1st July 2012  Draft
Middle East Peace Potential through Dynamics in Spherical GeometryEngendering connectivity from incommensurable 5fold and 6fold conceptual frameworks /  Introduction Produced on the occasion of worldwide media focus on the patterned movement of a truncated icosahedron in Kiev, whilst the United Nations demonstrates its longstanding inability to elicit relevant intelligence in more creative response to ongoing massacres in the Middle East. IntroductionThis is an exploration of the hypothesis that unique belief systems depend for their coherence on distinctive patterns typically embodied in geometrical symbols in two dimensions. On the basis of that assumption, the case tentatively explored here is that of the "incommensurability" of the 5fold Star of Islam and the 6fold Star of David of Judaism  both symbols appearing on flags of the nations having those distinct faiths. Efforts at the reconciliation of the perspectives embodied in these faiths are seen here to be as questionable as that of endeavouring to fit together the pieces of a jigsaw puzzle on a flat surface  when the pieces do not "match" or "fit" together. That approach may be compared to the practice of the impatient in seeking to "force" the pieces together  even "banging" on them to ensure that they lock together. Even if "successful", the resulting picture is "wrong". This metaphor has both conceptual implications and territorial implications. To employ a chemical metaphor, in the case of Judaism the pieces might be understood as 6valent, in comparison with "5valent" pieces of Islam. The guiding assumptions for this exploration are the muchquoted statements of Albert Einstein:
The approach taken here explores the possibility that the "pieces" only fit together on a threedimensional surface, namely a sphere. It is the spherical geometry that then merits consideration, together with the challenge of how to get from any "misfitting" twodimensional layout to a threedimensional form. Of course, twodimensional layouts are far more readily comprehensible. Hence the focus on them. However the threedimensional layout has the potential of rendering comprehensible a far more elegant layout which may well exemplify intuitions characteristic of the faiths so dramatically opposed. The approach follows from various earlier explorations of the potential of mathematics to offer a new perspectives on these issues, including: Geometry of Thinking for Sustainable Global Governance (2009), Spherical Configuration of Categories  to reflect systemic patterns of environmental checks and balances (1994), Using Disagreements for Superordinate Frame Configuration (1993), and Mathematical Theology: future science of confidence in belief (2011). In the case of the Middle East, the argument has previously been developed that the opposing cultures are widely recognized as highly competent in number theory and geometry (cf. Wikipedia List of Jewish mathematicians; Keith Devlin, The Mathematical Legacy of Islam, 2002; Keith Critchlow, Islamic Patterns: an analytical and cosmological approach, 1999; Issam ElSaid, Islamic Art and Architecture: the system of geometric design, 2008; Center for South Asian and Middle Eastern Studies, Islamic Mathematics; 2000, G. Donald Allen, Islamic Mathematics and Mathematicians, 2000; Wikipedia, Mathematics in Medieval Islam). Carra de Vaux continues to be quoted by Islamic scholars to the effect that : they (the Muslims) were indisputably the founders of plane and spherical geometry, which did not, strictly speaking, exist among the Greeks (The Philosophers of Islam, 1921). These skills do not however seem to have been brought to bear on the psychosocial dynamics which divide them in relation to territory  other than with respect to ballistics. Nor does there appear to have been any interest in suggesting that this possibility merits exploration, as previously discussed (And When the Bombing Stops? Territorial conflict as a challenge to mathematicians, 2000). What follows is essentially the description of the development of an interactive visualization experiment to illustrate possibilities accessible through any web browser using scalable vector graphics. The focus is on the experimental design and modification of the visualization, effectively enabling imaginative exploration of possibilities  both playfully and as offering a pattern language through which differences and their implications can be disputed in terms of design aesthetics. The experiment also highlights the potential of a visualization package, the appropriately named Stella Polyhedron Navigator, through which the "global" implications of such patterns can be explored, as previously suggested (Polyhedral Pattern Language: software facilitation of emergence, representation and transformation of psychosocial organization, 2008). Fundamental cognitive patterns assumed to be characteristic of belief systemsThe