IN RHOMBIC DODECA
"But it must be remembered that the centers of volume of the rhombic dodecahedral spherics are also the centers of each of all the closest-packed spheres of unit radius, and their volumetric centers are also omnicongruent with all the vertexes of all isotropic vector matrixes. The Couplers literally couple "everything," while alternatively permitting all the varieties of realizable events experienced by humans as the sensation of "free will."-R. Buckminster Fuller
CUBIC / KITE
Just as there are two ways in which MITES can pack into couplers (OCTANT / CROSSOCTANT), there are two ways in which couplers can pack into rhombic dodecahedra in allspace :
(1) CUBIC: Six whole couplers can pack by aligning their square equators with the square faces of a cube frame as in:
(2)KITE: Twelve half couplers (kites or kats) can pack by aligning rhombic equator with the rhombic faces of a rhombic dodecahedron frame as in:
Twelve whole couplers pack into the stellated rhombic dodecahedron in the same way as (2) above:
I have come to realize that no matter how Mites are oriented into the octants of the coupler in an octant packing, edge AE of the B module always coincides with one of the edges of the couplers' square equator. In the KITE packing of the octant-packed coupler, the square equator of the coupler always coincides with the plane of the long diagonal of the rhombic face through the volumetric center of the rhombic dodeca; and the BCE Bmod face, or edge AC, is on the plane through the rhombic equator of the coupler. Similarly, in a cubic packing of the crossoctant-packed coupler, the edge AE of the Bmod always coincides with the short axis (m) of the coupler and the BCE Face of the Bmod is always on the obverse or reverse surface of the coupler.
Fuller's net of the Bmod: Fig. 916.01
There are two 45 degree angles in face ACE of the B module. In any KITE packing of OCTANT-PACKED couplers (or CUBIC packing of CROSS-OCTANT packed couplers), either vertex E or vertex A of the Bmod will be at the volumetric center of the RD. If it is Vertex E then edge CA will coincide with (1/2) the long diagonal of the rhombic face and the Bmods are isolated in a packing as in RD1. If it is vertex A, then face BCE will be on the surface of the RD aligned on one side of (1/2) the long diagonal of the rhombic face as in RD2. In the models below, RD1 is constructed with all Vertex E oriented Bmods (no BCE faces) ; RD2 , from all Vertex A oriented Bmods (48 BCE faces); and RD3, 24 each of Vertex A and VertexE oriented Bmods (24 BCE faces).
In RD packings, spheric centers are only interconnected by Bmods when BCE faces are on the surface aligned with long diagonal of the rhombic face (see). In a CUBIC (or kate) packing of OCTANT-PACKED couplers or in a KITE packing of CROSS-OCTANT packed couplers, Vertex C is oriented toward but not coincident with the volumetric center of the RD and face ABE is aligned along the short diagonal of the rhombic face as in RD4 and RD5. Note that in packings like RD4 & RD5, Bmods do not interconnect spheric centers. Bmods are the "IVM" in RD packings and their positioning dictates how spherics are interconnected.In Synergetics, Fuller postulates that couplers serve to interconnect the spheric (rhombic dodeca) centers in an allspace packing similar to the way in which sphere centers are interconnected by VE radials in the IVM (see 954.47). My focus in this essay is on (2) above (the "kite or kat" packing), because this method of packing orients the couplers "between" rather than "within" rhombic dodeca in an allspace packing.
It seems that all RD packings with BCE Bmod faces could be classified according to the number, sign (+or-), and position of these faces on the surface of the RD from 0 (RD1) to 48 (RD2). In a packing of RD2 there are 48 Bmod face bonds or 48 Bmod vectors into the heart of each RD. Similarly, packings with ABE Bmod faces could be classified according to the number, sign and position of ABE faces on the surface of the RD from O(RD1) to 24 (RDs 4&5) although spheric centers are not interconnected by Bmods in these packings.
I am going to focus on the Bmod BCE faces as a metaphor for the IVM and try to understand how a single RD can be networked through the Bmod (BCE trivalently face-bonded) spheric center connections in an allspace packing: