Mapping the Hidden Patterns in Sphere Packing
by Russell Z Chu
Connections between 4 fcc lattices
Secondary structural systems
Fig. 13 is showing the various possible relationships between the 4 fcc lattices. These are tetrahedral relationships.
The primary system is the face center cubic lattice (fcc), which is composed of four fcc lattices- the tetrahedral A lattice, the tetrahedral B lattice, the octahedral C lattice and the cuboctahedral D lattice.
The secondary systems are compound structures of the 4 primary fcc lattices.
To follow are all the possible combinations, or relationships between four systems.
Between four systems there are 6 possible edge connections of two systems, 4 possible triangular connections of 3 systems, and 1 tetrahedral connection of all four systems.
Six - two fcc lattice connections: A+B, C+D, A+C, A+D, B+C, B+D. They are divided into two groups:
- (2) Simple cubic structures: A+B and C+D
- (4) Diamond net structures: A+C, A+D, B+C, B+D
Four - three fcc lattice connections: A+B+C, A+B+D, A+C+D, B+C+D.
One - four fcc lattice connection: A+B+C+D.
It appears that nature tries all possible combinations. Structures evolve through generations of self assembly.
We do see unexpected results when we combine the 4 fcc lattices. Unexpected because we are not used to getting different results when making similar operations except in chemistry, where the synergistic combinations create added value to the new system that are greater than the sum of the individual parts considered separately.
We find that, within the 6 - two fcc lattice combinations, there is a division into 2 cubical net combinations and 4 diamond net combinations.
If we look back at Chart 1 we notice that there are some patterns that might be significant in predicting future patterns:
- Frequencies of A and B go into each other.
- Frequencies of C go into D only.
- Frequencies of A go into C and D.
- Frequencies of B go into C and D.
From the combinations of 2 lattices we have the 2 cubic patterns and the 4 diamond patterns.
Combinations of 2 fcc lattices
Chart 2. The simple cubic structural growth patterns
Chart 2 and Chart 3 go together.
Chart 2 is showing the growth patterns of the compound structures A+B and C+D (simple cubic). This chart represents the simple cubic group, the other group of 2 fcc lattices is the diamond net structures shown in Chart 3.
The patterns are being shown in a schematic form of the cube, it is one of the faces of the cube. Please compare the schematic pattern with Fig.15a, 15b, and 15c. The black squares represent the A+B cubic net, the yellow diagonals represent the yellow tetrahedral A lattice and the green diagonals represent the green tetrahedral B lattice.
The same schematic pattern applies to the C+D cubic net in Fig. 15b. The aqua squares represent the C+D cubic net, the red diagonals represent the red octahedral lattice and the blue diagonals represent the blue cuboctahedral D lattice.
We can see from Chart 2 that the cubical frequency progression jumps back and forth from cubic AB to cubic CD. The cubic AB (black) are all odd frequencies and cubic CD (aqua) are all even frequencies. This alternating pattern is related to the caging pattern or nesting of cubes, as shown in Fig. 15c.
The simple cubic pattern is also the same as the octahedral void filling pattern. When A and B lattices combine they can be seen as a simple cubic net or the vertexes of one lattice occupying the center of the other lattice’s octahedrons. This also applies to C+D combinations.
Simple Cubic Structures
A+B and C+D
Two interpenetrating fcc lattices or simple cubic structure
Figure 14a, 14b.
Fig. 14a is the simple cubic structure of A+B.
Fig. 14b is the simple cubic structure of C+D.
The Unified View
The cubic net and the 2 interpenetrating fcc lattices.
Figure 15a, 15b and 15c.
Fig. 15a is the unified view of the AB cubic net with the fcc A lattice and the fcc B lattice.
Fig. 15b is the unified view of the CD cubic net with the fcc C lattice and the fcc D lattice. It is a 2 frequency cube with 8 unit cubes.
Fig. 15c is the 3 frequency cube (back) caging inside the 2 frequency cube (aqua). It shows the alternating pattern between frequencies.
It is important to note that the integrity of the structural system of each lattice, their pattern, are maintained even though they have combined into a new system.
Tensegrity and Synergy
The unification of the 2 interpenetrating fcc lattices and the cubical net allow us to see the simple cubic structure as a tensegrity structure. With the tetrahedral A lattice and the tetrahedral B lattice bound together by the cubical net Fig. 15a. Fig. 15b is the same tensegrity structure as Fig. 15a.
In Fig. 15a we can see a yellow tetrahedron, a green tetrahedron and a black cube. What we have is a tensegrity structure where the yellow tetrahedron and the green tetrahedron become compression members when the cube net goes into tension. It may be easier to visualize the face of the cube as a tensegrity, with the 2 diagonal rods secured at the ends with a string, in the shape of a square. These stable square faces then can be assembled into a tensegrity cube.
When the edge length of the cube net decreases (go into tension) the edge length of the tetrahedral A and B lattices would tend to decrease (go into compression) until a new equilibrium is established. This is a dynamic structural view of two chemical elements of fcc lattice type brought together and forming a simple cubic net tensegrity structure. We can visualize the expansion and contraction of materials with change of energy, temperature, pressure etc...
