{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,14]],"date-time":"2025-10-14T00:52:44Z","timestamp":1760403164668,"version":"build-2065373602"},"reference-count":23,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2022,1,11]],"date-time":"2022-01-11T00:00:00Z","timestamp":1641859200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11761039, 62062042"],"award-info":[{"award-number":["11761039, 62062042"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Science and Technology Project of the Education Department of Jiangxi Province of China","award":["GJJ190542"],"award-info":[{"award-number":["GJJ190542"]}]},{"name":"Natural Science Foundation of Jiangxi Province of China","award":["20202ACBL211001"],"award-info":[{"award-number":["20202ACBL211001"]}]},{"name":"Base and Talent Project of Guangxi Science and Technology","award":["2021AC20010"],"award-info":[{"award-number":["2021AC20010"]}]},{"DOI":"10.13039\/501100004607","name":"Guangxi Natural Science Foundation","doi-asserted-by":"publisher","award":["2018GXNSFBA281185"],"award-info":[{"award-number":["2018GXNSFBA281185"]}],"id":[{"id":"10.13039\/501100004607","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Undergraduate Teaching Reform Project of Guangxi Higher Education","award":["2021JGA224"],"award-info":[{"award-number":["2021JGA224"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Spirals, tilings, and hyperbolic geometry are important mathematical topics with outstanding aesthetic elements. Nonetheless, research on their aesthetic visualization is extremely limited. In this paper, we give a simple method for creating Escher-like hyperbolic spiral patterns. To this end, we first present a fast algorithm to construct Euclidean spiral tilings with cyclic symmetry. Then, based on a one-to-one mapping between Euclidean and hyperbolic spaces, we establish two simple approaches for constructing spiral tilings in hyperbolic models. Finally, we use wallpaper templates to render such tilings, which results in the desired Escher-like hyperbolic spiral patterns. The method proposed is able to generate a great variety of visually appealing patterns.<\/jats:p>","DOI":"10.3390\/sym14010134","type":"journal-article","created":{"date-parts":[[2022,1,11]],"date-time":"2022-01-11T20:33:04Z","timestamp":1641933184000},"page":"134","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Visualization of Escher-like Spiral Patterns in Hyperbolic Space"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6452-0468","authenticated-orcid":false,"given":"Chongyang","family":"Qiu","sequence":"first","affiliation":[{"name":"Human Sciences Department, Gannan University of Science and Technology, Ganzhou 341000, China"}]},{"given":"Xinfei","family":"Li","sequence":"additional","affiliation":[{"name":"School of Science, Guangxi University of Science and Technology, Liuzhou 545006, China"}]},{"given":"Jianhua","family":"Pang","sequence":"additional","affiliation":[{"name":"School of Science, Guangxi University of Science and Technology, Liuzhou 545006, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0447-3190","authenticated-orcid":false,"given":"Peichang","family":"Ouyang","sequence":"additional","affiliation":[{"name":"School of Science, Guangxi University of Science and Technology, Liuzhou 545006, China"},{"name":"School of Mathematics and Physics, Jinggangshan University, Ji\u2019an 343009, China"}]}],"member":"1968","published-online":{"date-parts":[[2022,1,11]]},"reference":[{"key":"ref_1","unstructured":"Cook, T.A. 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Available online: https:\/\/isohedral.ca\/escher-like-spiral-tilings\/."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/1\/134\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T14:28:19Z","timestamp":1760365699000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/1\/134"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,1,11]]},"references-count":23,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2022,1]]}},"alternative-id":["sym14010134"],"URL":"https:\/\/doi.org\/10.3390\/sym14010134","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,1,11]]}}}