The Study About The Number Of The Interior H-points Of Convex H-polygon On The Regular Hexagonal Archimedean Tiling

[6.6.6]-tiling is a planar Archimdean tiling by regular hexagons with unit edge.Let H be the set of vertices of[6.6.6]-tiling.A point of H is called an H-point,a convex polygon in R2 whose corners lie in H is called a convex H-polygon.Let C denote the set of all centers of the[6.6.6]-tiling.A point of C is called a C-point.A convex polygon in R2 whose corners lie in C is called a convex C-polygon.Clearly,HYC forms a planar Archimdean triangular-tiling with unit edge.Let T be the set of vertices of the triangular-tiling,a point of T is called a T-point,that is T?HYC.For an H-polygon K we denote bH(K)=|HI(?)K| and iH(K)=|HI int K|,where bH(K)is the number of boundary H-points of K and iH(K)is the number of interior H-points of K.Let K be a convex H-polygon in[6.6.6]-tiling.Denote by H(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 a convex hull of the C-points in the interior of K.We define the counting function G(v)= min{iH(K):vH(K)= v},where vH(K),iH(K)denote the number of vertices and the number of interior H-points of H-polygon K respectively.This paper proves and gets a range 10?G(11)?12 of minimum counting function G(11)for the number of interior H-points of a convex H-hendecagon K in[6.6.6]-tiling by the relation between the number of vertices of convex C-polygon Q and interior hull H(K)of K,and theoretical analysis of the number of boundary H-points or the interior H-points of Q.On the basis of studying,this paper gives the minimum counting function G(12)= 12 for a convex H-dodecagon K contains the H-points in[6.6.6]-tiling by integrating new lemmas and proof methods.