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geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, close-packing of equal
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
s is a dense arrangement of congruent spheres in an infinite, regular arrangement (or
lattice Lattice may refer to: Arts and design * Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material * Lattice (music), an organized grid model of pitch ratios * Lattice (pastry), an orna ...
).
Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
proved that the highest average density – that is, the greatest fraction of space occupied by spheres – that can be achieved by a
lattice Lattice may refer to: Arts and design * Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material * Lattice (music), an organized grid model of pitch ratios * Lattice (pastry), an orna ...
packing is :\frac \approx 0.74048. The same
packing density A packing density or packing fraction of a packing in some space is the fraction of the space filled by the figures making up the packing. In simplest terms, this is the ratio of the volume of bodies in a space to the volume of the space itself. I ...
can also be achieved by alternate stackings of the same close-packed planes of spheres, including structures that are aperiodic in the stacking direction. The
Kepler conjecture The Kepler conjecture, named after the 17th-century mathematician and astronomer Johannes Kepler, is a mathematical theorem about sphere packing in three-dimensional Euclidean space. It states that no arrangement of equally sized spheres filling s ...
states that this is the highest density that can be achieved by any arrangement of spheres, either regular or irregular. This conjecture was proven by T. C. Hales. Highest density is known only for 1, 2, 3, 8, and 24 dimensions. Many
crystal A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macro ...
structures are based on a close-packing of a single kind of atom, or a close-packing of large ions with smaller ions filling the spaces between them. The cubic and hexagonal arrangements are very close to one another in energy, and it may be difficult to predict which form will be preferred from first principles.


FCC and HCP lattices

There are two simple regular lattices that achieve this highest average density. They are called face-centered cubic (FCC) (also called cubic close packed) and
hexagonal In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. Regular hexagon A '' regular hexagon'' has ...
close-packed (HCP), based on their
symmetry Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
. Both are based upon sheets of spheres arranged at the vertices of a triangular tiling; they differ in how the sheets are stacked upon one another. The FCC lattice is also known to mathematicians as that generated by the A3
root system In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representatio ...
.


Cannonball problem

The problem of close-packing of spheres was first mathematically analyzed by
Thomas Harriot Thomas Harriot (; – 2 July 1621), also spelled Harriott, Hariot or Heriot, was an English astronomer, mathematician, ethnographer and translator to whom the theory of refraction is attributed. Thomas Harriot was also recognized for his con ...
around 1587, after a question on piling cannonballs on ships was posed to him by Sir
Walter Raleigh Sir Walter Raleigh (; – 29 October 1618) was an English statesman, soldier, writer and explorer. One of the most notable figures of the Elizabethan era, he played a leading part in English colonisation of North America, suppressed rebelli ...
on their expedition to America. Cannonballs were usually piled in a rectangular or triangular wooden frame, forming a three-sided or four-sided pyramid. Both arrangements produce a face-centered cubic lattice – with different orientation to the ground. Hexagonal close-packing would result in a six-sided pyramid with a hexagonal base. The
cannonball problem In the mathematics of figurate numbers, the cannonball problem asks which numbers are both square and square pyramidal. The problem can be stated as: given a square arrangement of cannonballs, for what size squares can these cannonballs also be a ...
asks which flat square arrangements of cannonballs can be stacked into a square pyramid.
Édouard Lucas __NOTOC__ François Édouard Anatole Lucas (; 4 April 1842 – 3 October 1891) was a French mathematician. Lucas is known for his study of the Fibonacci sequence. The related Lucas sequences and Lucas numbers are named after him. Biography Lucas ...
formulated the problem as the
Diophantine equation In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a ...
\sum_^ n^2 = M^2 or \frac N(N+1)(2N+1) = M^2 and conjectured that the only solutions are N = 1, M = 1, and N = 24, M = 70. Here N is the number of layers in the pyramidal stacking arrangement and M is the number of cannonballs along an edge in the flat square arrangement.


