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There's more to the Mite than meets the eye. Figure 13-5a shows a corresponding face of both a positive and a negative Mite. As every Mite incorporates both a positive and a negative A-module, it is the unpaired B-moduleslanting off to one side or the otherthat determines the charge of the overall system.
From the above recipe, it appears evident that this minimum space-filling tetrahedron is either right- or left-handed. This was also evident from our initial generating procedure, subdividing the IVM to carve out the smallest repeating unit with mirror symmetry, which is then split into equivalent mirror-image (positive and negative) Mites.
However, a surprising and easily overlooked result of joining two A's and a B modifies the above conclusion. An unexpected mirror plane runs through the middle of this irregular tetrahedron that we have assumed must be either right- or left-handed. This means that a Mite is actually its own mirror image. The positive and negative versions are identical. The slanting orientation depicted in Figure 13-5a obscures this fact; however, it turns out that the Mite has two isoceles faces, a fact that indicates that the remaining two faces must be identical (but mirror-image) triangles. Therefore, the outside container of the Mite (ignoring the arrangement of its modular ingredients) incorporates a subtle mirror plane and is actually exactly the same shape as its mirror image. A+ A- B+ equals A+ A- B- (Fig. 13-5b). This extraordinary fact means that we don't need positive and negative Mites to fill all space; one or the other version or both in random combinationswill suffice. As its own mirror image, any Mite can fill either position, positive or negative.
We can conclude therefore that the two versions are identical; however, Bucky points out that different internal configurations cannot be ignored, for they point to different energy characteristics:
Though outwardly conformed identically with one another, the Mites are always either positively or negatively biased internally with respect to their energy valving (amplifying, chocking, cutting off, and holding) proclivities. (954.43)
Cubes into Mites
Once the A- and B-game gets going, significant relationships are uncovered at every move. The game thus becomes more and more fascinating as it is played. For example, we might carve open a cube, generating six square-based pyramids, one from each face to the cg. Then slice each square pyramid into quarters, like a peanut-butter sandwich. We thereby rediscover the Mite, observing that the cube consists of twenty-four Mites (Fig. 13-6a). Oriented with one of its isosceles triangles (45° -45° -90°) on the cube's surface, the Mite simply rotates to transform Octet Trusses into boxes.
Next, we apply the LCD procedure to the rhombic dodecahedron. Its twelve faces each frame a diamond-shaped valley ending at the cg point. Each of these inverted pyramids can be split into four (two positive, two negative) asymmetrical tetrahedra (Fig. 13-6b). This is the LCD of Fuller's "spheric," and it is none other than the Mite, the LCD of the isotropic vector matrix. Forty-eight Mites in yet another orientation make up this space-filling shape. Not surprisingly, space-filling and Mites have an important connection.
No longer caught off guard by such interconnectedness, we conclude by observing that the Mitefaithfully representing octet symmetryis also an integral component of the cube and the rhombic dodecahedron.
Fuller's coupler is an irregular octahedron made of eight Mitesor sixteen A-modules and eight B-modules (Fig. 13-7). Given this composition, the "semisymmetrical" coupler is clearly a space filler. Because of the Mite's newfound mirror symmetry, the coupler does not have to have equal numbers of positive and negative B-modules; any eight Mites will make a coupler. This special octahedron has two equal-length axes (which will be referred to as x and y) and a third shorter axis (z). The point of intersection of the three axes will be called K.
The equal x and y axes outline a square equatorial cross-section, which we can now identify as the face of a cube. In fact, the coupler is two sixth-cube pyramids back to back. (3)
The other cross-sections (xz and yz) are diamondsin fact, the exact shape of a rhombic dodecahedron's face. At this point we shall not be surprised to learn that splitting the coupler in half along either diamond cross-section isolates one-twelfth of the rhombic dodecahedron. Six half couplers make a cube; twelve half couplers (split the other way) make a rhombic dodecahedron.
"Couplers literally couple 'everything'" (954.50). Aptly named, the new octahedron joins together both pairs of cubes and pairs of rhombic dodecahedra. Fuller's nomenclature proves quite logical:
We give it the name the Coupler because it always occurs between the adjacently matching diamond faces of all the symmetrical allspace-filling rhombic dodecahedra, the "spherics"...·(954.47)
The coupler's different pairs of opposite vertices reach to the geometric centers of two adjacent polyhedra (cube or spheric, depending on the orientation), incorporating their shared face as a cross-section (Fig. 13-8a, b). Half a coupler belongs to one cube (or spheric), and the other half to its neighbor. So the coupler literally couples-well, not "everything" but-a couple of space-fillers.
Fuller continues: The coupler's role is cosmically relevant, for "rhombic dodecahedra are the unique cosmic domains of their respectively embraced unit radius closest-packed spheres" (954.47). The coupler therefore connects the centers of gravity of adjacent spheres, and its domain includes both the spheres and the intervening (dead air) space. This observation explains why the coupler's volumetric center was labeled K: it marks the exact "kissing point" between tangent spheres in a closepacked array. Now we have the complete story behind Fuller's somewhat dense explanation of his coupler:
... The uniquely asymmetrical octahedra serve most economically to join, or couple, the centers of volume of each of the 12 unit radius spheres tangentially closest packed around every closest packed sphere in Universe, with the center of volume of that omnisymmetrical, ergo nuclear, sphere. (954.48)
Perhaps the most intriguing aspect of the coupler is its similar role in the cube and in the rhombic dodecahedron, thereby linking (or coupling) the two space fillers in a new partnership.
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