{"title":"TILINGS OF THE SPHERE BY CONGRUENT QUADRILATERALS II: EDGE COMBINATION WITH RATIONAL ANGLES","authors":"Yixi Liao, Erxiao Wang","doi":"10.1017/nmj.2023.20","DOIUrl":null,"url":null,"abstract":"\n Edge-to-edge tilings of the sphere by congruent quadrilaterals are completely classified in a series of three papers. This second one applies the powerful tool of trigonometric Diophantine equations to classify the case of \n \n \n \n$a^3b$\n\n \n -quadrilaterals with all angles being rational degrees. There are \n \n \n \n$12$\n\n \n sporadic and \n \n \n \n$3$\n\n \n infinite sequences of quadrilaterals admitting the two-layer earth map tilings together with their modifications, and \n \n \n \n$3$\n\n \n sporadic quadrilaterals admitting \n \n \n \n$4$\n\n \n exceptional tilings. Among them only three quadrilaterals are convex. New interesting non-edge-to-edge triangular tilings are obtained as a byproduct.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":"1 1","pages":""},"PeriodicalIF":0.8000,"publicationDate":"2022-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Nagoya Mathematical Journal","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/nmj.2023.20","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 3
Abstract
Edge-to-edge tilings of the sphere by congruent quadrilaterals are completely classified in a series of three papers. This second one applies the powerful tool of trigonometric Diophantine equations to classify the case of
$a^3b$
-quadrilaterals with all angles being rational degrees. There are
$12$
sporadic and
$3$
infinite sequences of quadrilaterals admitting the two-layer earth map tilings together with their modifications, and
$3$
sporadic quadrilaterals admitting
$4$
exceptional tilings. Among them only three quadrilaterals are convex. New interesting non-edge-to-edge triangular tilings are obtained as a byproduct.
期刊介绍：
The Nagoya Mathematical Journal is published quarterly. Since its formation in 1950 by a group led by Tadashi Nakayama, the journal has endeavoured to publish original research papers of the highest quality and of general interest, covering a broad range of pure mathematics. The journal is owned by Foundation Nagoya Mathematical Journal, which uses the proceeds from the journal to support mathematics worldwide.
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