Tuesday, April 10, 2012

How to Assemble Tetrahedrons from Building Blocks:

when the building blocks are made of closest packed spheres

It is simple to assemble tetrahedrons of different sizes when the building block is one sphere, but it is quite different if we try to assemble with bigger building blocks. The pictures below illustrate the building block composition of the tetrahedrons as they increase in size.

(click on pictures to enlarge)

A        B           C              D                   A1                       B1...
Fig. 1  Tetrahedral growth

Fig. 2   Exposed centers

Fig. 3   Octahedral growth

Fig. 4   One-sphere growth
(conventionally known as cuboctahedral)

NoteIn order to make it easier to understand the patterns, I decided to use the polyhedral nomenclature to describe the building blocks - such as the tetrahedron and octahedron, even though building blocks do not behave as polyhedrons. 

Shown in Fig.1 is a series of tetrahedrons that are growing in size. The edges of the tetrahedrons are increasing by one sphere at a time. 
In Fig. 2 the top and front tetrahedrons have been removed so the centers can be seen.

As you can see in Fig.1, the composition of the tetrahedrons are size dependent. The building blocks differ as the size of the tetrahedrons increase. There are several recursive patterns.

What can we tell about these series of structures?
  1. The overall repeating pattern in (Fig. 1) is that of the first four tetrahedrons labeled A, B, C, and D, which repeats  as it grows bigger - A1, B1, C1, D1,  A2,  B2, ... . The centers of each of the four tetrahedrons (A, B, C, D) are different building blocks, and they are color coded white, black, blue and red, as shown in (Fig. 1, 2, 3)
  2. Embedded in the tetrahedral growth series are also the growth of the octahedral and the one-sphere (cuboctahedral) growth series (Fig. 3, 4). 
  3. The relationship between the 4 center building blocks is that of 2 pairs of duals that are evolving (double duals): A-C and B-D. White A (up-tetrahedron) combines with blue C (down-tetrahedron) and black B (octahedron) combines with red (one-sphere, cuboctahedron). As shown in (Fig. 1) the 2 pairs of duals alternate in the progression and the duals themselves also alternate. Duals A-C are even number edge-spheres tetrahedrons and duals B-D are odd numbers edge-spheres tetrahedrons.
  4. In Fig. 1, every other tetrahedron (B, D, B1, D1...) have the familiar pattern of the octet (octahedron-tetrahedron) combination. Note that the octahedrons are always one-edge sphere bigger than the complementary tetrahedron. It is not equal in edge length as in the octet truss.
  5. The other half of the series is composed of 'up'-tetrahedron and its complementary 'down'-tetrahedron. As fig. 2 shows - the first white tetrahedron (A) is pointing up, and in the third (C) the blue tetrahedron in the center is pointing down, next in (A1) the tetrahedron in the center is white (up) again, and it keeps alternating in this manner.
  6. These 4 building blocks re-occur in the 4 isotropic vector matrix (4 fcc lattices) in "Mapping the Hidden Patterns in Sphere Packing" R Chu 2003 Chart 1 .
  7. These patterns are like 3dimensional fractals. We can substitute the octahedron in D (Fig. 1) with smaller building blocks or substitute the tetrahedrons in A1 (in Fig. 1) in the manner shown in (Fig. 5) below. As the tetrahedrons get bigger we can progressively substitute with smaller equivalent building blocks. This process of substituting with smaller building blocks could continue until it is composed of only single sphere building blocks.
  8. Fig. 4 shows the one-sphere growth surrounded by 12 spheres in the second layer (cuboctahedron). It is Fuller's vector equilibrium (VE). 
  9. All the possible building block combinations of tetrahedrons are the  symmetries within the tetrahedron. 

Fig. 5 

In Fig. 5, the 5sphere-edge tetrahedron and the 6sphere-edge tetrahedron show an increased level of complexity compared to D and E1 in (Fig. 1). As the tetrahedrons increase in size the levels of complexity increases.  So far we have only been looking at closest packed building blocks. A great variety of lesser density packing can be considered.
I do not have the skills to write these patterns as fractals or describe them mathematically. If any one is interested I would be happy to collaborate. Comments or questions are welcomed.
Russell Chu, 206-313-5708 (verbchu@gmail.com)

Monday, October 3, 2011

Building Block Geometry and Closest Packing of Spheres

The display of oranges in the grocery stores or street markets is a common example of closest packing of spheres. They are usually of the cubic closest packed (ccp) type, as the chemists named it. The other, they called it hexagonal closest packed (hcp). These stacking of oranges and cannon balls got a lot of attention through out the years. Johann Kepler conjectured that it was the closest one could pack globules or spheres. He also thought that it had something to do with how nature made snow flakes with hexagonal shapes. Many years later Buckminster Fuller thought that "Nature's Coordinate System" could be understood by investigating the closest packing of spheres and a complex of related geometrical structures. My introduction to geometry was through Fuller’s Synergetics. Maybe with building block geometry of closest packing of spheres we could get a little closer to understanding how nature builds.