Fig. 15a shows a 1f cube, it is the first possible cube. The cube exists only as a compound structure.
Synergy is defined as the behavior of whole systems unpredicted by the behavior of their parts taken separately (Fuller, Synergetics 101.01).
Syn.er.gism – the joint action of different substances in producing an effect greater than the sum of the effects of all the substances acting separately (dictionary).
I am proposing that the added effect of synergy can be associated with the added structural patterns, tension components, of tensegrity structures. In the case of the simple cubic structure it is the cubic net AB added to the A and B fcc lattices, as shown in Fig. 15a and 15b.
This effect of synergy and tensegrity could be seen further in the structures to follow.
Tensegrity: "The word 'tensegrity' is an invention: a contraction of 'tensional integrity.' Tensegrity describes a structural-relationship principle in which structural shape is guaranteed by the finitely closed, comprehensively continuous, tensional behaviors of the system and not by the discontinuous and exclusively local compressional member behaviors. Tensegrity provides the ability to yield increasingly without ultimately breaking or coming asunder." (Fuller, Synergetics 700.011)
For more on tensegrity: in ‘The Architecture of Life’ Dr. Donald E. Ingber wrote about ‘What is tensegrity’ with great insights.
All structures are tensegrities. What differentiate tensegrity structures are the different levels of separation between the patterns of compression members and the patterns of tension members within each structure. We start from the prime structural systems where the patterns of tension and compression (push and pull) are the same, the structural members are both compression and tension members, as we see in the fcc lattice (structural systems), next we see a separation of tension and compression patterns as in the compound structure of simple cubic as shown above Fig. 15a, b. This progression increases until compression members do not touch each other or we have continuous tension and islanded compression.
The different degrees of separation in tensegrities give different qualities to structures, from rigid to bouncy.
The minimum omni symmetrical prime structural systems are the tetrahedron, octahedron and icosahedron. Whenever we work with cubes, dodecahedra and other non-triangulated polyhedra we would be able to find the other parts of the structural system that have been overlooked.
Chart 3. The diamond net structural growth patterns
Chart 3 is a graphical representation of the 3D structures shown below in Figures 16 through 19. The squares are a representation of the tetrahedrons that are occupied by the diamond net.
We can see that all 4 combinations contain tetrahedral A or tetrahedral B lattice. The tetrahedral lattice influences the alternating pattern and 3D mirror like pattern between the A and B columns.
The 2f A+C is shown in Fig. 16b and 17a.
The 2f B+C is shown in Fig. 16c and 17b.
The 2f A+D is shown in Fig. 18a and 19a.
The 2f B+D is shown in Fig. 18b and 19b.
The 3f A+C is shown in Fig. 17c.
The 3f B+D is shown in Fig. 19c.
Diamond Net Structures
Fcc lattices A+C and B+C
Figure 16a, 16b, 16c, and 16d.
Fig. 16a is showing the diamond net (blue). We can see the diamond net in Fig. 16b connecting the yellow and red lattices.
Fig. 16b is showing the unified view of the 2 interpenetrating fcc lattices (A+C) with the blue diamond net. When we compare with Fig. 14d we see that it is the same structure except that it is rotated to the point of view of the tetrahedron. It is easier to see the yellow 1f tetrahedron caged inside the red 2f tetrahedron.
Fig. 16c is showing the 2 interpenetrating fcc lattices (B+C) with the gray diamond net. Notice that lattice B occupies the tetrahedrons not occupied by lattice A in Fig. 16b.
Diamond net pattern is the same as tetrahedral void filling pattern. The 2 interpenetrating lattice method let us see why some tetrahedrons are not filled and exactly where they are filled or not.
Looking at Fig. 16d we see the yellow tetrahedron caged inside the red 2f tetrahedron from the C lattice. The yellow tetrahedron does not touch any part of the red tetrahedron. We see the blue diamond net connecting the two fcc lattices together. This is also a tensegrity relationship where the 2 fcc lattices would contract when the diamond net contracts or when the diamond net goes into tension the 2 lattices would go into compression. Which also means when the two fcc lattices expand it would stretch the diamond net and if it stretches beyond the bond can hold the compound structure would fall apart, melt, dissolve, etc…
Figure 17a, 17b and 17c.
Fig. 17a is the 2f A+C structure in Chart 3. The tetrahedral bond is emphasized so it could be seen more clearly. The cubic grid is to help visualize spatial location. See Fig. 17c which does not have the aid of cubic grid.
Fig. 17b is the 2f B+C structure.
Fig. 17c is the 3f A+C structure.
Diamond Net StructuresFcc lattices A+D and B+D
Figure 18a, 18b.
Fig. 18a is showing the A+D fcc lattices and the interconnecting diamond net (aqua), same as Fig. 19a.
Fig. 18b is showing the B+D fcc lattices and the interconnecting diamond net (purple), same as Fig. 19b.
Figure 19a, 19b and 19c.
Fig. 19a is showing the 2f A+D with the tetrahedral or diamond net emphasized for better seeing.
Fig. 19b is showing the 2f B+D with the diamond net.
Fig. 19c is showing the 3f B+D with the diamond net. Notice the caging at 2f the D lattice is outside and at 3f the B lattice is outside.
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