Packing problems

The problem of finding the smallest ball such that k≤n+1, and in an infinite-dimensional Hilbert space with no restrictions. It is worth describing in detail here, to give a flavor of the general problem. In this case, a configuration of (k−1) 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 k open unit balls centered at x1,…,xk be the centers of x0. Consider the map from the finite set xj in the corresponding 1≤i<j≤k, 1≤j≤k one has k disjoint unit open balls in a ball of radius r if and only if {ej}j is an orthonormal basis, are disjoint and included in a ball of radius r is

People determine the number of spherical objects of given diameter R that will pack R the spheres arrange to ordered structures, called columnar structures.

People determine the minimum radius dn, between points. Optimal solutions have been proven for dn, between points. To convert between these two formulations of the problem, the square side for unit circles will be n < 300.

In tiling or tessellation problems, there are to be no gaps, nor overlaps. Many of the puzzles of this type involve packing rectangles or polyominoes into a larger rectangle or other square-like shape.

There are significant theorems on tiling rectangles (and cuboids) in rectangles (cuboids) with no gaps or overlaps:

An a × b rectangle can be packed with 1 × n strips if and only if n divides a or n divides b.

de Bruijn's theorem: A box can be packed with a harmonic brick a × a b × a b c if the box has dimensions a p × a b q × a b c r for some natural numbers p, q, r (i.e., the box is a multiple of the brick.)

The study of polyomino tilings largely concerns two classes of problems: to tile a rectangle with congruent tiles, and to pack one of each n-omino into a rectangle.

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.

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.

• Bin packing problem
• Close-packing of equal spheres
• Conway puzzle
• Covering problem
• Cutting stock problem
• Ellipsoid packing
• Kissing number problem
• Knapsack problem
• Random close pack
• Set packing
• Slothouber–Graatsma puzzle
• Strip packing problem
• Tetrahedron packing
• Tetris

• Weisstein, Eric W. "Klarner's Theorem". MathWorld.
• Weisstein, Eric W. "de Bruijn's Theorem". MathWorld.

• Optimizing Three-Dimensional Bin Packing

Many puzzle books as well as mathematical journals contain articles on packing problems.

• Links to various MathWorld articles on packing
• MathWorld notes on packing squares.
• Erich's Packing Center
• www.packomania.com A site with tables, graphs, calculators, references, etc.
• "Box Packing" by Ed Pegg, Jr., the Wolfram Demonstrations Project, 2007.
• Best known packings of equal circles in a circle, up to 1100

• Circle packing challenge problem in Python