Positioning and spacing

In both the FCC and HCP arrangements each sphere has twelve neighbors. For every sphere there is one gap surrounded by six spheres (
octahedral In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet a ...
) and two smaller gaps surrounded by four spheres (tetrahedral). The distances to the centers of these gaps from the centers of the surrounding spheres is for the tetrahedral, and for the octahedral, when the sphere radius is 1. Relative to a reference layer with positioning A, two more positionings B and C are possible. Every sequence of A, B, and C without immediate repetition of the same one is possible and gives an equally dense packing for spheres of a given radius. The most regular ones are *FCC = ABC ABC ABC... (every third layer is the same) *HCP = AB AB AB AB... (every other layer is the same). There is an uncountably infinite number of disordered arrangements of planes (e.g. ABCACBABABAC...) that are sometimes collectively referred to as "Barlow packings", after crystallographer William Barlow. In close-packing, the center-to-center spacing of spheres in the ''xy'' plane is a simple honeycomb-like tessellation with a pitch (distance between sphere centers) of one sphere diameter. The distance between sphere centers, projected on the ''z'' (vertical) axis, is: :\text_Z = \sqrt \cdot \approx0.816\,496\,58 d, where ''d'' is the diameter of a sphere; this follows from the tetrahedral arrangement of close-packed spheres. The
coordination number In chemistry, crystallography, and materials science, the coordination number, also called ligancy, of a central atom in a molecule or crystal is the number of atoms, molecules or ions bonded to it. The ion/molecule/atom surrounding the central io ...
of HCP and FCC is 12 and their
atomic packing factor In crystallography, atomic packing factor (APF), packing efficiency, or packing fraction is the fraction of volume in a crystal structure that is occupied by constituent particles. It is a dimensionless quantity and always less than unity. In atomi ...
s (APFs) are equal to the number mentioned above, 0.74.


Lattice generation

When forming any sphere-packing lattice, the first fact to notice is that whenever two spheres touch a straight line may be drawn from the center of one sphere to the center of the other intersecting the point of contact. The distance between the centers along the shortest path namely that straight line will therefore be ''r''1 + ''r''2 where ''r''1 is the radius of the first sphere and ''r''2 is the radius of the second. In close packing all of the spheres share a common radius, ''r''. Therefore, two centers would simply have a distance 2''r''.


