{"title":"Tilings of the sphere by congruent regular triangles and congruent rhombi","authors":"Qi Yuan, Erxiao Wang","doi":"10.1016/j.jalgebra.2025.03.033","DOIUrl":null,"url":null,"abstract":"<div><div>Abstract All edge-to-edge tilings of the sphere by congruent regular triangles and congruent rhombi are classified as: (1) a 1-parameter family of protosets each admitting a unique <span><math><mo>(</mo><mn>2</mn><msup><mrow><mi>a</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>,</mo><mn>3</mn><msup><mrow><mi>a</mi></mrow><mrow><mn>4</mn></mrow></msup><mo>)</mo></math></span>-tiling like a triangular prism; (2) a 1-parameter family of protosets each admitting 2 different <span><math><mo>(</mo><mn>8</mn><msup><mrow><mi>a</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>,</mo><mn>6</mn><msup><mrow><mi>a</mi></mrow><mrow><mn>4</mn></mrow></msup><mo>)</mo></math></span>-tilings like a cuboctahedron and a triangular orthobicupola respectively; (3) a sequence of protosets each admitting a unique <span><math><mo>(</mo><mn>2</mn><msup><mrow><mi>a</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>,</mo><mo>(</mo><mn>6</mn><mi>n</mi><mo>−</mo><mn>3</mn><mo>)</mo><msup><mrow><mi>a</mi></mrow><mrow><mn>4</mn></mrow></msup><mo>)</mo></math></span>-tiling like a generalized anti-triangular prism for each <span><math><mi>n</mi><mo>≥</mo><mn>3</mn></math></span>; (4) 26 sporadic protosets, among which nineteen admit a unique tiling, three admit 2 different tilings, one admits 3 different tilings, one admits 5 different tilings, two admit too many tilings to count (such polymorphic phenomena is unseen in monohedral spherical tilings with small total number of tiles). The moduli of parameterized tilings and all geometric data are provided.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"676 ","pages":"Pages 150-188"},"PeriodicalIF":0.8000,"publicationDate":"2025-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Algebra","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0021869325001723","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
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Abstract
Abstract All edge-to-edge tilings of the sphere by congruent regular triangles and congruent rhombi are classified as: (1) a 1-parameter family of protosets each admitting a unique -tiling like a triangular prism; (2) a 1-parameter family of protosets each admitting 2 different -tilings like a cuboctahedron and a triangular orthobicupola respectively; (3) a sequence of protosets each admitting a unique -tiling like a generalized anti-triangular prism for each ; (4) 26 sporadic protosets, among which nineteen admit a unique tiling, three admit 2 different tilings, one admits 3 different tilings, one admits 5 different tilings, two admit too many tilings to count (such polymorphic phenomena is unseen in monohedral spherical tilings with small total number of tiles). The moduli of parameterized tilings and all geometric data are provided.
期刊介绍：
The Journal of Algebra is a leading international journal and publishes papers that demonstrate high quality research results in algebra and related computational aspects. Only the very best and most interesting papers are to be considered for publication in the journal. With this in mind, it is important that the contribution offer a substantial result that will have a lasting effect upon the field. The journal also seeks work that presents innovative techniques that offer promising results for future research.
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