Poliedros regulares e semirregulares

Detalhes bibliográficos
Autor(a) principal: Resende, Luana Pimenta Muniz de
Data de Publicação: 2023
Tipo de documento: Trabalho de conclusão de curso
Idioma: por
Título da fonte: Repositório Institucional da UFU
Texto Completo: https://repositorio.ufu.br/handle/123456789/40981
Resumo: In this Undergraduate Final Project we present a study of the five regular polyhedra (or Platonic Solids) and the thirteen semi-regular polyhedra (or Archimedean Solids). This study was divided into two parts: theoretical part, with deductions from formulas and classification of above polyhedra and; practical part, with the dynamic construction of semi-regular polyhedra from regular polyhedra using the GeoGebra software. It is important to emphasize the impossibility of dynamically constructing semi-regular polyhedra in GeoGebra without some of the formulas related to regular polyhedra that were deduced in the theoretical part. The formulas deduced were: (i) Euler Relation for convex polyhedra; (ii) measurements of the central and dihedral angles of a regular polyhedron depending on its genus of faces and its genus of vertices; (iii) radii of spheres inscribed and circumscribed by a regular polyhedron depending on the genus of faces, the genus of vertices and the length of the edges; (iv) apothem and radius of the circle circumscribed to the face of a regular polyhedron depending on its genus of faces and its edge length and; (v) area and volume of a regular polyhedron as a function of its genus of faces, its genus of vertices, its number of faces and its edge length. It is also important to highlight that the constructions of semi-regular polyhedra were made through three geometric operations on regular polyhedra: (1) simple truncation; (2) composite truncation and; (3) snubification. In this work we hope to be able to contribute to the theory of regular and semi-regular polyhedra, as well as the dynamic geometric constructions of such polyhedra in the GeoGebra software
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spelling Poliedros regulares e semirregularesRegular and semiregular polyhedraClassificação de poliedrosClassification of polyhedraGeoGebraGeometria dinâmicaDynamic geometryTruncamentoTruncationCNPQ::CIENCIAS EXATAS E DA TERRA::MATEMATICA::GEOMETRIA E TOPOLOGIAIn this Undergraduate Final Project we present a study of the five regular polyhedra (or Platonic Solids) and the thirteen semi-regular polyhedra (or Archimedean Solids). This study was divided into two parts: theoretical part, with deductions from formulas and classification of above polyhedra and; practical part, with the dynamic construction of semi-regular polyhedra from regular polyhedra using the GeoGebra software. It is important to emphasize the impossibility of dynamically constructing semi-regular polyhedra in GeoGebra without some of the formulas related to regular polyhedra that were deduced in the theoretical part. The formulas deduced were: (i) Euler Relation for convex polyhedra; (ii) measurements of the central and dihedral angles of a regular polyhedron depending on its genus of faces and its genus of vertices; (iii) radii of spheres inscribed and circumscribed by a regular polyhedron depending on the genus of faces, the genus of vertices and the length of the edges; (iv) apothem and radius of the circle circumscribed to the face of a regular polyhedron depending on its genus of faces and its edge length and; (v) area and volume of a regular polyhedron as a function of its genus of faces, its genus of vertices, its number of faces and its edge length. It is also important to highlight that the constructions of semi-regular polyhedra were made through three geometric operations on regular polyhedra: (1) simple truncation; (2) composite truncation and; (3) snubification. In this work