{"title":"凯勒猜想重访","authors":"P. Horák, Dongryul Kim","doi":"10.36890/iejg.984269","DOIUrl":null,"url":null,"abstract":"In 1930 Keller conjectured that each tiling of Rn by unit cubes contains a pair of cubes sharing a complete (n-1)-dimensional face. This conjecture was solved only 50 years later by Lagarias and Shor who found a counterexample for all n >= 10. In this paper we show that neither a modification of Keller's when the unit cube is substituted by a \ntile of more complex shape is true.","PeriodicalId":43768,"journal":{"name":"International Electronic Journal of Geometry","volume":" ","pages":""},"PeriodicalIF":0.4000,"publicationDate":"2022-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Keller's Conjecture Revisited\",\"authors\":\"P. Horák, Dongryul Kim\",\"doi\":\"10.36890/iejg.984269\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In 1930 Keller conjectured that each tiling of Rn by unit cubes contains a pair of cubes sharing a complete (n-1)-dimensional face. This conjecture was solved only 50 years later by Lagarias and Shor who found a counterexample for all n >= 10. In this paper we show that neither a modification of Keller's when the unit cube is substituted by a \\ntile of more complex shape is true.\",\"PeriodicalId\":43768,\"journal\":{\"name\":\"International Electronic Journal of Geometry\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2022-05-20\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"International Electronic Journal of Geometry\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.36890/iejg.984269\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"International Electronic Journal of Geometry","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.36890/iejg.984269","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
In 1930 Keller conjectured that each tiling of Rn by unit cubes contains a pair of cubes sharing a complete (n-1)-dimensional face. This conjecture was solved only 50 years later by Lagarias and Shor who found a counterexample for all n >= 10. In this paper we show that neither a modification of Keller's when the unit cube is substituted by a
tile of more complex shape is true.
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