Resonance, fullerenes and the Middle East?

Middle East Peace Potential through Dynamics in Spherical Geometry (Part #11)

As noted above, reference is often made to football design, when describing carbon buckyballs or the root structure of geodesic domes, as an illustration of the truncated icosahedron. "Buckyballs", in honour of R. Buckminister Fuller, was the early term for a "new" form of carbon -- only recently recognized by chemists in 1985. The general term for such cage-like polyhedra molecular structures is now fullerenes of which the most important is C₆₀ -- as with the number of vertices in a truncated icosahedron, to whose polyhedral structure it corresponds (see especially (cf Fullerenes, Kirk-Othmer Encyclopedia of Chemical Technology). It has been inferred that fullerenes C_n can be constructed for n=20 and for all even n= 24. They have n vertices (i.e. C-atoms), 3n/2 edges, 12 pentagonal and (n-20)/2 hexagonal faces. Other forms include C₇₀, C₇₆, C₈₂ and C₈₄ -- with predictions extending to C₃₉₉₆. The smallest "fullerene" is said to be C₂₀, which has no hexagonal faces, and with its atoms positioned at the vertices of a pentagonal dodecahedron.

Of the greatest importance to the structural viability of the simplest molecules essential to life is the phenomenon of resonance whereby the possible bonding between the carbon atoms in the structure takes a dynamic alternating form. It is this dynamic form which is understood as being energetically the most efficient and economic -- giving rise to structures known as resonance hybrids. The structure is then understood to be represented by several contributing structures (also called resonance structures or canonical forms).

The recognition of the fullerenes resulted in early investigation of the nature of resonance within C₆₀. For example Harald Fripertinger (The Cycle Index of the Symmetry Group of the Fullerene C60, 1996), in a section entitled The resonance structure of the fullerene C60, notes that it was already known that the fullerene C₆₀ had 12500 resonance structures (D.J. Klein, T.G. Schmalz, G.E. Hite, and W.A. Seitz. Resonance in C60, Buckminsterfullerene. Journal American Chemical Society, 108, 1986, pp. 1301 - 1302). Fripertinger produces a valuable tabular summary indicating those which are essentially different.

There is now a very extensive mathematical and chemical literature on the nature of the connectivity within the truncated icosahedral form, and especially C₆₀. This research engenders visualizations which are potentially of great relevance to exploring structural configurations of psychosocial significance. A number are noted in the references (below), but an especially helpful example is that of Heping Zhang and Dong Ye (Cyclical Edge-connectivity and Resonance of Fullerenes, 2007).

The key question for this argument -- given the truncated icosahedral pattern explored above -- is whether "resonance" in some form, and "cyclical edge-connectivity", have implications for the viability of structures reconciling the differences between the "hexagonal" and "pentagonal" mindsets assumed here to be fundamental to the dynamics in the Middle East. The challenge might well be framed as one of reframing the pattern of edges to form a larger whole (cf Beyond Edge-bound Comprehension and Modal Impotence: combining q-holes through a pattern language, 1981). Simply stated, should the challenges of the Middle East be understood as a problem of resonance -- calling for the quality of thinking applied to resonant structures? Understood in this way, the very recent recognition of the existence of fullerenes -- as a "new" ordering pattern for one of the commonest elements vital to living structures -- suggests that current thinking with respect to the Middle East may as yet be insensitive to the viability of more complex possibilities.

Of particular interest to this approach is the use of a Schlegel diagram by those exploring resonance within the truncated icosahedron as the polyedral form of the basic fullerene C₆₀. Such diagrams provide a tool for studying combinatorial and topological properties of polytopes. For three-dimensional polyhedra, they offer a projection into a plane. In the case of a four-dimensional polychoron, this is projected into 3-space, and is therefore commonly used as a means of visualizing four-dimensional polytopes. Such a diagram is constructed by a perspective projection viewed from a point outside of the polyhedron, above the center of a selected facet. All vertices and edges of the polytope are projected onto a hyperplane of that facet.

Hyperplane perspective of truncated icosahedron via Schlegel diagrams with colours distinguishing 12 pentagonal and 20 hexagonal forms, necessarily distorted in the projection. NB: In each case, the form as a whole is to be counted as one of the facets. (images initially generated using Stella Polyhedron Navigator, "rectified" and coloured using Adobe)
Centred over a pentagonal facet	Centred over a hexagonal facet

Experimentally the 5-star and 6-star star images can be presented from the hyperplane perspective (above) as indicated below -- with particular lighting effects.

