Mapping the Hidden Patterns in Sphere Packing
by Russell Z Chu
Combinations of 3 fcc lattices
There are four possible combinations of 3 fcc lattice connections. See section on Connections between 4 fcc lattices.
1. A+B+C
2. A+B+D
3. A+C+D
4. B+C+D
Due to the increased complexity of patterns, not all the lattices are being shown at the same time. I also tried enhancing some polyhedral faces with color. All of the patterns, shown in this paper, are taken from one single computer model separated into 1, 2, 3, and 4 frequency models. They all contain the following patterns in them, the 4 fcc lattices, the 2 cubic nets and the 4 diamond nets.
The rhombic dodecahedron nets, that we are going to see below, are combinations of 2 diamond nets. The two colors of the rhombic dodecahedron net are the same as the color of the diamond nets. See color below the color chart.
The color code used for lattices and nets:
The 4 fcc lattices:
- Tetrahedral A lattice (yellow)
- Tetrahedral B lattice (green)
- Octahedral C lattice (red)
- Cuboctahedral D lattice (blue)
The 2 cubic nets:
- Cubic AB net (black) from the A+B lattice combination.
- Cubic CD net (Aqua) from the C+D lattice combination.
The 4 diamond nets:
- Diamond AC net (blue) from the A+C lattice combination.
- Diamond BC net (gray) from the B+C lattice combination.
- Diamond AD net (aqua) from the A+D lattice combination.
- Diamond BD net (purple) from the B+D lattice combination.
The Rhombic Dodecahedron Net structural growth patterns
Also associated with Fluorite type structures.
Chart 4.
Figure 20a, 20b, 20c and 20d.
Figure 21a, 21b, 21c and 21d.
Figure 22a, 22b.
Fig. 20a is showing the first possible rhombic dodecahedron. The rhombic dodecahedron is a compound structure. It takes three different fcc lattices to make a rhombic dodecahedron. As shown at top of Chart 4, it is a combination of A+B+C lattices. The two diamond nets are the blue AC net and the gray BC net. The diamond nets are the same as the diamond nets shown in the previous diamond net patterns.
In the first row of figures, only the rhombic dodecahedron nets are being shown. Not shown are the 3 fcc lattices and the other cubic net.
All 4 combinations shown above are rhombic dodecahedron nets. It becomes more apparent when looking at the second and third row. The first row is a 2 frequency cube grid, the second row is a 3 frequency cube and the third row is a 4 frequency cube.
In the first column, the combination A+B+C lattices has a rhombic dodecahedron at the center of the cube, which is the origin, the location of the first sphere of the D lattice. As the structure grows to the 4 frequency level there will be 12 more rhombic dodecahedrons, one attached to each face of the original rhombic dodecahedron. If this reminds you of the cuboctahedron you are right, it has the same orientation as the cuboctahedral D lattice.
Next in the combination A+B+D lattices there are six partial rhombic dodecahedron with their 4 edge corner sharing the origin. Note that in Fig. 21b the 6 partial rhombic dodecahedron’s openings are oriented towards the faces of the cube. They make the octahedron. They have the orientation of the octahedral C lattice.
The third combination of A+C+D is easier to see. There are 4 rhombic dodecahedrons, they make the shape of a tetrahedron. It is in the orientation of the tetrahedral B lattice.
In the last combination B+C+D it is also 4 rhombic dodecahedrons and this time it is in the tetrahedral A lattice orientation. It is possible to tell which tetrahedral orientation it has because I kept the orientation of the model the same throughout this project.
Some emerging patterns
The orientation pattern – if we look at the A+B+C lattice combination we see that the D lattice is missing and that happens to be the orientation pattern of the rhombic dodecahedrons. We have a rhombic dodecahedral net in the pattern of the cuboctahedral D fcc lattice. They are closest packing. The same follows for the other 3 combinations. We note that the rhombic dodecahedrons are empty. We will see next that in the last combination A+B+C+D that the D lattice will occupy the centers of the rhombic dodecahedral voids.
This is the 1st occurrence, in this system, of a polyhedron adopting the fcc lattice pattern. We know that others like icosahedrons and fullerenes can adopt the the structure of fcc pattern.
The rhombic dodecahedron net is the new set of repeating patterns of 4 fcc lattices.
Rhombic dodecahedron tensegrity
Figure 23.
We are looking at the 3 fcc lattices A+B+C (yellow, green and red), the cube net (black), and the rhombic dodecahedral net (blue). Again this is a tensegrity structure and it is increasingly more complex. With each added layer there is an added web of connectivity. We can see how much information we are missing when we look at this structure as only a rhombic dodecahedral net or a cubic net.
January 21, 2003
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