The Study About The Boundary H-points Of H-polygon On Regular Hexagonal Archimdean Tiling

The edge-by-edge planar tiling whose tiling elements are regular polygons and vertex feature is same is called planar Archimdean tiling.If the tiling elements around any vertex of planar tiling are n1-polygon,n2-polygon and so on by a arrangement of ring order successively,then the vertex is called[n1.n2.…]type.We know there are 11 kind of planar Archimdean tiling.[6.6.6]tiling is a planar Archimdean tiling by regular hexagons with unit edge.H is the set of vertices of[6.6.6]tiling.A simple polygon in R2 whose corners lie in H is called an H-polygon.In this paper,we study the problem of H-points on the boundary of H-polygon with given internal H-points.Some conclusions have been reached on this issue:the number of H-points on the boundary of H-triangle with 1 H-point is not more than 10;the number of H-points in the boundary of H-triangle with 3 H-points is not more than 16;any H-triangle with exactly k interior H-points can have at most 3k+7 boundary H-points and cannot have 3k+6 H-points on its boundary.They also proved that when k?5,there is no H-triangle with the boundary H-points of 3k,and H-quadrilateral with 1 interior H-points,its boundary H-points is not more than 10.However,the research on H-triangles and H-quadrangles is not very comprehensive.So we need study it further.In this note,we will continue to explore whether the number of H-points on the boundary of the H-triangle can be 3k?6.In the[6.6.6]tiling graph,we can determine when 3?k?10 the conjecture is correct by drawing.We use the notion of level of an H-triangle and the properties of triples(?,?,?)to prove the conjecture.We prove that it wrong for k?11,that is to say,if H-triangle contains k?11 H-points,then the number of boundary H-points cannot be 3k-6.As for the conjecture of the number of H-points on the H-quadrilateral boundary containing k H-points.In this paper,we will use the theories of planar tiling to explore the H-quadrilateral boundary H-points with 2 or 3 H-points in its relative interior.In other words,if H-quadrilateral with 2 H-points in its relative interior,its boundary H-points is not more than 13,and H-quadrilateral with 3 H-points in its relative interior,its boundary H-points is not more than 16.