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Closest Packings of Bowling Balls and Other Spheres

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A closest packing is just another way to say 'the densest packing' of objects in a room - like in Tetris, useful when figuring out the best way to load the boot of a car. The closest packings of rigid spheres, like bowling balls, have gotten some special attention from scientists and mathematicians in the last few centuries. This is not because they were trying to figure out the most stable way to load their car's boots with bowling balls, but because rigid spheres are quite good approximations for little particles from dust grains to atoms1. Before we go on with the Entry let it be said that the densest packing is not necessarily the most effective, especially in quotidian life. For example, when loading a cubic box with spheres, an ordinary stack may allow you to put more spheres into the box than the densest packing would. If there is not such a box, the densest packing is also the most effective and stable, an example for this is the well-known pile of cannonballs.

In the next sections it will be shown how the closest packings can be formed and that there are two special cases for a closest packing of rigid spheres: the hexagonal closest packing and the cubic closest packing. In literature one will often bump into the abbreviations 'hcp', which stands for 'hexagonal closest packing', and 'fcc', which stands for 'face centred cubic' - which is totally equivalent to 'ccp' or 'cubic closest packing', but which almost nobody uses. Many metals crystallise in these structures (eg crystalline gold has an 'fcc' structure, crystalline Magnesium an 'hcp' structure).

The best way to understand how these two packings form and to understand the differences between them is to build a model using small spheres (perhaps try using marbles instead of bowling balls). The worst way to understand how these structures look is by trying to conceive them from plain text - even with the aid of rudimentary ASCII drawings.

Closest Packings Explained in Plain Text with Rudimentary ASCII Drawings

The best way to start explaining three dimensional close packings is probably to start from the two dimensional model - which is a lot easier to understand. The closest packing of (identical) spheres in two dimensions has a hexagonal arrangement of spheres. In this packing, all spheres have the maximum number of neighbours, namely 6 (it is not possible to place more than six spheres around a central sphere so that they all touch). It looks more or less like this:

o - o - o o o \ / \ / o - o o o o \ / o o o o o o o o o o

This two-dimensional closest packing of rigid spheres can now be arranged in layers resulting in a three-dimensional packing of rigid spheres. If the next layer is placed so that each of the spheres lies in a groove formed by three spheres of the lower layer, then the new three-dimensional structure will be again a closest packing.

There are two different sets of grooves on which to place the next layer of spheres. In the ASCII drawing below, note that the pattern in the marked triangles change. The triangles marked with (°) and with (.) are - each in their own layer - still arranged in the same way as the first layer (the two-dimensional closest packing), but shifted differently on the second layer. Note that the triangles marked with (°) are shifted to the right and down, whereas in the second possibility the triangles marked with a (.) are just shifted down.

POSSIBILITY 1 o - o - o \°/ \°/ ° o - o \°/ ° o ° POSSIBILITY 2 o - o - o .\ /.\ /. o - o .\ /. o .

To form bigger three-dimensional bodies, a third layer of spheres can now be placed on top of the second layer's sets of grooves - again with the same two possibilities as above. One option would be to arrange them so that their positions coincide with the positions of the spheres on the first layer. If we denote the first layer's spheres as being at a set of positions 'A' and the second layer's spheres at positions 'B', we would then have the sequence A-B-A. The second option would be to arrange them according to the second possibility, leading to a pattern A-B-C.

A fourth layer could now be placed on the third layer, and there are again two possibilities, and so on for the remaining layers. It happens that a layer of the type 'A' (the triangles in the ASCII drawing above) has grooves at the positions of the spheres in layer 'B' and in 'C'. A layer of type 'B' has grooves leading to 'A' and 'C'. Finally, the layer of type 'C' has grooves for the types 'A' or 'B'. In conclusion there are three types of layers: A, B and C. To get a closest packing there is only one rule: The next layer must be of a different type.

There are an infinite number of ways to arrange the layers, periodically A-B-A-C-A-B-A-C or randomly A-B-A-B-C-A-C-B-A-C-A-B. All of those are closest packings. There are two special cases, however, namely the alternation of A and B (A-B-A-B...) and the repetition of the pattern A-B-C (A-B-C-A-B-C...). The first case leads to the aforementioned hexagonal closest packing (hcp), the second leads to the cubic closest packing (fcc, alias ccp).

The Hexagonal Closest Packing (hcp)

It is easy to understand why the hexagonal closest packing is called hexagonal (note the small hexagon in the lower right part of the ASCII drawing).

HEXAGONAL o - o - o o o ------ A \°/ \°/ ° ° ° o - o o - o o ° \°/ °/ ° \° o o o o o ° ° \° °/ ° ---- B o o o - o o ° ° ° ° °

The Cubic Closest Packing (ccp)

It is a lot more difficult to see the cube in the 'ccp' schematics. To do that one would have to draw a cube with a sphere at each edge and add another sphere right in the middle of every face of the cube (hence the more commonly found designation 'fcc') - note: no sphere in the middle of the cube! The close packing layers are lying oblique to the cube's sides, perpendicular to the three dimensional diagonals of the cube. CUBIC o - o - o o o ---- A .\°/.\°/. ° . ° . ° o - o o o o ° .\°/. ° . ° . ° . ---- B o o o o o . ° . ° . ° . ° . ° ---- C o o o o o ° . ° . ° . ° . ° . A better view that allows you to 'see' the cube is the following (note that the positions are marked with A, B and C to denote the different layers): B-----------------A /| /| / | B / | / | / | C-----------------B | | | B | | | | | | | C | | B | | | | | | | C | | | C-------------|---B | / | / | / C | / |/ |/ A-----------------C

Final Remarks

Close packings are the tightest way to pack spheres. Atoms, being nothing but tiny spheres (of some hundred picometres in diameter) often arrange in this way. Furthermore it is also possible to deduce atomic structures of metal alloys, salts and oxides by filling the voids of close packed spheres.

1A lot of information, for example the rigidity of a metal alloy, can be predicted by treating these materials as a pile of differently sized bowling balls en miniature. One would not have to synthesize the alloy to see what happens.

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