Exploring symmetry in rosettes of Truchet tiles
Autor(a) principal: | |
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Data de Publicação: | 2019 |
Outros Autores: | , |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
Texto Completo: | http://hdl.handle.net/10773/25785 |
Resumo: | In 1704 the French priest S ebastien Truchet published a paper where he explored and counted patterns made up from a square divided by a diagonal line into two colored parts,, now known as a Truchet tile. A few years later, Father Dominique Douat continued Truchet's work and published a book in 1722 containing many more patterns and further counts of con gurations. In this paper, we extend the work introduced by Truchet and Douat by considering all possible rosettes made up of an mXn array of square or non-square Truchet tiles. We then classify the rosettes according to their symmetry group and count all the distinct rosettes in each group, for all possible sizes. The results are summarized in a separate section where we further analyze the asymptotic behavior of the counts for square arrays. Finally, some applications are shown using two types of square flexagons. |
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Exploring symmetry in rosettes of Truchet tilesTruchet tilesRosette symmetry groupsRecreational mathematicsIn 1704 the French priest S ebastien Truchet published a paper where he explored and counted patterns made up from a square divided by a diagonal line into two colored parts,, now known as a Truchet tile. A few years later, Father Dominique Douat continued Truchet's work and published a book in 1722 containing many more patterns and further counts of con gurations. In this paper, we extend the work introduced by Truchet and Douat by considering all possible rosettes made up of an mXn array of square or non-square Truchet tiles. We then classify the rosettes according to their symmetry group and count all the distinct rosettes in each group, for all possible sizes. The results are summarized in a separate section where we further analyze the asymptotic behavior of the counts for square arrays. Finally, some applications are shown using two types of square flexagons.Taylor & Francis2020-12-01T00:00:00Z2019-01-01T00:00:00Z2019info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articleapplication/pdfhttp://hdl.handle.net/10773/25785eng1751-347210.1080/17513472.2019.1581963Hall, AndreiaAlmeida, Paulo J.Teixeira, Ricardoinfo:eu-repo/semantics/openAccessreponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãoinstacron:RCAAP2024-02-22T11:49:57Zoai:ria.ua.pt:10773/25785Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireopendoar:71602024-03-20T02:58:55.411325Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãofalse |
dc.title.none.fl_str_mv |
Exploring symmetry in rosettes of Truchet tiles |
title |
Exploring symmetry in rosettes of Truchet tiles |
spellingShingle |
Exploring symmetry in rosettes of Truchet tiles Hall, Andreia Truchet tiles Rosette symmetry groups Recreational mathematics |
title_short |
Exploring symmetry in rosettes of Truchet tiles |
title_full |
Exploring symmetry in rosettes of Truchet tiles |
title_fullStr |
Exploring symmetry in rosettes of Truchet tiles |
title_full_unstemmed |
Exploring symmetry in rosettes of Truchet tiles |
title_sort |
Exploring symmetry in rosettes of Truchet tiles |
author |
Hall, Andreia |
author_facet |
Hall, Andreia Almeida, Paulo J. Teixeira, Ricardo |
author_role |
author |
author2 |
Almeida, Paulo J. Teixeira, Ricardo |
author2_role |
author author |
dc.contributor.author.fl_str_mv |
Hall, Andreia Almeida, Paulo J. Teixeira, Ricardo |
dc.subject.por.fl_str_mv |
Truchet tiles Rosette symmetry groups Recreational mathematics |
topic |
Truchet tiles Rosette symmetry groups Recreational mathematics |
description |
In 1704 the French priest S ebastien Truchet published a paper where he explored and counted patterns made up from a square divided by a diagonal line into two colored parts,, now known as a Truchet tile. A few years later, Father Dominique Douat continued Truchet's work and published a book in 1722 containing many more patterns and further counts of con gurations. In this paper, we extend the work introduced by Truchet and Douat by considering all possible rosettes made up of an mXn array of square or non-square Truchet tiles. We then classify the rosettes according to their symmetry group and count all the distinct rosettes in each group, for all possible sizes. The results are summarized in a separate section where we further analyze the asymptotic behavior of the counts for square arrays. Finally, some applications are shown using two types of square flexagons. |
publishDate |
2019 |
dc.date.none.fl_str_mv |
2019-01-01T00:00:00Z 2019 2020-12-01T00:00:00Z |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/article |
format |
article |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
http://hdl.handle.net/10773/25785 |
url |
http://hdl.handle.net/10773/25785 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
1751-3472 10.1080/17513472.2019.1581963 |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
dc.format.none.fl_str_mv |
application/pdf |
dc.publisher.none.fl_str_mv |
Taylor & Francis |
publisher.none.fl_str_mv |
Taylor & Francis |
dc.source.none.fl_str_mv |
reponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação instacron:RCAAP |
instname_str |
Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
instacron_str |
RCAAP |
institution |
RCAAP |
reponame_str |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
collection |
Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
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Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
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1799137643396595712 |