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Other Space Fillers
The role played by tetrahedra and octahedra in an array of cubesnot to mention in the IVMdemonstrates that these two structures combine to create a variety of polyhedra. This observation suggests an operational strategy: experiment with various combinations in the hope of finding other space fillers.
Two tetrahedra are affixed to opposite faces of an octahedron with the same edge length. Supplementary dihedral angles cause adjacent triangular faces to be coplanar, thereby merging into six rhombic faces (Fig. 9-8). The resulting slanted structure, introduced in Chapter 9, is called a rhombohedron and can be thought of as a partially flattened cube. Six squares have simply been squashed into diamonds. It's not hard to imagine that an entire array of toothpick cubes could lean over in unison, transforming into an array of distorted cubes. Rhombohedra therefore fill space. The simplicity of this development hints at a starting point.
In his "Contribution to Synergetics," Loeb analyzes various polyhedra for their divisibility into tetrahedra and octahedra, and demonstrates that a shape will fill space if it consists of two tetrahedra for every one octahedron. (3) Suddenly Out of randomness an order emerges, and trial-and-error is replaced by a generalized principle. The ability to fill all space can be added to the list of descriptive properties (such as stability, symmetry, duality) by which we categorize the scattered cast of polyhedral characters. And now, armed with Loeb's conclusive analysis, we will continue to explore the puzzle from a slightly different frame of reference.
The Search Continues
The IVM simplifies our task, providing a frame of reference which itself fills space. All of the polyhedra covered so far, with the exception of the icosahedron and pentagonal dodecahedron, are outlined within the IVM and IVM' framework by various combinations of octahedral and tetrahedral components (both parts and wholes). We can therefore survey the matrix to ascertain which of these systems are able to meet face to face without intervening cavities.
Polyhedra that fit together without gaps must completely surround a common vertex. IVM vertices therefore provide a good starting point; we can systematically investigate the different types of nodes in the matrix, checking the surrounding cells for space-filling polyhedra. Our first node reveals an already familiar space-filling team: each IVM vertex joins six octahedra and eight tetrahedraas well as polyhedral combinations of the two. The most obvious of these combinations is the vector equilibrium, and our study of the IVM already made it clear that VEs do not fill space, but rather must cooperate with octahedra to create an uninterupted array, providing yet another example of complementarity. The necessity of this pairing follows directly from octahedron-tetrahedron interdependence. (Chapter 13 will elaborate on such space-filling teams, which arise out of the two-to-one ratio mentioned above.)
Our next step is to investigate other vertices. Interconnecting the centers of octahedra, we trace the array of minimum cubes. That one's easy. The fact that cubes fill space is not new; what else can we learn?
If octahedron centers provide the meeting point for eight cubes, what happens at the centers of tetrahedra? To begin with we observe that four quarter tetrahedra convene. The base of each shallow pyramid is the face of a neighboring octahedron, so we ask what shape is created by an octahedron framed by eight quarter tetrahedra. As seen in Chapter 9, this is equivalent to Loeb's "degenerate stellation"; the thin triangular faces of adjacent quarter tetrahedra become coplanar when surrounding an octahedron, and thereby merge into one diamond face. The result: a rhombic dodecahedron (Fig. 9-12). In IVM context, this means that every octahedron reaches out, incorporating eight neighboring quarter tetrahedra, so that the tetrahedra are completely used up (no leftovers). The entire matrix is thus involved, meaning of course that rhombic dodecahedra fill space. By interconnecting the centers of every tetrahedron in the IVM we automatically generate an array of rhombic dodecahedra.
The space-filling puzzle becomes more enticing with this new addition. With its many diamond faces and irregular surface angles, this shape is not, hke the cube, an obvious space filler. A casual observer would not suspect this intricate polyhedron of fitting so beautifully together. In fact, without the advantage of the IVM, it is difficult to picture how rhombic dodecahedra manage to fill space.
The Dual Perspective
Recalling from Chapter 4 that the rhombic dodecahedron is also a "degenerately stellated" cube, it is interesting to observe the relation-ship between the two packings. A cube with four body diagonals dividing the inside into six pyramids is shown in Figure 12-3a. These square-based pyramids are the exact shape required to degenerately stellate a second cube. We can "unwrap" that subdivided cube and place its components on the six faces of a second intact cubeonly to arrive once again at our diamond-faceted friend (Fig. 12-3b). We therefore have a new way to visualize the rhombic-dodecahedron packing. Start with an array of cubes, in which every other cube is subdivided by a central node, while the rest remain empty. We have just described an array of rhombic dodecahedra (Fig. 12-3c). A framework of cubes is so familiar and readily imagined that this exercise brings the rhombic dodecahedron's space-filling capability into easy grasp. (4)
Duality and Domain in Sphere Packing
Imagine that our closest-packed spheres are actually perfectly round balloons. Supposing they are all packed tightly into a closed container which insures that their positions are fixed, we then try to picture what would happen if more air were steadily pumped into each balloon. Remember that they are unable to movepreventing the natural reaction of collectively spreading out and taking up more space. Instead each individual balloon expands and presses more tightly against its neighbors so that the points of tangency merge into planes of tangency:
A bubble is only a spherical bubble by itself. The minute you get two bubbles together, they develop a plane between them. (536.44)
We allow the balloons to grow to the extent that all available space is occupied. What is the shape of the balloons once they merge together? Remember that twelve spheres pack tightly around one, and so the tangency points between spheres, which were located at the twelve (four-valent) vertices of vector equilibria, must now be replaced by the same number of four-valent faces. We are thus reminded that the domain of a sphere in closest packingin Fuller's words "the sphere and the sphere's own space"is a rhombic dodecahedron.
The above sequence provides an important insight into the space-filling ability of rhombic dodecahedra. By renaming this diamond-faceted polyhedron "spheric," Fuller places considerable emphasis on its relationship to closest packing. The spheric is thus presented as a cosmically significant shape, the domain of the generalized energy event, and consequently, the domain of every intersection in the omnisymmetrical vector matrix.
Synergetics thus arrives at its all-space fillers through investigation of nature's omnisymmetrical framework. IVM provides the context:
A "spheric" is any one of the rhombic dodecahedra symmetrically recurrent throughout an isotropic vector-matrix-geometry... (426.10)
426.20 Allspace Filling: ... Each rhombic dodecahedron defines exactly the unique and omnisimilar domain of every radiantly alternate vertex... as well as the unique and omnisimilar domains... of any aggregate of closest-packed uniradius spheres...
In a later section of Synergetics, Fuller expands upon the above observations with a more general statement about any point in space.
The most complete description of the domain of a point is not a vector equilibrium but a rhombic dodecahedron, because it would have to be allspace filling and because it has the most omnidirectional symmetry. The nearest thing you could get to a sphere in relation to a point, and which would fill all space, is a rhombic dodecahedron. (536.43)
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