Mapping the Hidden Patterns in Sphere Packing (2)

Mapping the Hidden Patterns in Sphere Packing

by Russell Z Chu

Connections between 4 fcc lattices

Secondary structural systems

Figure 13.

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:

1. (2) Simple cubic structures: A+B and C+D
2. (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:

1. Frequencies of A and B go into each other.
2. Frequencies of C go into D only.
3. Frequencies of A go into C and D.
4. 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.

Tensegrity relationships

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 Structures

Fcc 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.

January 21,2003

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Mapping the Hidden Patterns in Sphere Packing (1)

Mapping the Hidden Patterns in Sphere Packing

Lattices, nets, tensegrity structures and synergy



by Russell Z Chu

Falls Church, Virginia, January 2003

This is an exploration of patterns and structural systems in nanoarchitecture. These patterns are fundamental to the beginnings of physical structural systems.

In nature these patterns occur by self-assembly at the nano level, at the size of atoms. Sphere packing has been known since ancient times. By following the pattern development of the cubic closest packing, which is the face center cubic lattice (fcc), we will be able to see the richness inherent in such a simple pattern.

It is my hope that some of the patterns would be recognizable in your observations and that some of the insights presented here would be of help. This is mostly a visual presentation of computer models built using the SpringDance 3d program.


Introduction

The physical world surrounding us is made up of atoms in a variety of combinations. Each atom from the periodic table of elements is a unique system, which was created by the association of pre-atomic systems. In turn the synergetic association of atoms create new compound systems and these systems associate into new systems, etc... Each new system has qualities and properties unpredictable by their components.

Only recently we have been able to “see” atoms as fuzzy balls through the use of Scanning Tunneling Microscopes, which uses a small probe that “feels” the size of atoms and show them through the computer screen.

The original method used by crystallographers and others to determine the structures formed by the elements and their compounds was the x-ray scattered diffraction patterns which they would use to map in 3D the location of each atom.

These are examples of images that we have of atoms today.






Scanning Tunneling Microscope

Figure 1.


2-D network of 4 nm Au cluster array on GaAs (4 nanometer of Gold atoms).
Lee, Park STM






Field Ion and Field Emission Micrographs

Figure 2.



Helium FIM of single-walled carbon nanotube tip, at BIV of about 5700V.
Lovall/Graugnard, homebuilt FIM/FEM



Chemistry has used the close packing of spheres as a model for some of the simplest structures formed by elements, that of cubic close packing (ccp) and hexagonal close packing (hcp). Here we will follow the branch of cubic close packed structural systems. For more on hcp, see “The ccp and hcp family of structures” R. Chu, 1998.





A.F. Wells wrote in his preface to “Structural Inorganic Chemistry” 1984:



“However, while the incorporation of even the most meagre information about the structures of solids into the conventional teaching of chemistry is to be welcomed, a real understanding of the structures of crystals and of the relations between different structures is not possible without a knowledge of certain basic geometrical and topological facts and concepts. This essential background material includes the properties of polyhedra, the nature and symmetry of repeating patterns, and the ways in which spheres, of the same or different sizes, can be packed together.”





We will begin with the packing of spheres of the same size.


Hexagonal close packing of spheres (hcp)













Figure 3a, 3b, 3c, and 3d.


Fig. 3a is showing one sphere.


Fig. 3b is showing 12 spheres around one - a twist cuboctahedron.


Fig. 3c is showing the 2nd layer of 42 spheres. There is a loss of symmetry from the original square and triangular faces.


Fig. 3d is showing the hcp lattice, which is a vectorial representation of Fig. 3c.


Cubic close packing of spheres (ccp)

Also known as face center cubic lattice (fcc)











Figure 4a, 4b, 4c, and 4d.


Fig. 4a is showing one sphere.


Fig. 4b is showing 12 spheres around one – a cuboctahedron.


Fig. 4c is showing a second layer of 42 spheres packed around the 1st shell. This is a 2 shell or 2 frequency cuboctahedron.


Fig. 4d is showing the 2 frequency fcc lattice pattern, a vectorial representation of sphere packing in Fig. 4c.




The four face center cubic (fcc) lattices
Primary structural systems













Figure 5a, 5b, and 5c.


Tetrahedron A (yellow), tetrahedron B (green), octahedron C (red), cuboctahedron D (blue)


Fig. 5a is showing the 1st frequency of tetrahedral A lattice (yellow) with the tetrahedral B lattice (green) and their relationship to each other.


Fig. 5b is showing the 1st frequency of octahedral C lattice (red) in relation to A and B lattices.


Fig. 5c is showing the 1st frequency of cuboctahedral D lattice (blue) with the other 3 fcc lattices.


These four fcc lattices are topologically the same but their geometrical properties are different. All 4 lattices are concentric, i.e. they have a common center. Only the cuboctahedral D lattice is nucleated or it has a center sphere.

