle inequalities d...
Article
Full-text available
We generalize the Simson-Wallace locus in d-dimensional projective metric space, i.e. we look for the points whose orthogonal projections onto the hyperplanes of a fixed d-simplex lie on a hyperplane ((d−1)-plane). We show that this Simson–Wallace locus \({\mathcal{SW}}\) is a (hyper)surface of d + 1 degree, if the metric hyperplane to point polari...
Article
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...
Article
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 degre...
Article
Full-text available
After having investigated the geodesic and translation triangles and their angle sums in $\NIL$ and $\SLR$ 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 translation triangles in $\SOL$ geometry and prove that it can be larger or equal th...
Article
Full-text available
This presentation starts with the regular polygons, of course, then with the Platonic and Archimedean solids. The latter ones are whose symmetry groups are transitive on the vertices, and in addition, whose faces are regular polygons (see only I. Prok's home page [11] for them). Then there come these symmetry groups themselves (starting with the cu...
Article
Full-text available
A compact hyperbolic "cobweb" manifold (hyperbolic space form) of symbol $Cw(6,6,6)$ will be constructed in Fig.1,4,5 as a representant of a presumably infinite series $Cw(2p,2p,2p)$ $(3 \le p \in \bN$ natural numbers). This is a by-product of our investigations \cite{MSz16}. In that work dense ball packings and coverings of hyperbolic space $\HYP$...
Article
In this paper, we study regular prism tilings and corresponding least dense hyperball coverings in (Formula presented.)-dimensional hyperbolic space (Formula presented.)(Formula presented.) by congruent hyperballs. We determine the densities of the least dense hyperball coverings, we formulate two conjectures for the candidates of the least dense h...
Article
We construct ball packings of the universal cover of \({{\rm{SL}}_{2}({\mathbb{R}})}\) by geodesic balls and translation balls. The packings are generated by action of the prism groups \({{\mathbf{pq}}_{k}{\mathbf{o}}_{\ell}}\). We obtain volume formulae for calculations in geographical coordinates. Using these formulae we find numerically the maxi...
Article
Full-text available
Four packings of hyperbolic 3-space are known to yield the optimal packing density of $0.85328\dots$. They are realized in the regular tetrahedral and cubic Coxeter honeycombs with Schl\"afli symbols $\{3,3,6 \}$ and $\{4,3,6\}$. These honeycombs are totally asymptotic, and the packings consist of horoballs (of different types) centered at the idea...
Article
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...
Article
In this paper we study the locally optimal geodesic ball packings with equal balls to the S 2 × 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. Th...
Article
We study a series of 2-generator Sol-manifolds depending on a positive integer n, introduced by Molnár and Szirmai. We construct them as tetrahedron manifolds and show that they are twofold coverings of the 3-sphere branched over specified links. Finally, we give a surgery description of the considered 3-manifolds; indeed, they can be obtained by n...
Article
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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, Macmillian, New York, 1964). We give seven examples of horoball packing configurations that yield higher densities of 0.71644896…...
Article
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For one of Thurston model spaces, ${\widetilde{{\rm SL}_2({\mathbb{R}})}}$ , we discuss translation balls and packing that space by such balls in contrast to the packing by standard (geodesic) balls. We present an infinite family of packings generated by discrete groups of isometries, and observe numerical results on their densities. In particul...
Article
Full-text available
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 optim...
Article
Full-text available
After having investigated the regular prisms and prism tilings in the $\SLR$ space in the previous work \cite{Sz13-1} of the second author, we consider the problem of geodesic ball packings related to those tilings and their symmetry groups $\mathbf{pq2_1}$. $\SLR$ is one of the eight Thurston geometries that can be derived from the 3-dimensional L...
Article
Full-text available
In \cite{Sz13-1} we defined and described the {\it regular infinite or bounded} $p$-gonal prism tilings in $\SLR$ space. We proved that there exist infinitely 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 prism tilings $\cT_p(q)$ for integer parameters $p,q;~3...
Article
Full-text available
After having investigated the densest packings by congruent hyperballs to the regular prism tilings in the $n$-dimensional hyperbolic space $\mathbb{H}^n$ ($n \in \mathbb{N}, n \ge 3)$ we consider the dual covering problems and determine the least dense hyperball arrangements and their densities.
Article
\({{\widetilde{\bf SL_{2}R}}}\) geometry is one of the eight 3-dimensional Thurston geometries, it can be derived from the 3-dimensional Lie group of all 2 × 2 real matrices with determinant one. Our aim is to describe and visualize the regular infinite or bounded p-gonal prism tilings in \({{\widetilde{\bf SL_{2}R}}}\). For this purpose we introdu...
