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Sphere packing
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Sphere packing
In geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space. The spheres considered are usually all of identical size, and the space is usually three-dimensional Euclidean space. However, sphere packing problems can be generalised to consider unequal spheres, spaces of other dimensions (where the problem becomes circle packing in two dimensions, or hypersphere packing in higher dimensions) or to non-Euclidean spaces such as hyperbolic space.
A typical sphere packing problem is to find an arrangement in which the spheres fill as much of the space as possible. The proportion of space filled by the spheres is called the packing density of the arrangement. As the local density of a packing in an infinite space can vary depending on the volume over which it is measured, the problem is usually to maximise the average or asymptotic density, measured over a large enough volume.
For equal spheres in three dimensions, the densest packing uses approximately 74% of the volume. A random packing of equal spheres generally has a density around 63.5%.
A lattice arrangement (commonly called a regular arrangement) is one in which the points of the lattice form a very symmetric pattern that, in n-dimensional Euclidean space, needs only n vectors to be defined. Lattice arrangements are periodic, and have the property that when the lattice is translated (moved) so that one point is placed where another was, the arrangement is the same as before. Arrangements in which the points do not form a lattice can still be periodic, but also aperiodic, which includes random arrangements. Lattice packings are easier to classify than those that are not lattices due to their high degree of symmetry. Periodic lattices have well-defined densities.
In three-dimensional Euclidean space, the densest packing of equal spheres is achieved by a family of structures called close-packed structures. One method for generating such a structure is as follows. Consider a plane with a compact arrangement of spheres on it. Call it A. For any three neighbouring spheres, a fourth sphere can be placed on top in the hollow between the three bottom spheres. If we do this for half of the holes in a second plane above the first, we create a new compact layer. There are two possible choices for doing this, call them B and C. Suppose that we chose B. Then one half of the hollows of B lies above the centres of the balls in A and one half lies above the hollows of A that were not used for B. Thus the balls of a third layer can be placed either directly above the balls of the first one, yielding a layer of type A, or above the holes of the first layer that were not occupied by the second layer, yielding a layer of type C. Combining layers of types A, B, and C produces various close-packed structures.
Two simple arrangements within the close-packed family correspond to regular arrangements. One is called cubic close packing (or face-centred cubic, "FCC", which is a lattice)—where the layers are alternated in the ABCABC... sequence. The other is called hexagonal close packing ("HCP", which, although being a regular arrangement, is not a lattice), where the layers are alternated in the ABAB... sequence. But many layer stacking sequences are possible (ABAC, ABCBA, ABCBAC, etc.), and still generate a close-packed structure. In all of these arrangements each sphere touches 12 neighbouring spheres, and the average density is
In 1611, Johannes Kepler conjectured that this is the maximum possible density amongst both regular and irregular arrangements—this became known as the Kepler conjecture. Carl Friedrich Gauss proved in 1831 that these packings have the highest density amongst all possible lattice packings. In 1998, Thomas Callister Hales, following the approach suggested by László Fejes Tóth in 1953, announced a proof of the Kepler conjecture. Hales' proof is a proof by exhaustion involving checking of many individual cases using complex computer calculations. Referees said that they were "99% certain" of the correctness of Hales' proof. On 10 August 2014, Hales announced the completion of a formal proof using automated proof checking, removing any doubt.
Some other lattice packings are often found in physical systems. These include the cubic lattice with a density of π/6 ≈ 0.5236, the hexagonal arrangement with a density of π/√27 ≈ 0.6046 and the tetrahedral arrangement with a density of π√3/16 ≈ 0.3401.
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Sphere packing
In geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space. The spheres considered are usually all of identical size, and the space is usually three-dimensional Euclidean space. However, sphere packing problems can be generalised to consider unequal spheres, spaces of other dimensions (where the problem becomes circle packing in two dimensions, or hypersphere packing in higher dimensions) or to non-Euclidean spaces such as hyperbolic space.
A typical sphere packing problem is to find an arrangement in which the spheres fill as much of the space as possible. The proportion of space filled by the spheres is called the packing density of the arrangement. As the local density of a packing in an infinite space can vary depending on the volume over which it is measured, the problem is usually to maximise the average or asymptotic density, measured over a large enough volume.
For equal spheres in three dimensions, the densest packing uses approximately 74% of the volume. A random packing of equal spheres generally has a density around 63.5%.
A lattice arrangement (commonly called a regular arrangement) is one in which the points of the lattice form a very symmetric pattern that, in n-dimensional Euclidean space, needs only n vectors to be defined. Lattice arrangements are periodic, and have the property that when the lattice is translated (moved) so that one point is placed where another was, the arrangement is the same as before. Arrangements in which the points do not form a lattice can still be periodic, but also aperiodic, which includes random arrangements. Lattice packings are easier to classify than those that are not lattices due to their high degree of symmetry. Periodic lattices have well-defined densities.
In three-dimensional Euclidean space, the densest packing of equal spheres is achieved by a family of structures called close-packed structures. One method for generating such a structure is as follows. Consider a plane with a compact arrangement of spheres on it. Call it A. For any three neighbouring spheres, a fourth sphere can be placed on top in the hollow between the three bottom spheres. If we do this for half of the holes in a second plane above the first, we create a new compact layer. There are two possible choices for doing this, call them B and C. Suppose that we chose B. Then one half of the hollows of B lies above the centres of the balls in A and one half lies above the hollows of A that were not used for B. Thus the balls of a third layer can be placed either directly above the balls of the first one, yielding a layer of type A, or above the holes of the first layer that were not occupied by the second layer, yielding a layer of type C. Combining layers of types A, B, and C produces various close-packed structures.
Two simple arrangements within the close-packed family correspond to regular arrangements. One is called cubic close packing (or face-centred cubic, "FCC", which is a lattice)—where the layers are alternated in the ABCABC... sequence. The other is called hexagonal close packing ("HCP", which, although being a regular arrangement, is not a lattice), where the layers are alternated in the ABAB... sequence. But many layer stacking sequences are possible (ABAC, ABCBA, ABCBAC, etc.), and still generate a close-packed structure. In all of these arrangements each sphere touches 12 neighbouring spheres, and the average density is
In 1611, Johannes Kepler conjectured that this is the maximum possible density amongst both regular and irregular arrangements—this became known as the Kepler conjecture. Carl Friedrich Gauss proved in 1831 that these packings have the highest density amongst all possible lattice packings. In 1998, Thomas Callister Hales, following the approach suggested by László Fejes Tóth in 1953, announced a proof of the Kepler conjecture. Hales' proof is a proof by exhaustion involving checking of many individual cases using complex computer calculations. Referees said that they were "99% certain" of the correctness of Hales' proof. On 10 August 2014, Hales announced the completion of a formal proof using automated proof checking, removing any doubt.
Some other lattice packings are often found in physical systems. These include the cubic lattice with a density of π/6 ≈ 0.5236, the hexagonal arrangement with a density of π/√27 ≈ 0.6046 and the tetrahedral arrangement with a density of π√3/16 ≈ 0.3401.
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