Abstract
In this paper, tiling a plane with equilateral semi-regular convex polygons is considered, and, that is, tiling with equilateral polygons of 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 , whereare the non-negative integers which are not equal to zero at the same time, and 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.
Highlights
The analysis is performed by determining the set of all integer solutions of the corresponding Diophantine equation in the form of t ⋅α + s ⋅ β =2π, where t, s are the non-negative integers which are not equal to zero at the same time, and α, β are the interior angles of a semi-regular equilateral polygon from the characteristic angle
Tiling a plane with equilateral convex semi-regular polygons differs from tiling a plane with regular ones, and it belongs to a special group of tiling
Based on the characteristics of the semi-regular equilateral polygons, the following types of tiling a plane with semi-regular polygons can be differentiated: A) Tiling a plane with semi-regular polygons when the equal number of semi-regular polygons of the same type meet at each node; B) Tiling a plane with semi-regular polygons when semi-regular equilateral polygons of different types and equal sides meet at one node; C) Tiling a plane with semi-regular polygons when semi-regular equilateral polygons of different types and different sides meet at one node
Summary
The problem of tiling comes down to determining all possible divisions of a plane with the polygons: 1) The division of a plane with regular polygons, or when all the polygons and all the nodes are mutually equal. Such tiling is called a regular one. 2) The division of a plane in a way that several types of regular polygons meet at a node. Such tiling a plane is called an Archimedes one, or a semi-regular one.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
