Tessellating the plane with convex polygons
In this article we give a panoramic view over the classification of tilings of the euclidean plane by using copies of a single convex polygon (convex monohedral tilings). First, we show that a tiling with regular poligons is only possible by using triangles, squares and regular hexagons, a fact well...
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| Formato: | Artículo revista |
| Lenguaje: | Español |
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Unión Matemática Argentina - Facultad de Matemática, Astronomía, Física y Computación
2022
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| Acceso en línea: | https://revistas.unc.edu.ar/index.php/REM/article/view/37469 |
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I10-R366-article-37469 |
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I10-R366-article-374692022-05-17T18:49:50Z Tessellating the plane with convex polygons Teselando el plano con polígonos convexos Podestá, Ricardo A. Embaldosados Polígonos convexos Polígonos regulares Tilings Plane convex polygons regular polygons In this article we give a panoramic view over the classification of tilings of the euclidean plane by using copies of a single convex polygon (convex monohedral tilings). First, we show that a tiling with regular poligons is only possible by using triangles, squares and regular hexagons, a fact well known by the ancient greeks, and that if the polygon is not convex then there are infinite possible tilings. In this way, we focus on convex tilings withnon-regular polygons. First, we show that any triangle or quadrilateral tiles the plane. Then, we show that a polygon that tiles the plane must have at most 6 edges. Next, we consider the case of hexagons and show that there are only 3 different families of convex hexagons tiling the plane. Finally, we deal with pentagons, whose classification is more involved, and could be completed recently in 2017. We will show that there are 15 different families of pentagons tiling the plane. En este articulo damos un panorama sobre la clasificación de los embaldosados del plano euclídeo por copias de un único polígono convexo (teselados monoedralesconvexos). Primero mostramos que el teselado con polígonos regulares sólo es posible con triángulos, cuadrados y hexágonos, hecho ya conocido por los antiguos griegos, y que si el polígono es no-convexo entonces hay infinitos teselados posibles. Así, nos enfocamos en teselados convexos con polígonos no-regulares. Primero mostramos que cualquier triangulo o cuadrilátero tesela el plano. Después mostramos que un polígono que tesela el plano debe tener 6 lados o menos. A continuación, nos ocupamos de los hexágonos y mostramos que solo hay 3 familias distintas de hexágonos convexos que teselan el plano. Finalmente consideramos el caso de los pentágonos que es mas delicado, cuya clasificación completa pudo terminarse muy recientemente en 2017. Mostramos que hay solo 15 familias distintas de pentágonos que teselan el plano Unión Matemática Argentina - Facultad de Matemática, Astronomía, Física y Computación 2022-04-29 info:eu-repo/semantics/article info:eu-repo/semantics/publishedVersion Artículo evaluado por pares application/pdf https://revistas.unc.edu.ar/index.php/REM/article/view/37469 10.33044/revem.37469 Revista de Educación Matemática; Vol. 37 Núm. 1 (2022); 31-60 1852-2890 0326-8780 spa https://revistas.unc.edu.ar/index.php/REM/article/view/37469/37583 https://creativecommons.org/licenses/by-sa/4.0/ |
| institution |
Universidad Nacional de Córdoba |
| institution_str |
I-10 |
| repository_str |
R-366 |
| container_title_str |
Revista de Educación Matemática |
| language |
Español |
| format |
Artículo revista |
| topic |
Embaldosados Polígonos convexos Polígonos regulares Tilings Plane convex polygons regular polygons |
| spellingShingle |
Embaldosados Polígonos convexos Polígonos regulares Tilings Plane convex polygons regular polygons Podestá, Ricardo A. Tessellating the plane with convex polygons |
| topic_facet |
Embaldosados Polígonos convexos Polígonos regulares Tilings Plane convex polygons regular polygons |
| author |
Podestá, Ricardo A. |
| author_facet |
Podestá, Ricardo A. |
| author_sort |
Podestá, Ricardo A. |
| title |
Tessellating the plane with convex polygons |
| title_short |
Tessellating the plane with convex polygons |
| title_full |
Tessellating the plane with convex polygons |
| title_fullStr |
Tessellating the plane with convex polygons |
| title_full_unstemmed |
Tessellating the plane with convex polygons |
| title_sort |
tessellating the plane with convex polygons |
| description |
In this article we give a panoramic view over the classification of tilings of the euclidean plane by using copies of a single convex polygon (convex monohedral tilings). First, we show that a tiling with regular poligons is only possible by using triangles, squares and regular hexagons, a fact well known by the ancient greeks, and that if the polygon is not convex then there are infinite possible tilings. In this way, we focus on convex tilings withnon-regular polygons. First, we show that any triangle or quadrilateral tiles the plane. Then, we show that a polygon that tiles the plane must have at most 6 edges. Next, we consider the case of hexagons and show that there are only 3 different families of convex hexagons tiling the plane. Finally, we deal with pentagons, whose classification is more involved, and could be completed recently in 2017. We will show that there are 15 different families of pentagons tiling the plane. |
| publisher |
Unión Matemática Argentina - Facultad de Matemática, Astronomía, Física y Computación |
| publishDate |
2022 |
| url |
https://revistas.unc.edu.ar/index.php/REM/article/view/37469 |
| work_keys_str_mv |
AT podestaricardoa tessellatingtheplanewithconvexpolygons AT podestaricardoa teselandoelplanoconpoligonosconvexos |
| first_indexed |
2024-09-03T22:36:54Z |
| last_indexed |
2024-09-03T22:36:54Z |
| _version_ |
1809216191606030336 |