Building hyperbolic tessellations on the Poincaré disk with GeoGebra

Authors

DOI:

https://doi.org/10.23925/2237-9657.2022.v11i2p017-032

Keywords:

Hyperbolic geometry, Works by Escher, Mathematics Teaching

Abstract

We present in this work some characteristics of the Poincaré disk geometry, hyperbolic geometry in the plane, such as hyperbolic line, parallelism, distance between two points, hyperbolic polygons and circles, the sum of the internal angles and the area of a hyperbolic triangle. We associate some works by the Dutch graphic artist Maurits Cornelis Escher with tessellations on the circle, and we used GeoGebra’s hyperbolic tools to build two-dimensional figures and a tessellation with hyperbolic triangles on the Poincaré disk.  We conclude that GeoGebra is a great app to be explored in the study of non-Euclidean geometries in the plane, especially in Mathematics-Teaching Degree Course.

Downloads

Download data is not yet available.

Metrics

Metrics Loading ...

Author Biographies

Rudimar Luiz Nós, UTFPR - Universidade Tecnológica Federal do Paraná

Lattes: http://lattes.cnpq.br/4377393528295346

Alfred James Dias Albon

Link para o Lattes:

 http://lattes.cnpq.br/7658246345715043

References

ALBON, A. J. D. A geometria do disco de Poincaré. 93 f. Trabalho de Conclusão de Curso (Licenciatura em Matemática) – Universidade Tecnológica Federal do Paraná. Curitiba, 2021.

ANDRADE, P. F. Introdução à geometria hiperbólica: o modelo de Poincaré. 1. ed. Rio de Janeiro, SBM, 2013.

ANTUNES, A. K. L.; QUEIROZ, C. R. O. Q. Tesselações no plano hiperbólico. Sigmae, 6(2), 69-77, 2017.

BOYER, C. B.; MERZBACH, U. C. A history of mathematics. 3. ed. Hoboken, John Wiley & Sons, 2011.

BRANNAN, D. A.; ESPLEN, M. F.; GRAY, J. J. Geometry. 2. ed. Cambridge, Cambridge University Press, 2012.

BURTON, D. M. The history of mathematics: an introduction. 7. ed. New York, McGraw-Hill, 2011.

CASSELMAN, B. How did Escher do it? 2010. Disponível em: http://www.ams.org/publicoutreach/feature-column/fcarc-circle-limit. Acesso em: 20 set. 2021.

CHRISTERSSON, M. Non-Euclidean geometry. 2018. Disponível em: http://www.malinc.se/noneuclidean/en/index.php. Acesso em: 20 set. 2021.

DORIA, C. M. Geometrias: Euclidiana, esférica e hiperbólica. 1. ed. Rio de Janeiro, SBM, 2019.

ECO/UFRJ. Escher: quebrando as barreiras entre arte e matemática. 2013. Disponível em: https://comunicacaoeartes20122.wordpress.com/2013/02/19/escher-quebrando-as-barreiras-entre-arte-e-matematica/. Acesso em: 20 set. 2021.

ESCHER, M. C. The official website. 2021. Disponível em: https://mcescher.com/. Acesso em: 20 set. 2021.

EVES, H. Introdução à história da matemática. 5. ed. Campinas, Unicamp, 2011.

GEOGEBRA. GeoGebra: aplicativos matemáticos. 2021. Disponível em: https://www.geogebra.org/?lang=pt_BR. Acesso em: 20 set. 2021.

HALL, A.; PAIS, S. Learning and teaching symmetry by creating ceramic panels with Escher type tessellations. Indagatio Didactica, 10(2), 85-107, 2018.

KOVÁCS, Z. Poincaré disk model. 2015. Disponível em: https://www.geogebra.org/material/show/id/SQguSCzy. Acesso em: 07 jun. 2022.

MOTTA, G. P. Geometrias não Euclidianas no plano e geometria esférica. 115 f. Trabalho de Conclusão de Curso (Licenciatura em Matemática) – Universidade Tecnológica Federal do Paraná. Curitiba, 2018.

MURARI, C.; LAZARI, H. Tesselações hiperbólicas com régua e compasso. S. d. Disponível em: https://www.docsity.com/pt/hiperbolicas-com-regua-e-compasso/4744842/. Acesso em: 20 set. 2021.

RIBEIRO, R.; GRAVINA, M. Disco de Poincaré: uma proposta para explorar geometria hiperbólica no GeoGebra. Professor de Matemática Online, 1(1), 53-67, 2013.

RIBEIRO, G. F.; FERREIRA, L.; SANTOS, T. S. dos. Construções de macro ferramentas no GeoGebra para o ensino de geometria hiperbólica. Actas de la Conferencia Latinoamericana de GeoGebra, 182-189, 2012. Disponível em: http://www.geogebra.org.uy/2012/actas/15.pdf. Acesso em: 07 jun. 2022.

ROONEY, A. A história da matemática. 1. ed. São Paulo, M. Books, 2012.

SZILASSI, L. A dynamic visualization of the hyperbolic geometry. 2018. Disponível em: https://www.geogebra.org/m/ck6ecca5. Acesso em: 07 jun. 2022.

Downloads

Published

2022-11-12

How to Cite

Nós, R. L., & Dias Albon, A. J. (2022). Building hyperbolic tessellations on the Poincaré disk with GeoGebra. Journal of the GeoGebra International Institute of São Paulo, 11(2), 017–032. https://doi.org/10.23925/2237-9657.2022.v11i2p017-032

Issue

Vol. 11 No. 2 (2022)

Section

Artigos

License

Copyright (c) 2022 Revista do Instituto GeoGebra Internacional de São Paulo

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

Submission, processing, and publication of articles sent to the journal and registration of the DOI at Crossref is free of charge.

Authors retain their copyright and grant the journal the right of first publication of their article, which is simultaneously licensed under a Creative Commons - Attribution 4.0 International license CC BY that allows others to share the article by acknowledging its authorship and initial publication by the journal.

The GeoGebra journal encourages its authors to register their work with information and communication management systems aimed at researchers, such as Academia.edu, Mendeley, ResearchGate, etc.