assumption here is that an extremely fundamental cognitive identification is associated with the distinct patterns below  thereby constituting an expression of quite different faiths. These distinct patterns may imply a yet more fundamental identification which cannot be readily expressed through symbols in two dimensions. Whilst these symbols are indeed distinctly favoured in the flags of the cultures in conflict as considered here, the nature of their fundamental relationship to some form of cognitive identity could be questioned. The case of the Star of David is more straightforward. The use of the fivepointed star by Islamic nations is however subject to reservations  especially given its use in the flags of nonIslamic cultures and for other purposes. The Red Star has been widely used as a symbol of both Socialism and Communism. Use of the sixpointed star by Muslims  derives from a period when the Star of David and the Seal of Solomon were more harmoniously related in accord with medieval Jewish, Christian and Islamic legends. Of relevance to the argument here, the 5pointed star was a focal symbol of the Ottoman Empire. through which the crescent moon and star became affiliated with the Muslim world. This type of flag has become the de facto Islamic flag, and is used, with variations, by multiple Muslim lands such as Algeria, Azerbaijan, the Comoros, Malaysia, Maldives, Mauritania, Pakistan, Tunisia, Turkey, Turkmenistan, the Turkish Republic of Northern Cyprus, Uzbekistan, and the Western Sahara. As the crescent and star have no religious significance however, some Muslim scholars are against attaching these signs on mosques and minarets or using them to denote Muslim societies. The 5pointed star as used by Islam is not limited to red in colour, green may also be used, or white against a background such as red or green.
Mathematically these patterns cannot be readily combined. This issue is described in mathematics in terms of tiling, described generically as tesselation (cf Craig Kaplan, The Trouble with Five, Plus Magazine, 2007). This of course has immediate relevance to the attribution and layout of territory  with all the implications for politics and violence which have characterized the Middle East (And When the Bombing Stops? Territorial conflict as a challenge to mathematicians, 2000). It is noteworthy that in the quest for a key metaphor for the structural challenges of the immediate future, the tessellation of disparate orders was identified by the sesquicentennial symposium of Boston University (Lance Morrow, Metaphors of The World, Unite!, Time, 16 Oct. 1989). Such tilings of zones of order, might also be understood metaphorically as paving stones separated by cracks of varying sizes. For optimists these might be hairline cracks; for pessimists the degree of separation might be such as to prevent movement and to offer little protection from the disorderly 'gunge' rising from beneath. The approach taken here is to treat each of these patterns as "carried" by a polygon. This gives rise to the following, which still cannot be "fitted" together as a pattern in twodimensions..
A set of hexagons and pentagons can however be uniquely fitted together as a particular threedimensional polyhedron, namely the truncated icosahedron. This has a total of 32 faces: 20 hexagons and 12 pentagons. It has 60 vertices with 90 edges of the same length. These are of two different types according to their position in the structure: 60 of one type, and 30 of another.
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Animation of interaction and interlocking between cognitive patternsRather than introducing the visualization with a verbal explanation, this is presented immediately through the following screenshots of phases in the animation accessible here. A commentary is given below.
The animation progressively "populates" a space (a "territory") in two dimensions with instances of either a hexagon or a pentagon  as "vehicles" for the Star of David and the Islamic Star respectively. The polygonal vehicles are also usefully understood as jigs or templates enabling appropriate positioning  as in many processes for manufacturing objects which need to be held accurately in place and ensure pattern replicability. The final pattern reinforces the point that the two sets of polygons cannot be "fitted" into any coherent design  even though some edges between the two different shapes can be established as common, as "joining". The seemingly arbitrary pattern can, as a whole, be "folded" into a spherical form in three dimensions, as indicated by the following screen shots and the accompanying animation.
The screenshots above, and the animation below, are suggestive of the challenge and possibility of getting the contrasting 5fold and 6fold patterns to "fit together" in three dimensions. The subsequent commentary endeavours to clarify some of the implications.
