An Introduction to Synergetic Crystallography -February 1998

An Introduction to Synergetic Crystallography

by Russell Chu

in collaboration with Kirby Urner and Karl Erickson

First published Feb. 28, 1998

(left) Two frequency Isotropic Vector Matrix ( 2f IVM )

The purpose in this paper is to show that the various combinations of four interpenetrating Isotropic Vector Matrices ( IVMs ) are sufficient to map most basic crystal structures.


History of my Research

I introduced the idea of four interpenetrating IVMs in 1985 with the publication of an article on the isomatrix in Synergetica, a journal about Synergetics. In this study it was also shown some of the relationships to simple chemical structures.

It was only recently that it has been possible to illustrate these complex spatial relationships in a simple manner through the Java based program "Struck" created by Gerald de Jong.

Definition of Key Terms

The Isotropic Vector Matrix ( IVM ) is R B Fuller's description of the closest packing of spheres in the cubic close-packed (ccp) configuration. Fuller described the cuboctahedrally coordinated ccp as a vectorial matrix, with vectors connecting all vertices from the center of each sphere, and Isotropic for having a high degree of symmetry.

In a single layer of close-packed spheres each sphere is surrounded by 6 others (hexagonal coordination). When three or more layers are placed on each other we can have ABCABC sequence, the cubic close-packing (ccp) with each sphere surrounded by 12 others in a cuboctahedral coordination, and ABAB sequence, the hexagonal close-packing (hcp) with each sphere surrounded by 12 others in the anti-cuboctahedral coordination. Other more complex sequences are possible.

IVM can be seen as tetrahedra and octahedra filling space, with twice as many tetrahedra as octahedra.

The IVM in Engineering and Architecture is the Octet Truss, space frame, or tetrahedral truss.

Frequency in the IVM is equivalent to size/volume. As the size of the matrix increases we see progressively higher frequencies of concentric cuboctahedra shells.

Tetrahedron is Unit Volume (1) in Synergetics.

4 frequency Tetrahedron:

The concentric hierarchy is the occurrence of polyhedra around a common center point. In the IVM we have 12 spheres around one, which is the cuboctahedron with a center sphere. These concentric polyhedra progress from the tetrahedron to its dual (negative tetrahedron), which together comprise the duo-tet cube. The dual of the cube is the octahedron, and these two shapes (cube and octahedron) define the rhombic dodecahedron, and next comes the cuboctahedron which is the dual of the rhombic dodecahedron.

Volumetrically we have: T+ with volume 1, T- with volume 1, Duotet Cube with volume 3, Octahedron with volume 4, Rhombic Dodecahedron with volume 6, and cuboctahedron with volume 20. The polyhedra progression occurs within the four frequency IVM and repeats every fourth frequency.

The edges of the Cube and the Rhombic Dodecahedron would look as drawn only at a very high frequency. The cube is formed by the combination of T+ and T- ( the Duotet Cube) and the Rhombic Dodecahedron is formed by the combination of T+, T-, and O.

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Concentric Hierarchy

The 4 IVMs: Tetrahedron ( T+ ), ( T- ), Octahedron ( O ), and Cuboctahedron (CO)

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Why 4 IVMs?

Synergetics view:

The observation of concentric polyhedral growth in the IVM occurring every fourth frequency suggested the existence of three other IVMs, as described above in the concentric hierarchy. The second IVM being the Positive Tetrahedron (T+) centered; the third IVM is the Negative Tetrahedron (T-) centered; the fourth IVM is the Octahedron (O) centered, and the first IVM being the Cuboctahedron (CO) centered.

The 4 IVMs are necessary to account for all the frequencies of the Tetrahedra, Octahedra and Cuboctahedra. The Tetrahedra pulsate through all 4 IVMs; the Octahedra pulsates through the Octahedron centered IVM and the Cuboctahedron centered IVM; and the Cuboctahedra does not pulsate, it occurs in all its frequencies within its own IVM.

From the 4 frequency tetrahedron above we can see that a 2f tetrahedron (looking from the tip) is centered on a octahedron, a 3f tetrahedron is centered on a tetrahedron, and a 4f tetrahedron is centered on a center sphere or a cuboctahedron.

By following the tetrahedron ( T+ ), in the table below, through its frequencies from 1f through 4f we can see the pulsating from T+ to O to T+ to CO.

Frequency and Polyhedral Growth in the IVM

Frequency	( T+ ) IVM	( T- ) IVM	( O ) IVM	( CO ) IVM
1f	1f T+	1f T-	1f O	1f CO
2f			2f T+, 2f T-	2f O 2f CO
3f	3f T+	3f T-	3f O	3f CO
4f				4f T+, 4f T- 4f O 4f CO

The 4 IVMs can represent lattices by themselves or in combinations such that two, three or four IVMs interpenetrate each other in a concentric manner.

(T+) IVM ( A )	(T+) IVM = A (T-) IVM = B (O) IVM = C (CO) IVM = D Following are some translations from the cubic system to the IVM system: The 3 cubic Bravais Lattices: Cubic-P, simple cubic = A+B or C+D Cubic-I, body-centered cubic (bcc) = A+B+C+D Cubic-F, face-centered cubic (fcc) = C Diamond like structures= A+C or B+D
O IVM ( C ) this is the face-centered cubic	T- IVM ( B)
Simple Cubic ( C + D )	OC IVM ( D )
Body Centered Cubic (A+B+C+D)	( B+ C+D )