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The Voronoi Cell in a saturated Circle Packing and an elementary proof of Thue´s theorem 饱和圆包中的沃罗诺伊单元和图伊定理的基本证明
arXiv: History and Overview Pub Date : 2019-05-13 DOI: 10.15495/EPUB_UBT_00004374
Max Leppmeier
{"title":"The Voronoi Cell in a saturated Circle Packing and an elementary proof of Thue´s theorem","authors":"Max Leppmeier","doi":"10.15495/EPUB_UBT_00004374","DOIUrl":"https://doi.org/10.15495/EPUB_UBT_00004374","url":null,"abstract":"The famous Kepler conjecture has a less spectacular, two-dimensional equivalent: The theorem of Thue states that the densest circle packing in the Euclidean plane has a hexagonal structure. A common proof uses Voronoi cells and analyzes their area applying Jensen's inequality on convex functions to receive a local estimate which is globally valid. Based on the concept of Voronoi cells, we will introduce a new tessellation into so-called L-triangles which can be related to fundamental parallelograms of lattice circle packings. Therefore a globally disordered circle packing can be reduced to locally ordered configurations: We will show how the theorem of Lagrange on lattice circle packings can be applied to non-lattice circle packings. Thus we receive a new proof of Thue's theorem.","PeriodicalId":429168,"journal":{"name":"arXiv: History and Overview","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130104435","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Richard Dedekind y la arquitectura del continuo aritmético Richard Dedekind和算术连续体的架构
arXiv: History and Overview Pub Date : 2019-05-04 DOI: 10.47976/RBHM2013V13N2777-109
Luis Giraldo Gonz'alez Ricardo, Carlos S'anchez Fern'andez
{"title":"Richard Dedekind y la arquitectura del continuo aritmético","authors":"Luis Giraldo Gonz'alez Ricardo, Carlos S'anchez Fern'andez","doi":"10.47976/RBHM2013V13N2777-109","DOIUrl":"https://doi.org/10.47976/RBHM2013V13N2777-109","url":null,"abstract":"It is usually considered that the structuralist tendency in mathematics began in the twentieth century, at some point after the works on set theory and obtained its spreading through the works made by the Bourbaki group. In the present paper we argument the presence of this stylistic inclination in Richard Dedekind (1831-1916) when he made this dissertation for the habilitation as privatdozent at the University of Goettingen in 1854. Our main objective is to show how evolved the structuralist style in Dedekind's works, and to argument why we consider him significant to the architecture of arithmetical continuum.","PeriodicalId":429168,"journal":{"name":"arXiv: History and Overview","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125710904","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Model Theory, Arithmetic & Algebraic Geometry 模型理论、算术与代数几何
arXiv: History and Overview Pub Date : 2019-04-28 DOI: 10.13140/RG.2.2.17942.40009
Joel Torres del Valle
{"title":"Model Theory, Arithmetic & Algebraic Geometry","authors":"Joel Torres del Valle","doi":"10.13140/RG.2.2.17942.40009","DOIUrl":"https://doi.org/10.13140/RG.2.2.17942.40009","url":null,"abstract":"In this pages I give an overview of the relationship between Model Theory, Arithmetic and Algebraic Geometry. The topics will be the basic ones in the area, so this is just an invitation, in the presentation of topics I mainly follow the philosophy of the references.","PeriodicalId":429168,"journal":{"name":"arXiv: History and Overview","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116013190","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The Dodecahedron as a Voronoi Cell - and its (minor) importance for the Kepler conjecture 作为Voronoi细胞的十二面体——及其对开普勒猜想的(次要的)重要性
arXiv: History and Overview Pub Date : 2019-04-11 DOI: 10.15495/EPUB_UBT_00004313
Max Leppmeier
{"title":"The Dodecahedron as a Voronoi Cell - and its (minor) importance for the Kepler conjecture","authors":"Max Leppmeier","doi":"10.15495/EPUB_UBT_00004313","DOIUrl":"https://doi.org/10.15495/EPUB_UBT_00004313","url":null,"abstract":"The regular dodecahedron has a 2% smaller volume than the rhombic dodecahedron which is the Voronoi cell of a fcc packing. From this point of view it seems possible that the dodecahedral aspect which is the core of the so-called dodecahedral conjecture, will play a major part for an elementary proof of the Kepler conjecture. In this paper we will show that the icosahedral configuration caused by dodecahedron leads to tetrahedra with significantly larger volume than the fcc fundamental parallelotope tessellation tetrahedra. Therefore on the basis of a tetrahedral based point of view for sphere packing densities we will demonstrate the minor importance of the dodecahedron as a Voronoi cell for the Kepler conjecture.","PeriodicalId":429168,"journal":{"name":"arXiv: History and Overview","volume":"149 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134127758","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Matemáticas, espacios públicos e integración vecinal. El caso de Cuernavaca (México) 数学、公共空间和社区融合。库埃纳瓦卡案例(墨西哥)
arXiv: History and Overview Pub Date : 2018-11-23 DOI: 10.22323/3.03020210
