The Study About The H-Points Of Convex H-Polygon On Regular Hexagonal Archimedean Tiling

A planar tiling T is said to be an Archimedean tiling if it is an edge-to-edge tiling with a regular polygon as the tiling element and the vertex characteristics of each tiling vertex are the same.There 11 kinds of Archimedean tilings and the Archimedean tiling is generally marked by the vertex characteristics of the tiling.The hexagonal Archimedean tiling is a planar tiling consisting of regular hexagons with unit edge.Let H be the set of vertices of hexagonal Archimedean tiling,a point of H is called an H-point,a simple polygon whose corners lie in H is called an H-polygon.A polygon is a convex polygon if the line segment joining any two points on the polygon remains on the polygon.This paper studies the relationship between the interior H-points and boundary H-points of the H-triangle and Hquadrilateral in the hexagonal Archimedean tiling.For the conjecture Kolodziejczyk proposed in 2007 about the boundary H-points of H-polygons,that is,any convex Hpolygon P with exactly i(P)≥1 interior H-points can have at most 3i(P)+7 boundary H-points,which is proved to be correct.For the H-triangle with k interior H-points and 3k+5 boundary H-points,firstly,by analyzing the triple(α,β,γ)of H-triangle,determine the triples which meet the requirements.Then,excludes the impossible triples by using the theory of level and the distribution characteristics of tiling points.Finally,it shows that there exists H-triangle with k interior H-points and 3k+5 boundary H-points.And there are only two types of configurations,the specific constructions of these two configurations are given.For the study of the boundary H-points of H-quadrilateral P,firstly,the positions of Hpoints and boundary H-points of H-quadrilateral P are discussed by classification,and it is proved that any convex H-quadrilateral P with exactly i(P)≥4 interior H-points can have at most 3i(P)+7 boundary H-points by using existing conclusions and related argumentation methods.Then for the convex H-polygons with i(P)=1 and 2,it is proved that the number of their boundary H-points is 3i(P)+6 at most.That is,the convex Hquadrilateral with one interior H-point can have at most 9 boundary H-points,the convex H-quadrilateral with 2 interior H-points can have at most 12 boundary H-points.Thus,the related problems of the boundary H-number of H-polygons are perfected.