Simple HCP lattice

To form an A-B-A-B-... hexagonal close packing of spheres, the coordinate points of the lattice will be the spheres' centers. Suppose, the goal is to fill a box with spheres according to HCP. The box would be placed on the ''x''-''y''-''z'' coordinate space. First form a row of spheres. The centers will all lie on a straight line. Their ''x''-coordinate will vary by 2''r'' since the distance between each center of the spheres are touching is 2''r''. The ''y''-coordinate and z-coordinate will be the same. For simplicity, say that the balls are the first row and that their ''y''- and ''z''-coordinates are simply ''r'', so that their surfaces rest on the zero-planes. Coordinates of the centers of the first row will look like (2''r'', ''r'', ''r''), (4''r'', ''r'', ''r''), (6''r'' ,''r'', ''r''), (8''r'' ,''r'', ''r''), ... . Now, form the next row of spheres. Again, the centers will all lie on a straight line with ''x''-coordinate differences of 2''r'', but there will be a shift of distance ''r'' in the ''x''-direction so that the center of every sphere in this row aligns with the ''x''-coordinate of where two spheres touch in the first row. This allows the spheres of the new row to slide in closer to the first row until all spheres in the new row are touching two spheres of the first row. Since the new spheres ''touch'' two spheres, their centers form an equilateral triangle with those two neighbors' centers. The side lengths are all 2''r'', so the height or ''y''-coordinate difference between the rows is ''r''. Thus, this row will have coordinates like this: : \left(r, r + \sqrtr, r\right),\ \left(3r, r + \sqrtr, r\right),\ \left(5r, r + \sqrtr, r\right),\ \left(7r, r + \sqrtr, r\right), \dots. The first sphere of this row only touches one sphere in the original row, but its location follows suit with the rest of the row. The next row follows this pattern of shifting the ''x''-coordinate by ''r'' and the ''y''-coordinate by . Add rows until reaching the ''x'' and ''y'' maximum borders of the box. In an A-B-A-B-... stacking pattern, the odd numbered ''planes'' of spheres will have exactly the same coordinates save for a pitch difference in the ''z''-coordinates and the even numbered ''planes'' of spheres will share the same ''x''- and ''y''-coordinates. Both types of planes are formed using the pattern mentioned above, but the starting place for the ''first'' row's first sphere will be different. Using the plane described precisely above as plane #1, the A plane, place a sphere on top of this plane so that it lies touching three spheres in the A-plane. The three spheres are all already touching each other, forming an equilateral triangle, and since they all touch the new sphere, the four centers form a
regular tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all th ...
. All of the sides are equal to 2''r'' because all of the sides are formed by two spheres touching. The height of which or the ''z''-coordinate difference between the two "planes" is . This, combined with the offsets in the ''x'' and ''y''-coordinates gives the centers of the first row in the B plane: : \left(r, r + \frac, r + \frac\right),\ \left(3r, r + \frac, r + \frac\right),\ \left(5r, r + \frac, r + \frac\right),\ \left(7r, r + \frac, r + \frac\right), \dots. The second row's coordinates follow the pattern first described above and are: : \left(2r, r + \frac, r + \frac\right),\ \left(4r, r + \frac, r + \frac\right),\ \left(6r, r + \frac, r + \frac\right),\ \left(8r,r + \frac, r + \frac\right),\dots. The difference to the next plane, the A plane, is again in the ''z''-direction and a shift in the ''x'' and ''y'' to match those ''x''- and ''y''-coordinates of the first A plane. In general, the coordinates of sphere centers can be written as: : \begin 2i + ((j\ +\ k) \bmod 2)\\ \sqrt\left + \frac(k \bmod 2)\right\ \frack \endr where ''i'', ''j'' and ''k'' are indices starting at 0 for the ''x''-, ''y''- and ''z''-coordinates.


Miller indices

Crystallographic features of HCP systems, such as vectors and atomic plane families, can be described using a four-value
Miller index Miller indices form a notation system in crystallography for lattice planes in crystal (Bravais) lattices. In particular, a family of lattice planes of a given (direct) Bravais lattice is determined by three integers ''h'', ''k'', and ''� ...
notation ( ''hkil'' ) in which the third index ''i'' denotes a convenient but degenerate component which is equal to −''h'' − ''k''. The ''h'', ''i'' and ''k'' index directions are separated by 120°, and are thus not orthogonal; the ''l'' component is mutually perpendicular to the ''h'', ''i'' and ''k'' index directions.