Igor Barahona, Lucía López de Medrano, Barbara Mart'inez Moreno, Beatríz Limón Gutiérrez
{"title":"Matemáticas, espacios públicos e integración vecinal. El caso de Cuernavaca (México)","authors":"Igor Barahona, Lucía López de Medrano, Barbara Mart'inez Moreno, Beatríz Limón Gutiérrez","doi":"10.22323/3.03020210","DOIUrl":"https://doi.org/10.22323/3.03020210","url":null,"abstract":"We investigate the impact of mathematics on improving neighbourhood integration and perception of security. The main square of Chamilpa colony in Cuernavaca, M'exico is take as study case. This city is featured by its precarious recreational and leisurial infrastructure. Chamilpa is among the top of social outcast levels in the city. Data was collected through a questionnaire applied among attendees of the ARTEMAT festival. Results provide empirical evidence for supporting that performing mathematics activities on public spaces, increases the perception of security and improve the social cohesion.","PeriodicalId":429168,"journal":{"name":"arXiv: History and Overview","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123519485","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Chinese Names for Integers 整数的中文名称
arXiv: History and Overview Pub Date : 2018-10-26 DOI: 10.1007/978-3-319-10040-1_7
R'emi Anicotte
{"title":"Chinese Names for Integers","authors":"R'emi Anicotte","doi":"10.1007/978-3-319-10040-1_7","DOIUrl":"https://doi.org/10.1007/978-3-319-10040-1_7","url":null,"abstract":"","PeriodicalId":429168,"journal":{"name":"arXiv: History and Overview","volume":"80 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123057566","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 3
Elementare Zahlentheorie. 小学Zahlentheorie。
arXiv: History and Overview Pub Date : 2018-09-24 DOI: 10.1007/978-3-7643-7731-1
Yauheniya Abramchuk, Alina Bondarava, Matthias Kunik
{"title":"Elementare Zahlentheorie.","authors":"Yauheniya Abramchuk, Alina Bondarava, Matthias Kunik","doi":"10.1007/978-3-7643-7731-1","DOIUrl":"https://doi.org/10.1007/978-3-7643-7731-1","url":null,"abstract":"","PeriodicalId":429168,"journal":{"name":"arXiv: History and Overview","volume":"47 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115877926","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Mollweide's formula in teaching trigonometry Mollweide公式在三角学教学中的应用
arXiv: History and Overview Pub Date : 2018-08-24 DOI: 10.1039/teamat/hrr008
Natanael Karjanto
{"title":"Mollweide's formula in teaching trigonometry","authors":"Natanael Karjanto","doi":"10.1039/teamat/hrr008","DOIUrl":"https://doi.org/10.1039/teamat/hrr008","url":null,"abstract":"Trigonometry is one of the topics in mathematics that the students in both high school and pre-undergraduate levels need to learn. Generally, the topic covers trigonometric functions, trigonometric equations, trigonometric identities and solving oblique triangles using the Laws of Sines and Cosines. However, when solving the oblique triangles, Mollweide's formula is most likely to be omitted from the discussion. Mollweide's formula--which exhibits a cyclical nature--is particularly useful in checking one's result after solving an oblique triangle since all six components of the triangle are involved. It is interesting to note that proving Mollweide's formula can be performed without words. Furthermore, the Law of Tangents can be derived straightforwardly from this equation. In this article, we revisit Mollweide's formula and provide classroom examples where this equation comes into alive. In addition, we suggest that this seemingly less-known equation is to be included in the mathematics syllabus on the topic of Trigonometry.","PeriodicalId":429168,"journal":{"name":"arXiv: History and Overview","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131901930","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Geometry and Physics: An Overview 几何和物理:概述
arXiv: History and Overview Pub Date : 2018-05-17 DOI: 10.1090/CONM/718/14490
D. Morrison
{"title":"Geometry and Physics: An Overview","authors":"D. Morrison","doi":"10.1090/CONM/718/14490","DOIUrl":"https://doi.org/10.1090/CONM/718/14490","url":null,"abstract":"We present some episodes from the history of interactions between geometry and physics over the past century.","PeriodicalId":429168,"journal":{"name":"arXiv: History and Overview","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132065780","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Bolzano’s measurable numbers: are they real? 博尔扎诺的可测量数字:它们是真实的吗?
arXiv: History and Overview Pub Date : 2018-05-06 DOI: 10.1007/978-3-319-46615-6_4
S. Russ, Katevrina Trlifajov'a
{"title":"Bolzano’s measurable numbers: are they real?","authors":"S. Russ, Katevrina Trlifajov'a","doi":"10.1007/978-3-319-46615-6_4","DOIUrl":"https://doi.org/10.1007/978-3-319-46615-6_4","url":null,"abstract":"","PeriodicalId":429168,"journal":{"name":"arXiv: History and Overview","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133388621","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 7
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