On Covering And Packing Of Special Convex Bodies

Let D,Cn(n = 1,2,...)be plane convex bodies.We say that the sequence {Cn}permits a covering of D if D(?)U Cn.We say that {Cn} can be packed into C if D(?)? Cn and for arbitrary i,j?{1,2,…},i?j,int Ci ? int Cj=?.Denote by T the right triangle with leg lengths 1 and(?)3.Let ?n denote the least number r with the property that the triangle T can be covered by n closed circular discs of radius r.In chapter 1 we study loosest circle coverings of T and get the following results:?1 =1,?2=(?)3/3,?=1/2,?4=(?)3-(?)6/2,?5 =1/3,?6=(?)3/6,and ?5+2n<2/6+n(n?N*),Let ?n denote the greatest number r with the property that n congruent circular discs of radius r can be packed into the triangle T.In chapter 2 we consider densest packings T by n congruent circular discs and get the following results:r1 =(?)3-1/2,r2=2-(?)3,r3= 2(3)/11,r4=(?)3-1/4,r6=4(?)3/12,Denote by ?n(P)the greatest number ? such that n homothetic copies of P with homothety ratio A can be packed into P.In chapter 3 we consider the packings of P with 3 equal homothetic copies of P and get the following results:If P is a regular heptagon,then ?3(P)= 1+cos?/7+cos3?/14sin?/7/2(1|cox?/7|cos3?/14sin?/7|cot 3?/14sin?/7 sin?/7sin3?/14 tan?/7)?0.4606.If P is a regular octagon,then ?3(P)=5+2(?)2/17.