Packing problems
Many of these problems can be related to real-life packaging, storage and transportation issues.
Each packing problem has a dual covering problem, which asks how many of the same objects are required to completely cover every region of the container, where objects are allowed to overlap.
More commonly, the aim is to pack all the objects into as few containers as possible.
[1] In some variants the overlapping (of objects with each other and/or with the boundary of the container) is allowed but should be minimized.
This problem is relevant to a number of scientific disciplines, and has received significant attention.
The Kepler conjecture postulated an optimal solution for packing spheres hundreds of years before it was proven correct by Thomas Callister Hales.
Many other shapes have received attention, including ellipsoids,[2] Platonic and Archimedean solids[3] including tetrahedra,[4][5] tripods (unions of cubes along three positive axis-parallel rays),[6] and unequal-sphere dimers.
[7] These problems are mathematically distinct from the ideas in the circle packing theorem.
The related circle packing problem deals with packing circles, possibly of different sizes, on a surface, for instance the plane or a sphere.
Cubes can easily be arranged to fill three-dimensional space completely, the most natural packing being the cubic honeycomb.
No other Platonic solid can tile space on its own, but some preliminary results are known.
One of the best packings of regular dodecahedra is based on the aforementioned face-centered cubic (FCC) lattice.
Tetrahedra and octahedra together can fill all of space in an arrangement known as the tetrahedral-octahedral honeycomb.
The rectangular cuboids to be packed can be rotated by 90 degrees on each axis.
The problem of finding the smallest ball such that k disjoint open unit balls may be packed inside it has a simple and complete answer in n-dimensional Euclidean space if
It is worth describing in detail here, to give a flavor of the general problem.
In this case, a configuration of k pairwise tangent unit balls is available.
dimensional simplex with edge 2; this is easily realized starting from an orthonormal basis.
A small computation shows that the distance of each vertex from the barycenter is
Moreover, any other point of the space necessarily has a larger distance from at least one of the k vertices.
is an orthonormal basis, are disjoint and included in a ball of radius
People determine the number of spherical objects of given diameter d that can be packed into a cuboid of size
People determine the minimum radius R that will pack n identical, unit volume polyhedra of a given shape.
People are given n unit circles, and have to pack them in the smallest possible container.
People are given n unit squares and have to pack them into the smallest possible container, where the container type varies: In tiling or tessellation problems, there are to be no gaps, nor overlaps.
A classic puzzle of the second kind is to arrange all twelve pentominoes into rectangles sized 3×20, 4×15, 5×12 or 6×10.
Packing of irregular objects is a problem not lending itself well to closed form solutions; however, the applicability to practical environmental science is quite important.
For example, irregularly shaped soil particles pack differently as the sizes and shapes vary, leading to important outcomes for plant species to adapt root formations and to allow water movement in the soil.
[17] The problem of deciding whether a given set of polygons can fit in a given square container has been shown to be complete for the existential theory of the reals.
[18] Many puzzle books as well as mathematical journals contain articles on packing problems.