Article
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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...
Article
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$\NIL$ geometry is one of the eight 3-dimensional Thurston geometries, it can be derived from {W. Heisenberg}'s famous real matrix group. The aim of this paper is to study {\it lattice-like ball coverings} in $\NIL$ space. We introduce the notion of the density of the considered coverings and give upper and lower estimates to it, moreover in Secti...
Article
Full-text available
In Odehnal (2011) packings with geometric sequences of spheres to regular polyhedra in the Euclidean d-space with d ≥ 2 have been introduced. The packing densities depending on the dimension and the type of polyhedron have been determined. Some of these packings can also be performed in d-dimensional Euclidean space. In this paper we generalize the...
Article
Full-text available
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{C...
Article
Full-text available
The ball (or sphere) packing problem with equal balls, without any symmetry assumption, in a \(3\)-dimensional space of constant curvature was settled by Böröczky and Florian for the hyperbolic space \(\mathbf H ^3\), and, with the proof of the famous Kepler conjecture, by Hales for the Euclidean space \(\mathbf E ^3\). The goal of this paper is to...
Article
Full-text available
The $\mathbf{S}^2\!\times \!\mathbf{R}$ geometry can be derived by the direct product of the spherical plane $\mathbf{S}^2$ and the real line $\mathbf{R}$ . In (Beiträge zur Algebra und Geometrie (Contributions to Algebra and Geometry) 42:235–250, 2001), Farkas has classified and given the complete list of the space groups of $\mathbf{S}^2\...
Article
Nil geometry is one of the eight homogeneous Thurston 3-geometries: E-3, S-3, H-3, S-2 x R, H-2 x R, SL2R, Nil, Sol. Nil can be derived from W. HEISENBERG'S famous real matrix group. The notion of translation curve and translation ball can be introduced by initiative of E. MOLNAR (see [MS], [MSz], [Sz10]). P. SCOTT in [S] defined Nil lattices to wh...
Article
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W. Thurston classified the eight simply connected three-dimensional maximal homogeneous Riemannian geometries (see Thurston and Levy 1997, Scott 1983). One of these is the S 2 × R geometry which is the direct product of the spherical plane S 2 and the real line R. The complete list of the space groups of S 2 × R is given by Farkas (Beitr Algebra Ge...
Article
Full-text available
The S 2×R geometry can be derived by the direct product of the spherical plane S 2 and the real line R. In [1] J. Z. Farkas has classified and given the complete list its space groups. In [6] the second author has studied the geodesic balls and their volumes in S 2×R space, moreover he has introduced the notion of geodesic ball packing and its dens...
Article
In this paper we study the isoptic curves on the hyperbolic plane. This topic is widely investigated in the Euclidean geometry, but in the hyperbolic geometry there are only a few result. In [13] we have developed a method to investigate the isoptic curves in the hyperbolic geometry and we have applied it to line segments and ellipses.Our goal in t...
Article
The parallelohedron is one of basic concepts in the Euclidean geometry and in the 3-dimensional crystallography, has been introduced by the crystallographer E.S. Fedorov (1889). The 3-dimensional parallelohedron can be defined as a convex 3-dimensional polyhedron whose parallel copies tile the 3-dimensional Euclidean space in a face to face manner....
Article
Full-text available
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 loca...
Article
Full-text available
Sol geometry is one of the eight homogeneous Thurston 3-geometries $${\bf E}^{3}, {\bf S}^{3}, {\bf H}^{3}, {\bf S}^{2}\times{\bf R}, {\bf H}^{2}\times{\bf R}, \widetilde{{\bf SL}_{2}{\bf R}}, {\bf Nil}, {\bf Sol}.$$In [13] the densest lattice-like translation ball packings to a type (type I/1 in this paper) of Sol lattices has been determined. Som...
Article
Full-text available
The aim of this paper is 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 \(\overline{\mathbf{H}}^{3}\) extended with its absolute figure, where the ideal centers of horoballs give rise to vertices of a fully asymp...
Article
Full-text available
Applying d-dimensional projective spherical geometry PS d (R, V d+1 , V d+1), represented by the standard real (d + 1)-vector space and its dual up to positive real factors as ∼ equivalence, the Grassmann algebra of V d+1 and of V d+1 , respectively, represent the subspace structure of PS d and of P d . Then the central projection from a (d − 3)-ce...
