3,325 research outputs found
Density upper bound for congruent and non-congruent hyperball packings generated by truncated regular simplex tilings
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 and
in and -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.Comment: 24 pages, 5 figures. arXiv admin note: substantial text overlap with
arXiv:1505.03338, arXiv:1312.2328, arXiv:1405.024
On lattice coverings of Nil space by congruent geodesic balls
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 .
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.Comment: 23 pages, 7 figure
Packings with horo- and hyperballs generated by simple frustum orthoschemes
In this paper we deal with the packings derived by horo- and hyperballs
(briefly hyp-hor packings) in the -dimensional hyperbolic spaces \HYN
() which form a new class of the classical packing problems.
We construct in the and dimensional hyperbolic spaces hyp-hor
packings that are generated by complete Coxeter tilings of degree 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 () the density of the above
hyp-hor packings arbitrarily approximate the universal upper bound of the
hypercycle or horocycle packing density and in \HYP the
optimal configuration belongs to the Coxeter tiling with density
.
Moreover, we study the hyp-hor packings in truncated orthosche\-mes
(6< p < 7, ~ p\in \bR) whose density function is attained its maximum for a
parameter which lies in the interval and the densities for
parameters lying in this interval are larger that . That means
that these locally optimal hyp-hor configurations provide larger densities that
the B\"or\"oczky-Florian density upper bound for ball and
horoball packings but these hyp-hor packing configurations can not be extended
to the entirety of hyperbolic space .Comment: 27 pages, 9 figures. arXiv admin note: text overlap with
arXiv:1312.2328, arXiv:1405.024
The optimal hyperball packings related to the smallest compact arithmetic 5-orbifolds
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 ) in the five dimensional hyperbolic
space (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 . 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 .Comment: 15 pages, 4 figure
Triangle angle sums related to translation curves in \SOL geometry
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 than .
In our work we will use the projective model of \SOL described by E.
Moln\'ar in \cite{M97},Comment: 13 pages, 4 figure
Congruent and non-congruent hyperball packings related to doubly truncated Coxeter orthoschemes in hyperbolic -space
In \cite{Sz17-2} we considered hyperball packings in -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 hyperball packings in
truncated simplices. Therefore, in this paper we examine the doubly truncated
Coxeter orthoscheme tilings and the corresponding congruent and non-congruent
hyperball packings. We proved that related to the mentioned Coxeter tilings the
density of the densest congruent hyperball packing is that is
-- by our conjecture -- the upper bound density of the relating non-congruent
hyperball packings too.Comment: 24 pages, 6 figures. arXiv admin note: substantial text overlap with
arXiv:1803.0494
Horoball packings and their densities by generalized simplicial density function in the hyperbolic space
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
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
, and prove that, in this sense, {\it the well
known B\"or\"oczky--Florian density upper bound for "congruent horoball"
packings of does not remain valid to the fully asymptotic
tetrahedra.}
The density of this locally densest packing is , may be
surprisingly larger than the B\"or\"oczky--Florian density upper bound but our local ball arrangement seems not to have extension to the
whole hyperbolic space.Comment: 20 pages, 8 figure
Horoball packings related to hyperbolic cell
In this paper we study the horoball packings related to the hyperbolic 24
cell in the extended hyperbolic space 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
which is equal to the known greatest ball packing density in hyperbolic 4-space
given in \cite{KSz14}.Comment: 24 pages, 6 figures. arXiv admin note: text overlap with
arXiv:1401.608
Decomposition method related to saturated hyperball packings
In this paper we study the problem of hyperball (hypersphere) packings in
-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, 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.Comment: 13 pages, 3 figures. arXiv admin note: text overlap with
arXiv:1405.024
Geodesic ball packings generated by regular prism tilings in geometry
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
geometry. Packings are generated by action of the discrete prism
groups . We prove that these groups are realized by prism
tilings in space if 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 geometry. We get a dense (conjectured
locally densest) geodesic ball arrangement related to the parameters
where the kissing number of the packing is , similarly to the
densest lattice-like geodesic ball arrangement investigated by
the second author .Comment: arXiv admin note: text overlap with arXiv:1105.198
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