Partial Covering of a Circle by 6 and 7 Congruent Circles

Background: Some medical and technological tasks lead to the geometrical problem of how to cover the unit circle as much as possible by n congruent circles of given radius r, while r varies from the radius in the maximum packing to the radius...

Descripción completa

Guardado en:

Detalles Bibliográficos
Autores principales:	Zsolt Gáspár, Tibor Tarnai, Krisztián Hincz
Formato:	article
Lenguaje:	EN
Publicado:	MDPI AG 2021
Materias:	packing of equal circles covering by equal circles partial covering tensegrity optimization equilibrium paths Mathematics QA1-939
Acceso en línea:	https://doaj.org/article/edbee9e084774953989571de9d919abb
Etiquetas:	Agregar Etiqueta Sin Etiquetas, Sea el primero en etiquetar este registro!

id	oai:doaj.org-article:edbee9e084774953989571de9d919abb
record_format	dspace
spelling	oai:doaj.org-article:edbee9e084774953989571de9d919abb2021-11-25T19:06:59ZPartial Covering of a Circle by 6 and 7 Congruent Circles10.3390/sym131121332073-8994https://doaj.org/article/edbee9e084774953989571de9d919abb2021-11-01T00:00:00Zhttps://www.mdpi.com/2073-8994/13/11/2133https://doaj.org/toc/2073-8994Background: Some medical and technological tasks lead to the geometrical problem of how to cover the unit circle as much as possible by <i>n</i> congruent circles of given radius <i>r</i>, while <i>r</i> varies from the radius in the maximum packing to the radius in the minimum covering. Proven or conjectural solutions to this partial covering problem are known only for <i>n</i> = 2 to 5. In the present paper, numerical solutions are given to this problem for <i>n</i> = 6 and 7. Method: The method used transforms the geometrical problem to a mechanical one, where the solution to the geometrical problem is obtained by finding the self-stress positions of a generalised tensegrity structure. This method was developed by the authors and was published in an earlier publication. Results: The method applied results in locally optimal circle arrangements. The numerical data for the special circle arrangements are presented in a tabular form, and in drawings of the arrangements. Conclusion: It was found that the case of <i>n</i> = 6 is very complicated, whilst the case <i>n</i> = 7 is very simple. It is shown in this paper that locally optimal arrangements may exhibit different types of symmetry, and equilibrium paths may bifurcate.Zsolt GáspárTibor TarnaiKrisztián HinczMDPI AGarticlepacking of equal circlescovering by equal circlespartial coveringtensegrityoptimizationequilibrium pathsMathematicsQA1-939ENSymmetry, Vol 13, Iss 2133, p 2133 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	packing of equal circles covering by equal circles partial covering tensegrity optimization equilibrium paths Mathematics QA1-939
spellingShingle	packing of equal circles covering by equal circles partial covering tensegrity optimization equilibrium paths Mathematics QA1-939 Zsolt Gáspár Tibor Tarnai Krisztián Hincz Partial Covering of a Circle by 6 and 7 Congruent Circles
description	Background: Some medical and technological tasks lead to the geometrical problem of how to cover the unit circle as much as possible by <i>n</i> congruent circles of given radius <i>r</i>, while <i>r</i> varies from the radius in the maximum packing to the radius in the minimum covering. Proven or conjectural solutions to this partial covering problem are known only for <i>n</i> = 2 to 5. In the present paper, numerical solutions are given to this problem for <i>n</i> = 6 and 7. Method: The method used transforms the geometrical problem to a mechanical one, where the solution to the geometrical problem is obtained by finding the self-stress positions of a generalised tensegrity structure. This method was developed by the authors and was published in an earlier publication. Results: The method applied results in locally optimal circle arrangements. The numerical data for the special circle arrangements are presented in a tabular form, and in drawings of the arrangements. Conclusion: It was found that the case of <i>n</i> = 6 is very complicated, whilst the case <i>n</i> = 7 is very simple. It is shown in this paper that locally optimal arrangements may exhibit different types of symmetry, and equilibrium paths may bifurcate.
format	article
author	Zsolt Gáspár Tibor Tarnai Krisztián Hincz
author_facet	Zsolt Gáspár Tibor Tarnai Krisztián Hincz
author_sort	Zsolt Gáspár
title	Partial Covering of a Circle by 6 and 7 Congruent Circles
title_short	Partial Covering of a Circle by 6 and 7 Congruent Circles
title_full	Partial Covering of a Circle by 6 and 7 Congruent Circles
title_fullStr	Partial Covering of a Circle by 6 and 7 Congruent Circles
title_full_unstemmed	Partial Covering of a Circle by 6 and 7 Congruent Circles
title_sort	partial covering of a circle by 6 and 7 congruent circles
publisher	MDPI AG
publishDate	2021
url	https://doaj.org/article/edbee9e084774953989571de9d919abb
work_keys_str_mv	AT zsoltgaspar partialcoveringofacircleby6and7congruentcircles AT tibortarnai partialcoveringofacircleby6and7congruentcircles AT krisztianhincz partialcoveringofacircleby6and7congruentcircles
_version_	1718410314856267776

Partial Covering of a Circle by 6 and 7 Congruent Circles

Ejemplares similares