New lower bound for the optimal congruent geodesic ball packing density of screw motion groups in $\mathbf{H}^2\!\times\!\mathbf{R}$ space
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866911003891990528 |
|---|---|
| author | Yahya, Arnasli Szirmai, Jenő |
| author_facet | Yahya, Arnasli Szirmai, Jenő |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_21251 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | New lower bound for the optimal congruent geodesic ball packing density of screw motion groups in $\mathbf{H}^2\!\times\!\mathbf{R}$ space Yahya, Arnasli Szirmai, Jenő Metric Geometry 52C17, 52C22, 53A35, 51M20 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. |
| title | New lower bound for the optimal congruent geodesic ball packing density of screw motion groups in $\mathbf{H}^2\!\times\!\mathbf{R}$ space |
| topic | Metric Geometry 52C17, 52C22, 53A35, 51M20 |
| url | https://arxiv.org/abs/2407.21251 |