Packing problem: Wikis

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Encyclopedia

Packing problems are one area where mathematics meets puzzles (recreational mathematics). Many of these problems stem from real-life problems with packing items.

In a packing problem, you are given:

• one or more (usually two- or three-dimensional) containers
• several 'goods', some or all of which must be packed into this container

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, allowing touching, but without overlap.

Packing infinite space

Many of these problems, when the container size is increased in all directions, become equivalent to the problem of packing objects as densely as possible in infinite Euclidean space. This problem is relevant to a number of scientific disciplines, and has received significant attention. The Kepler conjecture postulated an optimal solution for spheres hundreds of years before it was proven correct by Hales. Many other shapes have received attention, including ellipsoids, tetrahedra, icosahedra, and unequal-sphere dimers.

Spheres into a Euclidean ball

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 $\scriptstyle k\leq n+1$, 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 pairwise tangent unit balls is available. Place the centers at the vertices a1,..,ak of a regular $\scriptstyle(k+1)$ 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 $\scriptstyle\sqrt{2\big(1-\frac{1}{k} \big)}$. Moreover, any other point of the space necessarily has a larger distance from at least one of the $\scriptstyle k$ vertices. In terms of inclusions of balls, the $\scriptstyle k$ open unit balls centered at $\scriptstyle a_1,..,a_k$ are included in a ball of radius $\scriptstyle r_k:=1+\sqrt{2\big(1-\frac{1}{k}\big)}$, which is minimal for this configuration.

To show that this configuration is optimal, let $\scriptstyle x_1,...,x_k$ be the centers of $\scriptstyle k$ disjoint open unit balls contained in a ball of radius $\scriptstyle r$ centered at a point $\scriptstyle x_0$. Consider the map from the finite set $\scriptstyle\{x_1,..x_k\}$ into $\scriptstyle\{a_1,..a_k\}$ taking $\scriptstyle x_j$ in the corresponding $\scriptstyle a_j$ for each $\scriptstyle 1\leq j\leq k$. Since for all $\scriptstyle 1\leq i, $\scriptstyle \|a_i-a_j\|=2\leq\|x_i-x_j\|$ this map is 1-Lipschitz and by the Kirszbraun theorem it extends to a 1-Lipschitz map that is globally defined; in particular, there exists a point $\scriptstyle a_0$ such that for all $\scriptstyle1\leq j\leq k$ one has $\scriptstyle\|a_0-a_j\|\leq\|x_0-x_j\|$, so that also $\scriptstyle r_k\leq1+\|a_0-a_j\|\leq 1+\|x_0-x_j\|\leq r$. This shows that there are $\scriptstyle k$ disjoint unit open balls in a ball of radius $\scriptstyle r$ if and only if $\scriptstyle r\geq r_k$. Notice that in an infinite dimensional Hilbert space this implies that there are infinitely many disjoint open unit balls inside a ball of radius $\scriptstyle r$ if and only if $\scriptstyle r\geq 1+\sqrt{2}$. For instance, the unit balls centered at $\scriptstyle\sqrt{2}e_j$, where $\scriptstyle\{e_j\}_j$ is an orthonormal basis, are disjoint and included in a ball of radius $\scriptstyle 1+\sqrt{2}$ centered at the origin. Moreover, for $\scriptstyle r<1+\sqrt{2}$, the maximum number of disjoint open unit balls inside a ball of radius r is $\scriptstyle\big\lfloor \frac{2}{2-(r-1)^2}\big\rfloor$.

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.

Packing circles

There are many other problems involving packing circles into a particular shape of the smallest possible size. Note that these problems are mathematically distinct from the ideas in the circle packing theorem.

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. [1]

Circles in circle

Some of the more non-trivial circle packing problems are packing unit circles into the smallest possible larger circle.

Minimum solutions:[citation needed]

1 1
2 2
3 2.154...
4 2.414...
5 2.701...
6 3
7 3
8 3.304...
9 3.613...
10 3.813...
11 3.923...
12 4.029...
13 4.236...
14 4.328...
15 4.521...
16 4.615...
17 4.792...
18 4.863...
19 4.863...
20 5.122...

Circles in square

Pack n unit circles into the smallest possible square. This is closely related to spreading points in a unit square with the objective of finding the greatest minimal separation, dn, between points[1]. To convert between these two formulations of the problem, the square side for unit circles will be L=2+2/dn.

