On Coverings And Packings Of Convex Bodies

Let C,D be planar convex polygons and let C1,C2,... be the homothetic copies of C. We say that the sequence{Cn} permits a covering of D if D(?)∪Cn. We say that {Cn} can be packed into D if D (?)∪Cn and that they have pairwise disjoint interiors.In chapter 2 we look at parallel covering and packing of a unit square with sequences of equilateral triangles and get the following results:Any (finite or infinite) sequence of equilateral triangles permits a parallel covering of the unit square S provided that the total area of the equilateral triangles is not less than 2+(?).Any (finite or infinite) sequence of equilateral triangles can be parallel packed into S provided that the total area of the equilateral triangles does not exceed(?)/6.In chapter 3 we consider the parallel covering of an isosceles trapezoid with bases of lengths 1 and 2, the height (?)/2 with sequences of squares and get the following resultsAny (finite or infinite) sequence of squares permits a parallel covering of the right triangle with leg lengths 1, (?) provided that the total area of the squares is not less than 4.Any (finite or infinite) sequence of squares permits a parallel covering of the isosceles trapezoid with bases of lengths 1 and 2, the height (?)/2 provided that the total area of the squares is not less than 4.A covering of H is said to be minimal if there is no other covering of H using a set of hypercubes S’, where tCn} with We denote by gd(n) the smallest size of a minimal covering using a set of n hypercubes. That is, gd(n)= min{s(C):C is a minimal covering of the unit hypercube with n hypercubes}.In chapter 4 we consider covering unit hypercube with small d-dimensional hyper-cubes and get the following results:If C is a minimal covering of the d-dimensional unit hypercube and C has n d-dimensional hypercubes, then gd(n)< s(C).For n≥2d+1, we have gd(n)≤ 2d-1+δ, where δ is a position value that can be made as close to 0 as desired. For any n≥2d,we have that gd(n)≥2d-1.