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Planes of Symmetry
The following observations pertain to an indefinite expanse of cubically closepacked spheres. For example, imagine a room full of Ping Pong balls so tightly nested together that every ball touches exactly twelve others as described above. The idea of a room full of balls at first suggests such chaos that the precise organization arising out of the requirements of closest packing is truly remarkable. The array contains seven planes of symmetry, characterized by two different types of cross-section. The first is obvious, because we generated the cubic packing with successive layers of triangulated planes. However, it is not necessarily obvious that there are four different orientations of triangulated planesparallel to the four faces of the tetrahedron (and therefore to the VE's four intersecting hexagons). Even though we built this array by stacking triangular layers of balls in only one direction, the result incorporates parallel triangulated layers in four different directions. (We further note that all four hexagons of the VE are preserved in cubic packing, whereas in hexagonal packing, hexagons are formed in only one orientation, the horizontal plane.)
Secondly, there are three distinct planes characterized by a square pattern of spheres. This discovery seems to contradict our expectations, for we have learned that squares are not closepacked. However, the emergence of three mutually perpendicular square patterns is an inescapable by-product of nesting triangular layers. The square planes correspond to the VE's square faces, which consist of three mutually perpendicular pairs of parallel facesjust like the faces of a cube.
In a space-filling array of closepacked spheres, these seven planes extend indefinitelywith neither curve nor bend. A cross-section of such a packing has a square or triangular arrangement, depending on which way the packing is sliced. So although we started with only triangulated layers (in order to create a maximally dense array of spheres), square cross-sections automatically arose, just as octahedral cavities automatically arose next to the radiating tetrahedra in the vector equilibrium. This is the shape of space.
It is interesting to note that, although when we stacked triangulated layers it was necessary to make a decision at the third layer that led to two different packings, if we were to start out instead with the square layers (unstable though they may be), there is only one way to proceed. Successive square layers can be placed on top of each other so that each ball lands in a square nest (as opposed to being placed directly on top of another ball, creating an array of cubes which would clearly not be closepacked). The internesting of layers stabilizes the otherwise unstable separate layers. Once two layers are packed together, every ball nests in a group of four balls on the adjacent layercreating half-octahedral pyramids (Fig. 8-6) separated by the inevitable by-product tetrahedra. Continuing to stack square layers in this way, cubic packingrather than hexagonalemerges. The vector equilibrium array is thus generated automatically, with no decisions along the way, by simply stacking square layers. Each and every ball is surrounded by exactly twelve others, in the more symmetrical of the two possibilities.
It is satisfying to reflect on the exquisite logic of this tradeoff: although balls arranged in square patterns are not as closely packed as triangular planes, the nests are deeper. A ball placed in any four-ball nest (to start a second layer) sinks deeply into the cluster; in comparison it seems perched on top of the tight triangular nest. Therefore, successive square layers, although inefficient in themselves, fit more closely together than triangulated layers.
Part of the challenge to mathematicians in proving that the hexagonal and cubic packings qualify as the solution to minimizing interstitial space is the fact that spheres in these two packings occupy just over 74% of the available space, while the four-ball tetrahedron alone is able to occupy 77.96% of its overall volume. (1) It is easy to accept that four balls cannot be pushed any closer together than the tetrahedral cluster and accordingly that the latter figure is the maximum density. Therefore, 74% seems insufficientnot easy to accept as the solution to the problem of closest packing. However, there is no getting around the fact that the six-ball octahedral cluster (Fig. 8-7) is less dense than the four-ball tetrahedronas the former has more leftover room in the middleand that the constraints of space are such that tetrahedral sphere clusters simply cannot be extended indefinitely by themselves. Attempting to fill space exclusively with tetrahedral groups, we quickly discover awkward leftover gapsspaces not quite big enough to contain another sphere. In order for spheres to be both consistently tangent and tightly nested together, we have to allow the naturally alternating tetrahedral and octahedral clusters. The problem would be remarkably easy if spheres could pack tetrahedrally in an indefinite array, but they cannot. No amount of force can change this constraint; space is simply not shaped that way. Twelve around one, creating 60-degree angles both radially and axially, with alternating tetrahedra and octahedra, is the closest packing.
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