Hyperplane perspective of truncated icosahedron using star images (addition of images to Schlegel diagram perspective using Stella Polyhedron Navigator)
Centred over a pentagonal facet	Centred over a hexagonal facet

The standard Schlegel diagram can be used to make evident a unique property of the resonance structure of C₆₀ -- of potential relevance to the above argument -- namely that it avoids double bonds within pentagons and maximizes the number of alternating single-double bonds within hexagons. Thus all hexagons have three conjugated double bonds while all pentagons are empty (as shown in the images below). With respect to connectivity across the pattern, one blogger discusses the possibility that bonding is in some way delocalized across the sphere and offers two alternative bonding patterns (C60 Double Bonding Networks, 2010).

Schlegel diagrams of truncated icosahedron distinguishing single and double bonds (orientations corresponding to those above)
Centred over a pentagonal facet	Centred over a hexagonal facet

Also of great potential interest is recent recognition of what might be understood as the stages in a process of "self-organization" within fullerene polyhedral cages (Stan Schein and Michelle Sands-Kidner, A Geometric Principle May Guide Self-Assembly of Fullerene Cages from Clathrin Triskelia and from Carbon Atoms. Biophys J. 94, 2008). Such organizational leads merit the most careful attention with respect to the Middle East.

In relation to the animation above, it is clear that the pattern of pentagons and hexagons could be drawn dynamically into a Schlegel configuration and then unfolded out of the hyperplane into spherical form. Whilst this is relatively straightforward using SVG, it is appropriate to note that the four-dimensional features of the Stella Polyhedron Navigator also permit a significant number of perspectival transformations, some of which specifically approximate the perspective from a hyperplane in the form of a Schlegal diagram. Other software exists to generate such diagrams (notably Polymake), but possibly less intuitively than the Stella application.

The relevance of a "hyperplane perspective" in this context raises the question as to how appropriate connectivity -- namely global "peace" and "sustainability" -- are best to be represented and comprehended, if not "envisaged". With respect to complex system dynamics for which mathematicians make use of a complex plane, a geometric representation of the complex numbers on a "real axis" and an orthogonal "imaginary axis". Its implications have been considered as a means of exploring the strategic relationship between problematique, resolutique, irresolutique and imaginatique (Imagining the Real Challenge and Realizing the Imaginal Pathway of Sustainable Transformation, 2007). The question would then be the relation between a hyperplane and a complex plane with respect to comprehension of "peace" and "sustainability".

Tiling: In contrast to the point made with respect to the "jigsaw" impossibility of appropriate tiling of two-dimensional space with respect to Middle East differences, the "hyperplane perspective" indicates how the polygonal "tiles", subject to perspectival distortion, may indeed be configured in alternative tesselations -- seemingly in two dimensions.

If the tiling metaphor is of relevance to the challenge of reconciling seemingly incompatible frameworks, a further lead is offered by the recent discovery of the unexpected order characteristic of aperiodic tiling patterns of quasicrystals whose 5-fold symmetry bears an unusual degree of resemblance to what have been recognized as "Islamic patterns" (as noted above). Their discovery revealed a new principle for packing of atoms and molecules -- again suggestive of a mindset necessary to discover new ways of "packing" incompatible perspectives. As with reference to a hyperplane perspective on polyhedra, mathematically, quasicrystals have been shown to be derivable by treating them as projections of a higher-dimensional lattice. There are two known types of quasicrystal:

polygonal (dihedral) quasicrystals, have an axis of eight, ten, or 12-fold local symmetry (octagonal, decagonal, or dodecagonal quasicrystals, respectively). They are periodic along this axis and quasiperiodic in planes normal to it.
icosahedral quasicrystals, are aperiodic in all directions.

Again the nature of the mathematical study of quasicrystals helps to frame the question of the nature of the "higher dimensionality" -- to which the mathematicians of both Islamic and Jewish faith would rspectively subscribe in principle -- and what are the possibilities for projections from that context into two dimensions? Of particular interest in the history of quasicrystals is the manner in which their discoverer, the Israeli chemist Dan Shechtman (Nobel Prize, 2011), was repeatedly deprecated for his investigations by a previous double Nobel laureate in that same domain. An example for which the alleged eppur si muove remains a symbol?

What explorations of other unusual patterns of order are similarly deprecated by those conventionally esteemed as the highest authorities in the matter? With respect to the psychosocial domain, such denial has been honoured by its own acronym TINA (There Is No Alternative). According to TINA, economic liberalism is the only valid remaining ideology. There is no scope for "new thinking" or the kind of paradigm shift signalled by quasicrystal discovery.

Images relevant to discussion of "tiling" according to the pattern of aperiodic quasicrystals (reproduced from Wikipedia entry)
Example of quasicrystal pattern	Example of Penrose tiling pattern

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Resonance, fullerenes and the Middle East?