These 4 lattices imply that the fcc lattice can only be of one kind at a time, i.e. a tetrahedral A lattice, a tetrahedral B lattice, an octahedral C lattice or a cuboctahedral D lattice. This means that the common view of seeing the fcc lattice as composed of tetrahedrons and octahedrons is misleading. This is understandable since the common view is that there is only one fcc lattice.




How can fcc lattices be the same and different?


It has to do with the level at which we are looking. We can look at humans as a species and we all look the same, but at closer examination we start seeing some uniqueness, male and female, etc... We can go further and find that a single individual is distinct from all the other individuals.

It is the same case with sphere packing and the fcc lattice. Until recently we did not have the tools to take a closer look at the more complex patterns of sphere packing. It is hard to imagine that such unique and complex relationships exist in such a simple pattern as the closest packing of spheres or stacking of oranges in the shape of pyramids.

I first observed the patterns within the cuboctahedral D lattice, see chart below, in 1983 when I built a model of Buckminster Fuller’s isotropic vector matrix out of toothpick and colored the different polyhedrons as in Chart 1 (picture of toothpick model). I was intrigued by the pattern and the absence of the other frequencies. This started my search for the other frequencies of the tetrahedron and octahedron arriving at the 4 fcc lattices.

To follow are the patterns of each of the 4 fcc lattices and the patterns from the combinations of these 4 fcc lattices.


Chart 1. The four fcc lattices and their polyhedral growth patterns

Frequency	Tetrahedral A (p4)	Tetrahedral B (p4)	Octahedral C (p2)	Cuboctahedral D (p1)
1f	1f A	1f B	1f C	1f D
2f			2f A 2f B	2f C 2f D
3f	3f B	3f A	3f C	3f D
4f				4f A 4f B 4f C 4f D
5f	5f A	5f B	5f C	5f D
6f			6f A 6f B	6f C 6f D
7f	7f B	7f A	7f C	7f D
8f				8f A 8f B 8f C 8f D

Overview

The common denominator between all patterns presented in this paper is that they are all concentric. All of the patterns share the same center in space. Otherwise, it would be very difficult to compare patterns. It also allows us to identify the different views of the same pattern.

Chart 1 identifies the 4 fcc lattices. All the patterns or structural systems, to follow, are derived from combinations of these four fcc lattices.

This chart has a purpose of giving us an overall view of all 4 fcc lattices relative to each other as well as the patterns particular to each individual lattice. It includes the periodic polyhedral growth by frequency within each of the 4 fcc lattices and the periodic patterns across the other lattices.

The fcc lattice has been looked at from many different points of view of repeating patterns, as stacking of layers of spheres, cubic cells, packing of tetrahedrons and octahedrons etc… What I found to be most useful was to look at the whole, from a spherical point of view, shells instead of planar layers or cells. We can build the closest packing of spheres starting from one sphere and packing 12 spheres around the one, and that would be the first shell in the shape of a cuboctahedron. The cuboctahedron can be inscribed inside a sphere. We can add another layer of spheres and we would have a second shell. This is how crystals grow.

We can use the word frequency to identify the number of shells, which would also correspond to the number of edge lengths of the polyhedron, or intervals between spheres. For example a 1 frequency (1f) tetrahedron would have an edge of 1 unit length and a 2f tetrahedron would have an edge of 2 unit length. In the case of spheres in the configuration of a 2f tetrahedron the outer edges would have 3 spheres.

The 4 primary close packed structural systems are of the Cubic Close Packed (ccp) configuration also known as the face center cubic (fcc) lattice and referred by Fuller as the isotropic vector matrix (ivm) in “Synergetics 1 & 2” 1975 and 1979.


Analysis of the 4 fcc chart:

In the first column we have the frequency series 1 to 8, followed by the Tetrahedral A lattice (yellow), the Tetrahedral B lattice (green), the Octahedral C lattice (red) and the Cuboctahedral D lattice (blue). Each column represents the polyhedral growth within that fcc lattice.

When we look at the chart we find that we can account for all the frequencies from 1 to 8 of all four A, B, C, and D lattices.

If we follow the frequency progression of each polyhedron we would find that the D lattice is the only one that contains all of its D frequencies (cuboctahedron) sequentially - 1f D, 2f D, 3f D, 4f D… within its own lattice. The other 3 lattices have different patterns of distribution as seen in the chart.

All the odd C frequencies (octahedron), 1f C, 3f C, 5f C… are in the C lattice, and the even frequencies, 2f C, 4f C, 6f C…are in the D lattice.

The tetrahedron A and tetrahedron B have a more complex pattern. They alternate odd frequencies between each other’s lattice and even frequencies between C lattice and D lattice. Such that 1f A is in A lattice, 2f A is in C lattice, 3f A is in B lattice, 4f A is in D lattice and 5f A is back in A lattice.

The periodicity is the number of frequencies at which they repeat their pattern. The periodicity for each polyhedron remains the same across the lattices. The tetrahedron A has a period of 4 frequency (p4), the tetrahedron B also has a period of 4 (p4), the octahedron C has a period of 2 (p2), and the cuboctahedron has a period of 1 (p1).

