{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,8]],"date-time":"2026-05-08T14:47:12Z","timestamp":1778251632325,"version":"3.51.4"},"reference-count":25,"publisher":"Springer Science and Business Media LLC","issue":"23","license":[{"start":{"date-parts":[[2020,8,29]],"date-time":"2020-08-29T00:00:00Z","timestamp":1598659200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2020,8,29]],"date-time":"2020-08-29T00:00:00Z","timestamp":1598659200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100007069","name":"Universit\u00e0 della Calabria","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100007069","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Soft Comput"],"published-print":{"date-parts":[[2020,12]]},"abstract":"Abstract<\/jats:title>Fibonacci numbers are the basis of a new geometric construction that leads to the definition of a family $$\\{C_n:n\\in \\mathbb {N}\\}$$<\/jats:tex-math>\n\n{<\/mml:mo>\n\nC<\/mml:mi>\nn<\/mml:mi>\n<\/mml:msub>\n:<\/mml:mo>\nn<\/mml:mi>\n\u2208<\/mml:mo>\nN<\/mml:mi>\n}<\/mml:mo>\n<\/mml:mrow>\n<\/mml:math><\/jats:alternatives><\/jats:inline-formula> of octagons that come very close to the regular octagon. Such octagons, in some previous articles, have been given the name of Carboncettus octagons<\/jats:italic> for historical reasons. Going further, in this paper we want to introduce and investigate some algebraic constructs that arise from the family $$\\{C_n:n\\in \\mathbb {N}\\}$$<\/jats:tex-math>\n\n{<\/mml:mo>\n\nC<\/mml:mi>\nn<\/mml:mi>\n<\/mml:msub>\n:<\/mml:mo>\nn<\/mml:mi>\n\u2208<\/mml:mo>\nN<\/mml:mi>\n}<\/mml:mo>\n<\/mml:mrow>\n<\/mml:math><\/jats:alternatives><\/jats:inline-formula> and therefore from Fibonacci numbers: From each Carboncettus octagon $$C_n$$<\/jats:tex-math>\n\nC<\/mml:mi>\nn<\/mml:mi>\n<\/mml:msub>\n<\/mml:math><\/jats:alternatives><\/jats:inline-formula>, it is possible to obtain an infinite (right) word $$W_n$$<\/jats:tex-math>\n\nW<\/mml:mi>\nn<\/mml:mi>\n<\/mml:msub>\n<\/mml:math><\/jats:alternatives><\/jats:inline-formula> on the binary alphabet $$\\{0,1\\}$$<\/jats:tex-math>\n\n{<\/mml:mo>\n0<\/mml:mn>\n,<\/mml:mo>\n1<\/mml:mn>\n}<\/mml:mo>\n<\/mml:mrow>\n<\/mml:math><\/jats:alternatives><\/jats:inline-formula>, which we will call the nth Carboncettus word<\/jats:italic>. The main theorem shows that all the Carboncettus words thus defined are Sturmian words except in the case $$n=5$$<\/jats:tex-math>\n\nn<\/mml:mi>\n=<\/mml:mo>\n5<\/mml:mn>\n<\/mml:mrow>\n<\/mml:math><\/jats:alternatives><\/jats:inline-formula>. The fifth Carboncettus word $$W_5$$<\/jats:tex-math>\n\nW<\/mml:mi>\n5<\/mml:mn>\n<\/mml:msub>\n<\/mml:math><\/jats:alternatives><\/jats:inline-formula> is in fact the only word of the family to be purely periodic: It has period 17 and periodic factor 000\u00a0100\u00a0100\u00a0010\u00a0010\u00a001. Finally, we also define a further word $$W_{\\infty }$$<\/jats:tex-math>\n\nW<\/mml:mi>\n\u221e<\/mml:mi>\n<\/mml:msub>\n<\/mml:math><\/jats:alternatives><\/jats:inline-formula> named the Carboncettus limit word<\/jats:italic> and, as second main result, we prove that the limit of the sequence of Carboncettus words is $$W_{\\infty }$$<\/jats:tex-math>\n\nW<\/mml:mi>\n\u221e<\/mml:mi>\n<\/mml:msub>\n<\/mml:math><\/jats:alternatives><\/jats:inline-formula> itself.<\/jats:p>","DOI":"10.1007\/s00500-020-05256-1","type":"journal-article","created":{"date-parts":[[2020,8,29]],"date-time":"2020-08-29T13:10:32Z","timestamp":1598706632000},"page":"17497-17508","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":18,"title":["New algebraic and geometric constructs arising from Fibonacci numbers"],"prefix":"10.1007","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4011-848X","authenticated-orcid":false,"given":"Fabio","family":"Caldarola","sequence":"first","affiliation":[]},{"given":"Gianfranco","family":"d\u2019Atri","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6208-1307","authenticated-orcid":false,"given":"Mario","family":"Maiolo","sequence":"additional","affiliation":[]},{"given":"Giuseppe","family":"Pirillo","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2020,8,29]]},"reference":[{"key":"5256_CR1","first-page":"48","volume":"21","author":"A Albano","year":"2015","unstructured":"Albano A (2015) The Fibonacci sequence and the golden section in a lunette decoration of the medieval church of San Nicola in Pisa. 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