Difference between revisions of "Packing"
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| − | + | {{MSC|52C20|52C22}} | |
| + | {{TEX|done}} \( \def \Z { {\cal Z}} \) | ||
| − | A packing of a (finite or infinite) family of sets $M_i$ in a set $A$ is, in its strict sense, | + | A '''packing''' of a (finite or infinite) family of sets $M_i$ in a set $A$ is, in its strict sense, |
any pairwise disjoint family of subsets $M_i\subset A$. | any pairwise disjoint family of subsets $M_i\subset A$. | ||
| − | However, in geometry, the sets $M_i$ are often closed domains, | + | However, in geometry, for packings in Euclidean spaces, |
| − | and then this condition is relaxed to requiring only | + | on an $n$-dimensional sphere, in a closed domanin, or in some other manifold, the sets $M_i$ are often closed domains, |
| − | that the interiors of the sets are pairwise disjoint. | + | and then this condition is relaxed to requiring only that the interiors of the sets are pairwise disjoint. |
| − | Lattice packings | + | ==== Lattice packings ==== |
As a special case, in vector spaces $V$, such as $\R^d$, | As a special case, in vector spaces $V$, such as $\R^d$, | ||
| − | packings of translates $ \{ M+v \mid v \in Z \} $, of a set $ M \subset V $ are considered. | + | packings of translates $ \{ M+v \mid v \in \Z \} $, of a set $ M \subset V $ |
| − | If the set $Z \subset V $ of translation vectors is a lattice, | + | (also called the packing $(M,\Z)$ of $M$ ''by'' $\Z$) are considered. |
| − | then the packing is called a lattice packing. | + | If the set $ \Z \subset V $ of translation vectors is a point lattice, |
| + | then the packing is called a '''[[lattice packing]]'''. | ||
In particular, such packings are investigated in the geometry of numbers and in discrete geometry. | In particular, such packings are investigated in the geometry of numbers and in discrete geometry. | ||
| − | Sphere packings | + | ==== Sphere packings ==== |
Packings of congruent spheres are considered both in the geometry of numbers | Packings of congruent spheres are considered both in the geometry of numbers | ||
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A central problem is finding the densest packing, | A central problem is finding the densest packing, | ||
and the densest lattice packing, of congruent spheres in $\R^d$. | and the densest lattice packing, of congruent spheres in $\R^d$. | ||
| − | For $d=3$, the problem (known as Kepler conjecture or Kepler problem) | + | For $d=3$, the problem (known as Kepler conjecture or [[Kepler problem]]) |
to decide whether there is a better packing than the densest lattice packing | to decide whether there is a better packing than the densest lattice packing | ||
was a famous open problem that was recently solved by Hales (1998). | was a famous open problem that was recently solved by Hales (1998). | ||
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However, Hales and his team are working on a computer-verifyable version of the proof. | However, Hales and his team are working on a computer-verifyable version of the proof. | ||
| − | Tilings | + | ==== Tilings ==== |
| − | A tiling is a packing without gaps, | + | A '''[[tiling]]''' is a packing without gaps, |
i.e., such that the $M_i$ are also a covering of $A$. | i.e., such that the $M_i$ are also a covering of $A$. | ||
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====References==== | ====References==== | ||
| − | + | {| | |
| + | |- | ||
| + | |valign="top"|{{Ref|Ba}}|| valign="top"| E.P. Baranovskiĭ, "Packings, coverings, partitionings and certain other arrangements in spaces with constant curvature" (1969) Algebra. Topology. Geometry. (1967) (Russian) pp. 189–225 Akad. Nauk SSSR Vsesojuz. Inst. Naučn. Tehn. Informacii, Moscow {{MR|0257885}} | ||
| + | |- | ||
| + | |valign="top"|{{Ref|CoSl}}||valign="top"| J.H. Conway, N.J.A. Sloane, "Sphere packing, lattices and groups", Springer (1988) {{MR|0920369}} | ||
| + | |- | ||
| + | | valign="top"|{{Ref|FeTo}}||valign="top"| L. Fejes Tóth, "Lagerungen in der Ebene, auf der Kugel und im Raum", Springer (1972) {{MR|0353117}} {{ZBL|0229.52009}} | ||
| + | |- | ||
| + | | valign="top"|{{Ref|Ro}}||valign="top"| C.A. Rogers, "Packing and covering", Cambridge Univ. Press (1964) {{MR|0172183}} {{ZBL|0176.51401}} | ||
| + | |- | ||
| + | |} | ||
Latest revision as of 09:47, 5 March 2012
2020 Mathematics Subject Classification: Primary: 52C20 Secondary: 52C22 [MSN][ZBL] \( \def \Z { {\cal Z}} \)
A packing of a (finite or infinite) family of sets $M_i$ in a set $A$ is, in its strict sense, any pairwise disjoint family of subsets $M_i\subset A$.
However, in geometry, for packings in Euclidean spaces, on an $n$-dimensional sphere, in a closed domanin, or in some other manifold, the sets $M_i$ are often closed domains, and then this condition is relaxed to requiring only that the interiors of the sets are pairwise disjoint.
Lattice packings
As a special case, in vector spaces $V$, such as $\R^d$, packings of translates $ \{ M+v \mid v \in \Z \} $, of a set $ M \subset V $ (also called the packing $(M,\Z)$ of $M$ by $\Z$) are considered. If the set $ \Z \subset V $ of translation vectors is a point lattice, then the packing is called a lattice packing. In particular, such packings are investigated in the geometry of numbers and in discrete geometry.
Sphere packings
Packings of congruent spheres are considered both in the geometry of numbers and in discrete geometry, and have applications in coding theory. A central problem is finding the densest packing, and the densest lattice packing, of congruent spheres in $\R^d$. For $d=3$, the problem (known as Kepler conjecture or Kepler problem) to decide whether there is a better packing than the densest lattice packing was a famous open problem that was recently solved by Hales (1998). With the help of massive computer calculations he showed that the densest lattice packing of spheres is optimal. This result is generally considered as correct but because of its size it has not yet been verified independently. However, Hales and his team are working on a computer-verifyable version of the proof.
Tilings
A tiling is a packing without gaps, i.e., such that the $M_i$ are also a covering of $A$.
References
| [Ba] | E.P. Baranovskiĭ, "Packings, coverings, partitionings and certain other arrangements in spaces with constant curvature" (1969) Algebra. Topology. Geometry. (1967) (Russian) pp. 189–225 Akad. Nauk SSSR Vsesojuz. Inst. Naučn. Tehn. Informacii, Moscow MR0257885 |
| [CoSl] | J.H. Conway, N.J.A. Sloane, "Sphere packing, lattices and groups", Springer (1988) MR0920369 |
| [FeTo] | L. Fejes Tóth, "Lagerungen in der Ebene, auf der Kugel und im Raum", Springer (1972) MR0353117 Zbl 0229.52009 |
| [Ro] | C.A. Rogers, "Packing and covering", Cambridge Univ. Press (1964) MR0172183 Zbl 0176.51401 |
How to Cite This Entry:
Packing. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Packing&oldid=20852
Packing. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Packing&oldid=20852
This article was adapted from an original article by A.V. Malyshev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article