Some tessellations with convex polygons were already discovered by ancient Greeks, but nowadays more and more mathematicians are engaged in solving problems of this branch of geometry. The Thesis presents the problem of tiling with convex polygons. Pentagons are the most interesting of all polygons. Until the year of 2017 we didn't know, whether we know all convex pentagons that tile the plane.
We divide convex pentagons that tile the plane into different types according to the pattern of tilings. Special conditions apply for convex pentagons of a certain type. Different tiles form different tilings depending on the properties of these pentagons. First types of convex pentagons were described by Reinhardt in 1918 and the last known type was discovered by Scientists of the University of Washington Bothell in 2015.
Nevertheless, the problem of finding all convex pentagons that tile the plane cannot be solved only by listing the cases. In order to understand the problem, it is necessary to observe the properties of both tiles and tilings. In 1985, M. D. Hirschhorn and D. C. Hunt were able to prove the completeness of the list of equilateral pentagons that tile the plane. Russian mathematician O. Bagina came to the same conclusion in 2004. The proof of the completeness of the list of pentagons, whose tilings are edge to edge, was presented independently from Sugimoto in 2012 and Bagina in 2011. In the thesis we show insights into proofs of Hirschhorn and Hunt, Bagina and Sugimoto, as their discoveries are groundbreaking in finding the answer to the question of tiling of the plane with convex pentagons.
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