we hope to be able to contribute to the theory of regular and semi-regular polyhedra, as well as the dynamic geometric constructions of such polyhedra in the GeoGebra softwarePesquisa sem auxílio de agências de fomentoTrabalho de Conclusão de Curso (Graduação)Neste Trabalho de Conclusão de Curso apresentamos um estudo dos cinco poliedros regulares (ou Poliedros de Platão) e dos treze poliedros semirregulares (ou Poliedros de Arquimedes). Tal estudo foi dividido em duas partes: parte teórica, com deduções de fórmulas e classificação dos referidos poliedros e; parte prática, com a construção dinâmica dos poliedros semirregulares a partir dos poliedros regulares utilizando o software GeoGebra. É importante enfatizar a impossibilidade de se construir dinamicamente os poliedros semirregulares no GeoGebra sem algumas das fórmulas relacionadas aos poliedros regulares que foram deduzidas na parte teórica. As fórmulas deduzidas foram: (i) Relação de Euler para poliedros convexos; (ii) medidas dos ângulos central e diedral de poliedro regular em função de seu gênero de faces e de seu gênero de vértices; (iii) raios de esferas inscrita e circunscrita a poliedro regular em função de seu gênero de faces, de seu gênero de vértices e de seu comprimento de arestas; (iv) apótema e raio de círculo circunscrito à face de poliedro regular em função de seu gênero de faces e de seu comprimento de arestas e; (v) área e volume de poliedro regular em função de seu gênero de faces, de seu gênero de vértices, de seu número de faces e de seu comprimento de arestas. Também é importante ressaltar que as construções dos poliedros semirregulares foram feitas por meio de três operações geométricas sobre os poliedros regulares: (1) truncamento simples; (2) truncamento composto e; (3) snubificação. Neste trabalho temos a expectativa de poder contruibuir na teoria dos poliedros regulares e semirregulares, bem como nas construções geométricas dinâmicas de tais poliedros no software GeoGebra.Universidade Federal de UberlândiaBrasilMatemáticaAgustini, Edsonhttp://lattes.cnpq.br/1537249856486330Marin, Douglashttp://lattes.cnpq.br/6734500640303971Lopes, Érika Maria Chiocahttp://lattes.cnpq.br/0024613652139150Resende, Luana Pimenta Muniz de2024-01-16T12:46:45Z2024-01-16T12:46:45Z2023-12-01info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/bachelorThesisapplication/pdfRESENDE, Luana Pimenta Muniz de. Poliedros regulares e semirregulares. 2023. 99 f. Trabalho de Conclusão de Curso (Graduação em Matemática) – Universidade Federal de Uberlândia, Uberlândia, 2024.https://repositorio.ufu.br/handle/123456789/40981porinfo:eu-repo/semantics/openAccessreponame:Repositório Institucional da UFUinstname:Universidade Federal de Uberlândia (UFU)instacron:UFU2024-01-17T06:17:23Zoai:repositorio.ufu.br:123456789/40981Repositório InstitucionalONGhttp://repositorio.ufu.br/oai/requestdiinf@dirbi.ufu.bropendoar:2024-01-17T06:17:23Repositório Institucional da UFU - Universidade Federal de Uberlândia (UFU)false
dc.title.none.fl_str_mv Poliedros regulares e semirregulares
Regular and semiregular polyhedra
title Poliedros regulares e semirregulares
spellingShingle Poliedros regulares e semirregulares
Resende, Luana Pimenta Muniz de
Classificação de poliedros
Classification of polyhedra
GeoGebra
Geometria dinâmica
Dynamic geometry
Truncamento
Truncation
CNPQ::CIENCIAS EXATAS E DA TERRA::MATEMATICA::GEOMETRIA E TOPOLOGIA
title_short Poliedros regulares e semirregulares
title_full Poliedros regulares e semirregulares
title_fullStr Poliedros regulares e semirregulares
title_full_unstemmed Poliedros regulares e semirregulares
title_sort Poliedros regulares e semirregulares
author Resende, Luana Pimenta Muniz de
author_facet Resende, Luana Pimenta Muniz de
author_role author
dc.contributor.none.fl_str_mv Agustini, Edson
http://lattes.cnpq.br/1537249856486330