Article
The Nil geometry, which is one of the eight 3-dimensional Thurston geometries, can be derived from W. Heisenberg’s famous real matrix group. Our goal in this work to develop a numerical procedure to study the equidistant surfaces in the Nil geometry and apply it to contract Dirichlet-Voronoi cells. For the computations, we use the projective model...
Article
Full-text available
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...
Article
Full-text available
There are 8 maximal homogeneous simply connected Riemannian 3-dimensional spaces: Sol R , Nil , L R , S R , H , S H , S , E 2 2 2 3 3 3 ~ × × for which there exist isometric symmetries and a group of isometries with discrete orbits, in addition with compact fundamental domain. So there arise similar problems to those of Euclidean crystallography. A...
Article
In the eight homogeneous Thurston 3-geometries – E 3 , S 3 , H 3 , S 2 ×ℝ, H 2 ×ℝ, SL ˜ 2 ℝ, Nil , Sol – the notions of translation curves and translation balls can be introduced in a unified way by initiative of E. Molnár (see [A. Bölcskei and B. Szilágyi, KoG 10, 27–32 (2006; Zbl 1147.53015); E. Molnár and B. Szilágyi, Publ. Math. 78, No. 2, 327–...
Article
Full-text available
Using projective metric geometry we develop a technique to describe homogeneous 3-dimensional metrics on cone-manifolds generated by two rotations. In particular, for some cone-manifolds with singularities along 2-bridge knots and links we give explicit descriptions of all possible geometries ( \mathbb S3{\mathbb S^3}, [(SL2(\mathbb R))\tilde]{\...
Article
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The trigonal Wankel engine is kinematically based on the motion where a circle pm of radius 3d, as the moving pole curve, rolles on the circle ps of radius 2d, as the standing pole curve in the interior. Then the regular trigonal rotor with circumcircle of radius ρ > 3d, fixed concentrically to the moving pole circle, describes its orbit curve c ρ...
Article
We investigate d-dimensional hyperbolic regular honeycombs, which are constructed by a class of “complete Coxeter orthoschemes”. Namely, we consider every tiling in the d-dimensional hyperbolic space ℍ d , d≥4, where a horosphere is inscribed in each regular polyhedron – so it is infinitely centred – moreover, its vertices are proper points or lie...
Article
We construct a very simple orientable 3-manifold with one cusp (Fig. 3) which cannot be geometrized in any Thurston geometry. It turns out in Theorem 4.1 that our manifold can be splitted into a Euclidean piece (Fig. 4) and an H 2 ×ℝ one (Fig. 5 and 6). In E. Molnár, J. Szirmai and J. R. Weeks [3-simplex tilings, splitting orbifolds and manifolds,...
Article
W. Heisenbergs famous real matrix group provides a non-commutative translation group of an affine 3-space. The Nil geometry which is one of the eight homogeneous Thurston 3-geometries, can be derived from this matrix group. E. Molnár proved in [2], that the homogeneous 3-spaces have a unified interpretation in the projective 3-sphere PS3(V4, V 4, ℝ...
Article
In a former paper [18] a method is described that determines the data and the density of the optimal ball or horoball packing to each Coxeter tiling in the hyperbolic 3-space. In this work we extend this procedure - based on the projective interpretation of the hyperbolic geometry - to higher dimensional Coxeter honeycombs in ℍd, (d = 4, 5), and de...
Article
We investigate the regular p-gonal prism tilings (mosaics) in the hyperbolic 3-space that were classified by I. Vermes in [12]and [13]. The optimal hyperball packings of these tilings are generated by the ``inscribed hyperspheres'' whose metric data can be calculated by our method -- based on the projective interpretation of the hyperbolic geometry...
Chapter
This is a survey on the tilings (T, Γ) in the title where the vertex stabilizers in are finite spherical S2 or infinite Euclidean E2 (cocompact) plane groups. The results are collected in figures and tables and illustrated by an infinite family series Family 30 in Section 4. The obtained orbifolds, maybe after splitting procedure, are realized in s...
Article
In this paper we look for densest ball packings of Euclidean space IE 3 to given symmetry groups. We restrict our investigation to the 36 space groups of the cubic crystal system, and we search for only those packings where the group acts simply transitively on the balls. In order to find the centre of a ball and its radius for the case of an optim...
Article
In this paper I describe a method - based on the projective interpre- tation of the hyperbolic geometry - that determines the data and the density of the optimal ball and horoball packings of each well-known Coxeter tiling (Coxeter honeycomb) in the hyperbolic space H 3 .