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=φ.Denote by T the right triangle with leg lengths 1 and(?)3.Let τn 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:τ1 =1,τ2=(?)3/3,τ=1/2,τ4=(?)3-(?)6/2,τ5 =1/3,τ6=(?)3/6,and τ5+2n<2/6+n(n∈N*),Let τn 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 λn(P)the greatest number λ 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 λ3(P)= 1+cosπ/7+cos3π/14sinπ/7/2(1|coxπ/7|cos3π/14sinπ/7|cot 3π/14sinπ/7 sinπ/7sin3π/14 tanπ/7)≈0.4606.If P is a regular octagon,then λ3(P)=5+2(?)2/17.