Marin, Douglas
http://lattes.cnpq.br/6734500640303971
Lopes, Érika Maria Chioca
http://lattes.cnpq.br/0024613652139150
dc.contributor.author.fl_str_mv Resende, Luana Pimenta Muniz de
dc.subject.por.fl_str_mv Classificação de poliedros
Classification of polyhedra
GeoGebra
Geometria dinâmica
Dynamic geometry
Truncamento
Truncation
CNPQ::CIENCIAS EXATAS E DA TERRA::MATEMATICA::GEOMETRIA E TOPOLOGIA
topic Classificação de poliedros
Classification of polyhedra
GeoGebra
Geometria dinâmica
Dynamic geometry
Truncamento
Truncation
CNPQ::CIENCIAS EXATAS E DA TERRA::MATEMATICA::GEOMETRIA E TOPOLOGIA
description In this Undergraduate Final Project we present a study of the five regular polyhedra (or Platonic Solids) and the thirteen semi-regular polyhedra (or Archimedean Solids). This study was divided into two parts: theoretical part, with deductions from formulas and classification of above polyhedra and; practical part, with the dynamic construction of semi-regular polyhedra from regular polyhedra using the GeoGebra software. It is important to emphasize the impossibility of dynamically constructing semi-regular polyhedra in GeoGebra without some of the formulas related to regular polyhedra that were deduced in the theoretical part. The formulas deduced were: (i) Euler Relation for convex polyhedra; (ii) measurements of the central and dihedral angles of a regular polyhedron depending on its genus of faces and its genus of vertices; (iii) radii of spheres inscribed and circumscribed by a regular polyhedron depending on the genus of faces, the genus of vertices and the length of the edges; (iv) apothem and radius of the circle circumscribed to the face of a regular polyhedron depending on its genus of faces and its edge length and; (v) area and volume of a regular polyhedron as a function of its genus of faces, its genus of vertices, its number of faces and its edge length. It is also important to highlight that the constructions of semi-regular polyhedra were made through three geometric operations on regular polyhedra: (1) simple truncation; (2) composite truncation and; (3) snubification. In this work we hope to be able to contribute to the theory of regular and semi-regular polyhedra, as well as the dynamic geometric constructions of such polyhedra in the GeoGebra software
publishDate 2023
dc.date.none.fl_str_mv 2023-12-01
2024-01-16T12:46:45Z
2024-01-16T12:46:45Z
dc.type.status.fl_str_mv info:eu-repo/semantics/publishedVersion
dc.type.driver.fl_str_mv info:eu-repo/semantics/bachelorThesis
format bachelorThesis
status_str publishedVersion
dc.identifier.uri.fl_str_mv RESENDE, Luana Pimenta Muniz de. Poliedros regulares e semirregulares. 2023. 99 f. Trabalho de Conclusão de Curso (Graduação em Matemática) – Universidade Federal de Uberlândia, Uberlândia, 2024.
https://repositorio.ufu.br/handle/123456789/40981
identifier_str_mv RESENDE, Luana Pimenta Muniz de. Poliedros regulares e semirregulares. 2023. 99 f. Trabalho de Conclusão de Curso (Graduação em Matemática) – Universidade Federal de Uberlândia, Uberlândia, 2024.
url https://repositorio.ufu.br/handle/123456789/40981
dc.language.iso.fl_str_mv por
language por
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 Universidade Federal de Uberlândia
Brasil
Matemática
publisher.none.fl_str_mv Universidade Federal de Uberlândia
Brasil
Matemática
dc.source.none.fl_str_mv reponame:Repositório Institucional da UFU
instname:Universidade Federal de Uberlândia (UFU)
instacron:UFU
instname_str Universidade Federal de Uberlândia (UFU)
instacron_str UFU
institution UFU
reponame_str Repositório Institucional da UFU
collection Repositório Institucional da UFU
repository.name.fl_str_mv Repositório Institucional da UFU - Universidade Federal de Uberlândia (UFU)
repository.mail.fl_str_mv diinf@dirbi.ufu.br
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