Filling the remaining space

The FCC and HCP packings are the densest known packings of equal spheres with the highest symmetry (smallest repeat units). Denser
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 pack ...
s are known, but they involve unequal sphere packing. A packing density of 1, filling space completely, requires non-spherical shapes, such as honeycombs. Replacing each contact point between two spheres with an edge connecting the centers of the touching spheres produces tetrahedrons and octahedrons of equal edge lengths. The FCC arrangement produces the
tetrahedral-octahedral honeycomb The tetrahedral-octahedral honeycomb, alternated cubic honeycomb is a quasiregular space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of alternating regular octahedra and tetrahedra in a ratio of 1:2. Other names i ...
. The HCP arrangement produces the
gyrated tetrahedral-octahedral honeycomb The tetrahedral-octahedral honeycomb, alternated cubic honeycomb is a quasiregular space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of alternating regular octahedra and tetrahedra in a ratio of 1:2. Other names in ...
. If, instead, every sphere is augmented with the points in space that are closer to it than to any other sphere, the duals of these honeycombs are produced: the
rhombic dodecahedral honeycomb The rhombic dodecahedral honeycomb (also dodecahedrille) is a space-filling tessellation (or honeycomb) in Euclidean 3-space. It is the Voronoi diagram of the face-centered cubic sphere-packing, which has the densest possible packing of equal s ...
for FCC, and the
trapezo-rhombic dodecahedral honeycomb The rhombic dodecahedral honeycomb (also dodecahedrille) is a space-filling tessellation (or honeycomb) in Euclidean 3-space. It is the Voronoi diagram of the face-centered cubic sphere-packing, which has the densest possible packing of equ ...
for HCP. Spherical bubbles appear in soapy water in a FCC or HCP arrangement when the water in the gaps between the bubbles drains out. This pattern also approaches the
rhombic dodecahedral honeycomb The rhombic dodecahedral honeycomb (also dodecahedrille) is a space-filling tessellation (or honeycomb) in Euclidean 3-space. It is the Voronoi diagram of the face-centered cubic sphere-packing, which has the densest possible packing of equal s ...
or
trapezo-rhombic dodecahedral honeycomb The rhombic dodecahedral honeycomb (also dodecahedrille) is a space-filling tessellation (or honeycomb) in Euclidean 3-space. It is the Voronoi diagram of the face-centered cubic sphere-packing, which has the densest possible packing of equ ...
. However, such FCC or HCP foams of very small liquid content are unstable, as they do not satisfy
Plateau's laws Plateau's laws describe the structure of soap films. These laws were formulated in the 19th century by the Belgian physicist Joseph Plateau from his experimental observations. Many patterns in nature are based on foams obeying these laws. Laws ...
. The Kelvin foam and the Weaire–Phelan foam are more stable, having smaller interfacial energy in the limit of a very small liquid content. There are two types of
interstitial hole In crystallography, interstitial sites, holes or voids are the empty space that exists between the packing of atoms (spheres) in the crystal structure. The holes are easy to see if you try to pack circles together; no matter how close you get ...
s left by hcp and fcc conformations; tetrahedral and octahedral void. Four spheres surround the tetrahedral hole with three spheres being in one layer and one sphere from the next layer. Six spheres surround an octahedral voids with three spheres coming from one layer and three spheres coming from the next layer. Structures of many simple chemical compounds, for instance, are often described in terms of small atoms occupying tetrahedral or octahedral holes in closed-packed systems that are formed from larger atoms. Layered structures are formed by alternating empty and filled octahedral planes. Two octahedral layers usually allow for four structural arrangements that can either be filled by an hpc of fcc packing systems. In filling tetrahedral holes a complete filling leads to fcc field array. In unit cells, hole filling can sometimes lead to polyhedral arrays with a mix of hcp and fcc layering.


See also

*
Cubic crystal system In crystallography, the cubic (or isometric) crystal system is a crystal system where the unit cell is in the shape of a cube. This is one of the most common and simplest shapes found in crystals and minerals. There are three main varieties ...
*
Hermite constant In mathematics, the Hermite constant, named after Charles Hermite, determines how short an element of a lattice in Euclidean space can be. The constant ''γn'' for integers ''n'' > 0 is defined as follows. For a lattice ''L'' in Euclidean space ...
*
Random close pack Random close packing (RCP) of spheres is an empirical parameter used to characterize the maximum volume fraction of solid objects obtained when they are packed randomly. For example, when a solid container is filled with grain, shaking the containe ...
*
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 pack ...
*
Cylinder sphere packing Sphere packing in a cylinder is a three-dimensional packing problem with the objective of packing a given number of identical spheres inside a cylinder of specified diameter and length. For cylinders with diameters on the same order of magnitude ...


Notes


External links


P. Krishna & D. Pandey, "Close-Packed Structures" International Union of Crystallography by University College Cardiff Press. Cardiff, Wales. PDF
{{DEFAULTSORT:Close-Packing Of Spheres Discrete geometry Crystallography Packing problems Spheres