Current best solutions:

Number of circles Square size dn[2]
1 2
2 3.414... 1.414...∗
3 3.931... 1.035...∗
4 4 1*
5 4.828... 0.707...∗
6 5.328... 0.601...∗
7 5.732... 0.536...∗
8 5.863... 0.518...∗
9 6 0.5 ∗
10 6.747... 0.421...
11 7.022... 0.398...
12 7.144... 0.389...
13 7.463... 0.366...
14 7.796... 0.345...∗
15 7.932... 0.337...
16 8 0.333...∗
17 8.532... 0.306...
18 8.656... 0.300...
19 8.907... 0.290...
20 8.978... 0.287...

∗ indicates that the solution is known to be optimal.

Circles in isosceles right triangle

Pack n unit circles into the smallest possible isosceles right triangle (lengths shown are length of leg)

Minimum solutions:[citation needed]

Number of circles Length
1 3.414...
2 4.828...
3 5.414...
4 6.242...
5 7.146...
6 7.414...
7 8.181...
8 8.692...
9 9.071...
10 9.414...
11 10.059...
12 10.422...
13 10.798...
14 11.141...
15 11.414...

Circles in equilateral triangle

Pack n unit circles into the smallest possible equilateral triangle (lengths shown are side length).

Minimum solutions:[citation needed]

Number of circles Length
1 3.464...
2 5.464...
3 5.464...
4 6.928...
5 7.464...
6 7.464...
7 8.928...
8 9.293...
9 9.464...
10 9.464...
11 10.730...
12 10.928...
13 11.406...
14 11.464...
15 11.464...

Circles in regular hexagon

Pack n unit circles into the smallest possible regular hexagon (lengths shown are side length).

Minimum solutions:[citation needed]

Number of circles Length
1 1.154...
2 2.154...
3 2.309...
4 2.666...
5 2.999...
6 3.154...
7 3.154...
8 3.709...
9 4.011...
10 4.119...
11 4.309...
12 4.309...
13 4.618...
14 4.666...
15 4.961...

Packing squares

Squares in square

A problem is the square packing problem, where one must determine how many squares of side 1 you can pack into a square of side a. Obviously, if a is an integer, the answer is a2, but the precise, or even asymptotic, amount of wasted space for a a non-integer is open.

Proven minimum solutions:[3]

Number of squares Square size
1 1
2 2
3 2
4 2
5 2.707 (2 + 2 −1/2)
6 3
7 3
8 3
9 3
10 3.707 (3 + 2 −1/2)

Other results:

• If you can pack n2 − 2 squares in a square of side a, then an.[4]
• The naive approach (side matches side) leaves wasted space of less than 2a + 1.[3]
• The wasted space is asymptotically o(a7/11).[5]
• The wasted space is not asymptotically o(a1/2).[6]
• 11 unit squares cannot be packed in a square of side less than $2+2\sqrt{4/5}$.[7]

Squares in circle

Pack n squares in the smallest possible circle.

Minimum solutions:[citation needed]

1 0.707...
2 1.118...
3 1.288...
4 1.414...
5 1.581...
6 1.688...
7 1.802...
8 1.978...
9 2.077...
10 2.121...
11 2.215...
12 2.236...

Tiling

In tiling or tesselation 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.

Rectangles in rectangle

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

Klarner's theorem: An a × b rectangle can be packed with 1 × n strips iff n | a or n | b.[8]
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.)

Polyominoes

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.

Notes

1. ^ Croft, Hallard T.; Falconer, Kenneth J.; Guy, Richard K. (1991). Unsolved Problems in Geometry. New York: Springer-Verlag. pp. 108–110. ISBN 0-387-97506-3.
2. ^ Croft, Hallard T.; Falconer, Kenneth J.; Guy, Richard K. (1991). Unsolved Problems in Geometry. New York: Springer-Verlag. p. 108. ISBN 0-387-97506-3.
3. ^ a b Erich Friedman, "Packing unit squares in squares: a survey and new results", The Electronic Journal of Combinatorics DS7 (2005).
4. ^ M. Kearney and P. Shiu, "Efficient packing of unit squares in a square", The Electronic Journal of Combinatorics 9:1 #R14 (2002).
5. ^ P. Erdős and R. L. Graham, "On packing squares with equal squares", Journal of Combinatorial Theory, Series A 19 (1975), pp. 119–123.
6. ^ K. F. Roth and R. C. Vaughan, "Inefficiency in packing squares with unit squares", Journal of Combinatorial Theory, Series A 24 (1978), pp. 170-186.
7. ^ W. Stromquist, "Packing 10 or 11 unit squares in a square", The Electronic Journal of Combinatorics 10 #R8 (2003).
8. ^ Wagon, Stan (August-September 1987). "Fourteen Proofs of a Result About Tiling a Rectangle". The American Mathematical Monthly 94 (7): 601-617. Retrieved 6 Jan 2010.