So, what is the connection between building blocks, closest packing of spheres, a stack of cannon balls and the Egyptian pyramids?  1. They all have the same shape (pyramids). 2. They are made of building blocks even though the shape of the building blocks are different. 3. The code or the geometry of how the building blocks go together is the same. The code: build a square base, add consecutive layers on top such that the number of bricks or oranges at the edges of the next layer is reduced by one, thus the edges of each layer move towards the center of the square by half the width of the building block. This is what naturally happens with the oranges, as you stack a layer on top of each other until the last layer has only one orange.

Most interpretations of closest packing of spheres are based on euclidean geometry such as faced centered cubic (fcc) lattices or cells. Fuller introduced the isotropic vector matrix and the octet truss, as a vectorial - synenergetic representation of closest packing composed of octahedron and tetrahedron.

The concept of building block associated with closest packing of sphere is not new. When we build a pyramid with oranges we add them one at a time - as a building block at a time. In essence, we are basically dividing the pyramid by one - a building block of one orange. What is new is that there is more to it. We can divide closest packing of spheres, ccp or hcp, by larger building blocks, and we get a geometry of evolving building blocks where ccp and hcp are the two possible ways the building blocks combine in a lock and key manner. In the case of closest packing, the building blocks are constrained to be closest packed and the combined structures are also closest packed. The building blocks evolve in multiples of one -- such as 1, 2, 3, 4,... Not all building blocks or combinations of building blocks will closest pack.

Building block geometry is non euclidean it is discrete. It is a geometry of relationships between space and building blocks. In the case of closest packing of hard spheres the ratio is 74% spheres and 26% space. This is where it differs from the euclidean all space filling (100%) solid geometry, the solid polyhedral geometry does not account for space. 

Below is a video showing the assembly of a 20 ping pong ball 'triangular pyramid' with 5 building blocks, each composed of 4 closest packed balls. This structure is cubic closest packed.

A Comparison of Polyhedral and Building Block Geometry

The two models are color coded the same (black and white). The left model is a polyhedral representation of the model on the right - which is the one in the video above. The left model has 20 vertexes and the right has 20 spheres. On the right it is composed of 5 building blocks - 4 white and 1 black. On the Left there are the corresponding 4 white and 1 black tetrahedron and in addition 6 more tetrahedrons and 4 octahedrons (in blue). The polyhedral representation comes from drawing lines interconnecting all the centers of the spheres, which Fuller named it octet truss also referred to as the isotropic vector matrix or face center cubic lattice (fcc).
We could say tetrahedral building blocks combine to closest pack and tetrahedron and octahedron combine to fill all space.

The importance of hcp and ccp/fcc in structures

When we look at the structures of the chemical elements at ground state (zero temperature and pressure) we would find that around 49% of the 92 elements have the hcp or ccp structure (Table 1.1 "Bonding and Structure of Molecules and Solids" David Pettifor - The most frequently occurring structure types are the closed packed metallic lattices hcp,fcc, and bcc (body-centred cubic) with packing fractions of 0.74, 0.74, and 0.68 respectively..). Closest packing of spheres is a minima, the lowest energy state. Building block geometry could also model the structures that are less dense.

Fuller emphasized the fact that nature does not have solids. Maybe the non solid building block geometry of sphere packing from closest packing to very loose packing such as linear, chains, and organic structures could be a better model for how nature builds.

Models of evolving building blocks and how they assemble in closest packing

The display below was made to be shown at the 2011 Design Science Symposium Exhibit, which will run from 6th to13th of November at the Waterman Gallery, Rhode Island School of Design. 

The top row shows evolving building blocks - beginning with one. The bottom row shows some of the assembled building blocks in the hcp or ccp configuration. You can play with the building blocks by dis-assembling and assembling the various structures.

This are sections of the above display. The row of numbers corresponds to the number of spheres in each building block. At the very bottom is the type of closest packed structures that can be built with the above building block - HCP and or CCP. We will label the 2 sphere building block (bb2). We get a tetrahedral structure by assembling two bb2. We get an octahedral structure by assembling two bb3. (bb3)s can only make hcp structures.

The bb4 can assemble into hcp or ccp structures. An equivalent polyhedral (octahedron + tetrahedron) structure is shown for comparison.

The bb6 can only assemble in hcp. On the right, the bb6+bb1 can only assemble in ccp. This is an assembly of two separate building blocks - bb6 in white and bb1 in black.

The bb8 can assemble in hcp or ccp. The bb4+bb19, bb10+bb16, and bb13+bb14 complement each other in assembling into ccp structures. bb13 is the lower one with all white balls (12 around 1). A model like bb13+bb14 was given to me by Timothy Cox in 2003 (patent Des. 405,385).

I would like to further explore and discuss the geometry of building blocks in this blog. I would appreciate your thoughts and comments. Please contact me if you are interested or know of someone that has done similar work. If you are near RISD, during Nov. 6 - 13 (2011), please check out Waterman Gallery, and I will also be at the Symposium.