In lattice A, the tetrahedron A has a period of 4, starting from 1f A and repeating at 5f A and 9f A etc… The tetrahedron B, in lattice A, starts at 3f B and repeats every 4 frequency at 7f B and 13f B etc…

The periodicity in tetrahedral B lattice is the same but transposed. Starting with tetrahedron 1f B and repeats at 5f B, 9f B etc… The tetrahedron A in lattice B starts at 3f A and repeats at 7f A, 13f A etc…

In the octahedral C lattice the octahedron repeats every odd frequency, starting from 1f C and repeating at 3f C, 5f C, 7f C etc… and the tetrahedron A and B in C lattice have a period of 4 frequency, both start at 2f and repeat at 6f, 10f etc…

The cuboctahedron D has a period of 1 frequency, all D frequencies occur in the D lattice, therefore caging does not occur. The octahedron C in D lattice has a period of 2, starting at 2f C and repeating at 4f C, 6f C, 8f C etc… The tetrahedron A and B both start at 4f and repeat at 8f, 12f, 16f etc…

A unique pattern occurs in the cuboctahedral D lattice where all 4 polyhedra occur with the same frequency, the 4f A, 4f B, 4f C, and 4f D. This pattern repeats every 4 frequency at 8f, 12f, 16f etc… As Fuller pointed out the 4 frequency tetrahedron is the first nucleated tetrahedron.

Could we see patterns that might influence or predict patterns in their future compound structures?
See patterns in the connections between fcc lattices and compare with Chart 1.



Tetrahedral A lattice caging





















Figure 6a, 6b.


The caging pattern is the view of the polyhedra sequence, across the columns, from frequency 1 through 8. The pattern of each fcc lattice can be seen below Fig.11a, 11b, 11c and 11d, and here in stereo.

Fig. 6a is showing the tetrahedral A progression from 1 to 3 frequency, notice that it is what is inside Fig. 6b.

Fig. 6b is showing the tetrahedral A progression from 1 to 5 frequency. We are seeing the tetrahedral A lattice (yellow) and the other 3 lattices inside of it.



Looking at Chart 1, we can follow the progression of tetrahedron 1f A to 5f A, going from A to C to B to D and back to A lattice. We can see the yellow 1f tetrahedron from A lattice, then the red 2f tetrahedron from C lattice, the green 3f tetrahedron from B lattice, the blue 4f tetrahedron from D lattice, and back to yellow 5f tetrahedron from A lattice.


The charts presented here and the understanding of these patterns would be useful in nanotechnology and in the use of self-assembly processes.


When we are building structures by manipulating one atom at a time we will find that the location of a particular atom or the overall shape would depend on what lattice type it belongs to. We will see that in the second generation of compound structures the geometrical patterns will increase in variety. The patterns would change from layer to layer or shell to shell.


The 3d models and the charts presented here show a guide for all the pattern variations within the shells and also the various planes within the lattices.


The tetrahedral caging pattern that we see here is composed of all 4 lattices, it is the same as the 4 interpenetrating lattices, which is the same as the body center cubic (bcc) structure.


Tetrahedral B lattice caging





















Figure 7a, 7b.


Fig. 7a is showing the tetrahedral B progression from 1f B through 3f B.


Fig. 7b is showing the progression from 1f B through 5f B.




Octahedral C lattice caging



















Figure 8a, 8b.


The octahedral C lattice caging alternates between the octahedral lattice (red) and the cuboctahedral D lattice (blue), it is an alternating odd, even frequency pattern.


The periodicity of octahedrons is 2 (p2). All odd frequencies occur within the octahedral C lattice and all even frequencies occur within the cuboctahedral D lattice.


Octahedron caging has the pattern of simple cubic structure.




Cuboctahedral D lattice

















Figure 9a, 9b.

Cuboctahedral D lattice is the only lattice that contains all of its own polyhedral frequencies, and it is the only lattice that is nucleated, Fig. 9a.


The periodicity of cuboctahedron is 1 (p1). In Fig. 9b we see frequency 1 and 2.



The four interpenetrating lattices



















Figure 10a, 10b.


In Fig. 10a we see the 1st frequency of all 4 fcc lattices: tetrahedral 1f A (yellow), tetrahedral 1f B (green), octahedral 1f C (red), and cuboctahedral 1f D (blue). They are in a concentric configuration or the 4 fcc lattices are interpenetrating.

Fig. 10b is showing the 2nd frequency, the 2f D lattice with the other 3 lattices inside.




View of A, B, C, and D fcc lattices from the cubical perspective
































Figure 11a, 11b, 11c, 11d.


Fig. 11a is the tetrahedral A fcc lattice (yellow). Fig. 11b is the tetrahedral B fcc lattice (green). Fig. 11c is the octahedral C fcc lattice (red). Fig. 11d is the cuboctahedral D fcc lattice (blue).






















Figure 12. Inside view of Fig.10b, 4 interpenetrating fcc lattices.





January 21, 2003

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