{"title":"New lower bound for the optimal congruent geodesic ball packing density of screw motion groups in $\\mathbf{H}^2\\!\\times\\!\\mathbf{R}$ space","authors":"Arnasli Yahya, Jenő Szirmai","doi":"arxiv-2407.21251","DOIUrl":null,"url":null,"abstract":"In this paper, we present a new record for the densest geodesic congruent\nball packing configurations in $\\mathbf{H}^2\\!\\times\\!\\mathbf{R}$ geometry,\ngenerated by screw motion groups. These groups are derived from the direct\nproduct of rotational groups on $\\mathbf{H}^2$ and some translation components\non the real fibre direction $\\mathbf{R}$ that can be determined by the\ncorresponding Frobenius congruences. Moreover, we developed a procedure to\ndetermine the optimal radius for the densest geodesic ball packing\nconfigurations related to the considered screw motion groups. The highest\npacking density, $\\approx0.80529$, is achieved by a multi-transitive case given\nby rotational parameters $(2,20,4)$. E. Moln\\'{a}r demonstrated that\nhomogeneous 3-spaces can be uniformly interpreted in the projective 3-sphere\n$\\mathcal{PS}^3(\\mathbf{V}^4, \\boldsymbol{V}_4, \\mathbf{R})$. We use this\nprojective model of $\\mathbf{H}^2\\!\\times\\!\\mathbf{R}$ to compute and visualize\nthe locally optimal geodesic ball arrangements.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Metric Geometry","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2407.21251","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
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. Moln\'{a}r 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.