<p align="justify"> <span style="font-family:Verdana;">In this paper, tiling a plane with equilateral semi-regular convex polygons is considered, and, that is, tiling with equilateral p...<p align="justify"> <span style="font-family:Verdana;">In this paper, tiling a plane with equilateral semi-regular convex polygons is considered, and, that is, tiling with equilateral polygons of</span><span style="font-family:Verdana;"> the same type. Tiling a plane with semi-regular polygons depends not only on the type of a semi-regular polygon, but also on its interior angles that join at a node. In relation to the interior angles, semi-regular equilateral polygons with the same or different interior angles can be joined in the nodes. Here, we shall first consider tiling a plane with semi-regular equilateral polygons with 2m-sides. The analysis is performed by determining the set of all integer solutions of the corresponding Diophantine equation in the form of <img alt="" src="Edit_c185b1c4-6b78-4af5-b1c2-4932af77bf65.png" />, where<img alt="" src="Edit_2e6548d5-3254-4005-b19e-9d49cd5d6f81.png" />are the non-negative integers which are not equal to zero at the same time, and <img alt="" src="Edit_a6dbde8a-5f3a-43d4-bc89-27dcc3057d23.png" />are the interior angles of a semi-regular equilateral polygon from the characteristic angle. It is shown that of all semi-regular equilateral polygons with 2m-sides, a plane can be tiled only with the semi-regular equilateral quadrilaterals and semi-regular equilateral hexagons. Then, the problem of tiling a plane with semi-regular equilateral quadrilaterals is analyzed in detail, and then the one with semi-regular equilateral hexagons. For these semi-regular polygons, all possible solutions of the corresponding Diophantine equations were analyzed and all nodes were determined, and then the problem for different values of characteristic elements was observed. For some of the observed cases of tiling a plane with these semi-regular polygons, some graphical presentations of tiling constructions are also given.</span> </p>展开更多
文摘<p align="justify"> <span style="font-family:Verdana;">In this paper, tiling a plane with equilateral semi-regular convex polygons is considered, and, that is, tiling with equilateral polygons of</span><span style="font-family:Verdana;"> the same type. Tiling a plane with semi-regular polygons depends not only on the type of a semi-regular polygon, but also on its interior angles that join at a node. In relation to the interior angles, semi-regular equilateral polygons with the same or different interior angles can be joined in the nodes. Here, we shall first consider tiling a plane with semi-regular equilateral polygons with 2m-sides. The analysis is performed by determining the set of all integer solutions of the corresponding Diophantine equation in the form of <img alt="" src="Edit_c185b1c4-6b78-4af5-b1c2-4932af77bf65.png" />, where<img alt="" src="Edit_2e6548d5-3254-4005-b19e-9d49cd5d6f81.png" />are the non-negative integers which are not equal to zero at the same time, and <img alt="" src="Edit_a6dbde8a-5f3a-43d4-bc89-27dcc3057d23.png" />are the interior angles of a semi-regular equilateral polygon from the characteristic angle. It is shown that of all semi-regular equilateral polygons with 2m-sides, a plane can be tiled only with the semi-regular equilateral quadrilaterals and semi-regular equilateral hexagons. Then, the problem of tiling a plane with semi-regular equilateral quadrilaterals is analyzed in detail, and then the one with semi-regular equilateral hexagons. For these semi-regular polygons, all possible solutions of the corresponding Diophantine equations were analyzed and all nodes were determined, and then the problem for different values of characteristic elements was observed. For some of the observed cases of tiling a plane with these semi-regular polygons, some graphical presentations of tiling constructions are also given.</span> </p>
基金Project(52225404)supported by the National Natural Science Foundation for the Distinguished Young Scholars of ChinaProject(BJJWZYJH01201911413037)supported by the Beijing Outstanding Young Scientist Program,China+1 种基金Project(202206430036)supported by the China Scholarship CouncilProject(2022YJSLJ02)supported by the Fundamental Research Funds for the Central Universities,China。
