The Study About The Number Of The Interior F-points Of Convex F-polygonon The [3.6.3.6]-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 are₁n-polygon,₂n-polygon and so on by a arrangement of ring order successively,then the vertex is called[n₁.n₂.n₃?43?]-type.There are 11 kinds of planar Archimdean tiling,this paper studies[.3.6.36]-tiling,that is a planar Archimdean tiling by regular triangles and regular hexagons with unit edge.Let F be the set of vertices of[.3.6.36]-tiling.A point of F is called an F-point,a convex polygon in R² whose corners lie in F is called a convex F-polygon.Let C denote the set of all centers of the regular hexagons in[.3.6.36]-tiling.A point of C is called a C-point.A convex polygon inR²whose corners lie in C is called a convex C-polygon.Clearly,F?C forms a planar Archimdean[3.3.3.3.3.3]-tiling with unit edge.Let Tbe the set of vertices of the[.3.33.3..33]-tiling,a point of T is called a T-point,that isT?28?F?C.For an F-polygon K we denote_Fb?K??28?F?40??Kandi_F?K??28?F?40?intK,whereb_F?K?is the number of boundary F-points of Kandi_F?K?is the number of interior F-points of K.LetKbe a convex F-polygon in[.3.6.36]-tiling.Denote byH?K?the interior hull of K,that is,the convex hull of the T-points in the interior of K.A convex C-polygon Q is the convex hull of the C-points in the interior of K.We define the counting functionf?v??28?min{i_F?K?:v_F?K??28?v},wherev_F?K?denote the number of vertices of F-polygon K.This paper proves the exact value of the minimum counting function of the number of interior F-points of a convex F-polygon K in[.3.6.36]-tiling by using the theories of planar tiling and convex sets to analyze the relation between the number of vertices of convex C-polygon Q and interior hullH?K?of K,and the number of boundary F-points or the interior F-points of Q theoretically.