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| discrete geometry | |
| tiling puzzles | |
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In a packing problem, you are given:
Usually the packing must be without gaps or overlaps, but in some packing problems the overlapping (of goods with each other and/or with the boundary of the container) is allowed but should be minimised. In others, gaps are allowed, but overlaps are not (usually the total area of gaps has to be minimised). PROBLEMS There are many different types of packing problems. Usually they involve finding the maximum number of a certain shape that can be packed into a larger, perhaps different shape. Sphere in Cuboid A classic problem is the Sphere Packing problem, where one must determine how many Spherical objects of given diameter ''d'' can be packed into a Cuboid of size ''a'' × ''b'' × ''c''. This is one of the hardest problems in this category . Packing Circles There are many other problems involving packing circles into a particular shape of the smallest possible size. Hexagonal packing Circles (and their counterparts in other dimensions) can never be packed with 100% efficiency in Dimensions larger than one (in a one dimensional universe, circles merely consist of two points). That is, there will always be unused space if you are only packing circles. The most efficient way of packing circles, Hexagonal Packing produces approximately 90% efficiency. {Link without Title} Circles in circle Some of the more non-trivial circle packing problems are packing unit circles into the smallest possible larger circle. Proven Minimum Solutions: Circles in square Pack ''n'' unit circles into the smallest possible square. Proven Minimum Solutions: Circles in isosceles right triangle Pack ''n'' unit circles into the smallest possible isosceles right triangle (lengths shown are length of leg) Proven Minimum Solutions: Circles in equilateral triangle Pack ''n'' unit circles into the smallest possible equilateral triangle (lengths shown are side length). Proven Minimum Solutions: Circles in regular hexagon Pack ''n'' unit circles into the smallest possible regular hexagon (lengths shown are side length). Proven Minimum Solutions: Packing squares Squares in square A problem is the square packing problem, where one must determine how many Square s of side 1 you can pack into a square of side ''a''. Obviously, here if ''a'' is an integer, the answer is ''a''2, but the precise, or even Asymptotic , amount of wasted space for ''a'' a non-integer is open. Proven Minimum Solutions: Other known results:
Squares in circle Pack ''n'' squares in the smallest possible circle. Proven Minimum Solutions: TILING In this type of problem there are to be no gaps, nor overlaps. Most of the time this involves packing Rectangles or Polyominoes into a larger rectangle or other square-like shape. rectangles with rectangles There are significant theorems on tiling rectangles (and cuboids) in rectangles (cuboids) with no gaps or overlaps: |
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