by Russell Z Chu
Combination of all 4 fcc lattices
This is the last combination. We started by identifying the 4 fcc lattices and showed all the possible combinations, the six combinations of 2 fcc lattices and the four combinations of 3 fcc lattices. This is the combination of all 4 fcc lattices, making a total of 11 combinations.
The combination of 4 interpenetrating fcc lattices or all of the fcc lattices together is the body center cubic lattice (bcc). It is the combination of the 4 fcc lattices with the 2 cubic nets and the 4 diamond nets or 4 rhombic dodecahedron nets, all together.
Different views of the bcc lattice
Figure 24a. The 4 fcc lattices.
Figure 24b. The 2 cubic nets.
Figure 25a. The 4 fcc lattices with the 2 interconnecting cubic nets.
Figure 25b. The 4 diamond nets or 4 interpenetrating rhombic dodecahedron nets.
Figure 26a. The 2 cubic nets with the 4 diamond nets (which are the same as the 4 rhombic dodecahedral nets.)
Figure 26b. The body center cubic lattice with all of the 4 fcc lattices and the interconnecting nets.
Figure 27a. The ‘coupler’ net is another possible view of the bcc lattice.
Figure 27b. This is one view of the ‘syte’ net.
Richard Hawkins has pointed out that Fig. 27b is one of the modules that Fuller named ‘SYTE’ (SYmetrical TEtrahedron) and that this particular syte is named ‘RITE’.
The coupler and syte are part of Fuller’s allspace-filling quanta module system. ”All the geometries in the cosmic hierarchy (see Table 982.62) emerge from the successive subdividing of the tetrahedron and its combined parts.” Synergetics 100.105.
The coupler is an asymmetrical octahedron. It is a 1/3 of the cube, which can be seen divided in 6 parts in fig. 26a by the 4 diagonals, or diamond nets.
The syte is an asymmetrical tetrahedron. It is a 1/4 of the coupler. Both couplers and sytes are considered ‘all space fillers’.
Figure 28. The bcc lattice with 4 fcc lattices and all the interconnecting nets.
Overview and Conclusion
This exploration of the patterns in sphere packing was confined to the cubic closest packing branch and we only explored the growth of patterns from the face centered cubic (fcc) lattice to the body centered cubic (bcc) lattice. The evolution from fcc to bcc structures.
Most of the patterns exhibited here are known patterns, but often only partially observed.
In review, it became apparent that Fuller’s use of concentric pattern growth denoted in frequencies, in the isotropic vector matrix, was a key to facilitating my finding the 4 fcc lattices. I identified the other 3 fcc lattices from the growth patterns of the isotropic vector matrix which I labeled the cuboctahedral D fcc lattice.
The face center cubic (fcc) lattice, with emphasis on the 8 spheres at the corners of the cube and the 6 spheres at the center of the cubic faces (the octahedron), is the octahedral C lattice.
Only recently I learned that Richard Hawkins had identified 4 interpenetrating rhombic dodecahedral lattices from a different path. “I arrived at this understanding by realizing that one coupler can integrate either whole or half into six different rhombic dodecahedral domains in an RD packing and that since the domains on either side of the couplers 2 rhombic equators are part of the same lattice, there are four rhombic dodecahedral lattices in allspace.” R. Hawkins, 2003. You may view his work here.
It is heartening to see that others have come to the same conclusions from different paths.
We notice that different patterns of various combinations of fcc lattices, including the 4 fcc lattices correlate to known structural patterns of elemental and compound chemical structures. A hierarchy between the different patterns was established by showing that they could be derived from the associations of the 4 prime fcc lattices. This followed the order of the simpler to the complex by adding two, three, and 4 lattices together.
We note that in the evolutionary patterns from fcc structures to bcc structures (as shown in this paper) the original length between sphere centers of the fcc lattices remained the same. What this means is that the size of spheres decreased with each structural combination. If the sizes of spheres were to have remained the same the overall structure would have expanded, i.e. the packing would be less dense. In reality the combinations can be of different size atoms, or variations on the sizes of atoms such as a decrease in size due to added structural connections.
As proposed by Donald Ingber in “The Architecture of Life” we found that the underlying principle of tensegrity does extend to the assembly of atoms. That the new interconnections between the self-assembly of atoms, as in compound structures, work together with the original structural systems to form a new tensegrity system.
It is proposed by R. Chu that synergy could be associated with the added interconnections in the new compound structures, in particular, the new structural elements of the new tensegrity system. This was first shown in Fig. 15a.
The above implies that, in compound structures, the parts of the new system still maintain their original structural integrity. If this is true it would mean that some chemical elements, that exhibit compound patterns, would have differences between the atoms. I.e. diamond structures are composed of two fcc lattices, this could mean that there are two kinds of carbons, and bcc structures are composed of 4 fcc lattices, etc…
When we look at the structures of the chemical elements we find that the three most common structures are hcp, fcc, and bcc. From Table 1.1 of “Bonding and Structure of Molecules and Solids” by David Pettifor (1995), we find that, out of the 92 elements: 25 are hcp type, 20 are fcc type, 12 are bcc type, 3 are diamond type, one is simple cubic type, and 31 are of other types.
Towards the end of this exploration it became evident that there are many parallels with Fuller’s quanta modules, which are being explored by Richard Hawkins.
Further explorations of structural patterns derived from Fuller’s isotropic vector matrix could extend our knowledge of the evolution of structures.
January 21, 2003
______________________________________________________________________________________________
Previous Page Home Page Page 1, 2, 3 Next Page
No comments:
Post a Comment