Parallel Packing And Covering Of Plane Convex Bodies With Squares

Let C,D be plane convex polygons and let G1,C2,… be the homothetic copies of C.We say that the sequence {Cn} permits a covering of D if D C(?)Cn,We say that{Cn} can be packed into D if D(?)Cn and that they have pairwise disjoint interiors.In particular,a covering or packing of D with {Cn} is called parallel if there is a side of D such that each Cn has a side parallel to this side of D.In chapter 1 we look at parallel covering of an isosceles trapezoid with sequences of squares and get the following results:Any finite or infinite sequence of squares permits a parallel covering of the isosceles trapezoid T with bases of lengths c and 2a + c,with height a provided that the total area of the squares is not less than(2a + c)2.Any finite or infinite sequence of squares permits a parallel covering of the isosceles trapezoid T with bases of lengths c and 2a + c?and with height b provided that the total area of the squares is not less than(2a + c)2,where b<a.In chapter 2 we consider the parallel covering and packing of' a golden triangle with base length 1 and with height h =(?)with sequences of squares and get the following results:Any finite or infinite sequence of squares permits a parallel covering of the golden triangle provided that the total area of the squares is not less than h2.Any finite or infinite sequence of squares permits a parallel packing of the golden triangle provided that the total area of the squares does not exceed 2h2/(2h+1)2.