The stages of the animation are suggestive of an intuitive relation between the Star and Crescent of Islam. Disposed together otherwise, the elements of the unfolded net could well be indicative of either the 5fold or 6fold stars. An association football is constructed by stitching together 32 panels of waterproofed leather or plastic: 12 regular pentagons and 20 regular hexagons. As noted, the 32panel configuration is the spherical polyhedron corresponding to the truncated icosahedron; it is spherical because the faces bulge from the pressure of the air inside. The basketball and volleyball are not constructed in this way. Reference is often made to the football design, when describing carbon buckyballs or the root structure of geodesic domes, as an illustration of the truncated icosahedron. Interactive design of cognitive pattern animationIt should be stressed that the approach taken here is a "proof of concept" with respect to such an animation and the possibilities of its design. The result demonstrated has a variety of inadequacies. These can be variously attributed to:
The advantage of the approach taken in using SVG is that  with a minimum of skill  the code can be readily modified to change the animation. SVG code can be changed using any text editor. It does not need to be compiled. Whilst modifying the structure of the animation as a whole may require more skill, parameters such as the following can be readily changed (as discussed more systematically below):
Changing duration means that the rates of one phase relative to another can be modified, offering the possibilities of:
Clearly these changes may serve primarily to suggest:
The design context is usefully clarified by the information in the following box insert  which may be readily ignored, but is useful to enable others of greater competence to improve on the approach and the design.
The note above shows how the Stella Polyhedron Navigator was used to facilitate production of the first phase of the animation using SVG. There is a case for considering how this application could be used by itself to experiment with useful animations  avoiding the issues with respect to technical competence in SVG and the inability of some browsers to handle SVG animation.
Visualizing alternative stories through manipulation of animation design optionsThe purpose is to enable people to experiment interactively with various "stories", especially in the twodimensional territorial occupation phase. Clearly some options privilege one framework and place the other at a disadvantage. The options could could be seen like a musical keyboard in which the challenge is to render elegant the relationships between the two by manipulating the many combinations of parameters. The animation offered as a demo is the consequence of one set of design choices. Clearly many others are possible. The animation could itself be improved in a variety of ways with greater expertise..
Polyhedral catalysts of global imaginationThe following images display the truncated icosahedron from the animation above. The 5pointed star and 6pointed star are however displayed in white backgrounds with an alternative colouring scheme. The remaining images show various geometrical transformations of the form using the features of the Stella Polyhedron Navigator. The purpose is to encourage imaginative reflection through alternative windows on the relationship between the 5fold and the 6fold  as it may apply in particular to the Middle East. As with the various map projections of the globe, the images are included to provoke the question as to what they might variously offer, if anything, as a new way of seeing the relationships there.
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The implications of any morphing process can also be explored from the perspective that the truncated icosahedron is a form of "compromise" between an embedded dodecahedron with 20 vertices (touching the 20 hexagonal faces) and an embedded icosahedron with 12 vertices (touching the 12 pentagonal faces). With one or other emerging dynamically to a greater degree through morphing, this recalls the importance attached by R. Buckminster Fuller to a related form, the cuboctahedron, which he described as being a vector equilibirum because of the "pumping" process through which it could be transformed (Vector Equilibrium and its Transformation Pathways, 1980). Selected faces of the truncated icosahedron can also be "augmented" as illustrated by the following pair of images.
The truncated icosahedron may be transformed into a variety of geodesic spheres  according to the frequency, namely the number of subdivisions along each edge of that original polyhedron.
The original truncated icosahedron may also be transformed into a zonohedron according to various rules, as illustrated by the following.
The original truncated icosahedron may be subjection to a stellation process, namely producing a new polyhedron with faces that lie in the same planes as the faces of that original. The process consists of extending elements such as edges or face planes, usually in a symmetrical way, until they meet each other again. There are 1117 stellations identified by Stella Polyhedron Navigator of which particular examples are given below.
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In addition to the final product of the stellation process (shown below), various transformations of the original truncated icosahedron are possible within Stella Polyhedron Navigator in order to give a "glimpse" of their structure projected in four dimensions. One of these is presented below.
Design, pattern language and geometryThe visualization gives focus to the possibilities of design in relation to pattern language and geometry. Design and pattern language have been a primary focus of Christopher Alexander in the quest for the qualities characteristic of an attractive place to be (A Pattern Language, 1977). This suggests the extension of "pattern language" to nonmaterial domains, as experimentally elaborated separately (5fold Pattern Language, 1984) as part of larger exercise (Patterns of Conceptual Integration, 1984) including a review of Patterns of Nfoldness: Comparison of integrated multiset concept schemes as forms of presentation (1980). .In the light of Alexander's more recent work ((The Nature of Order: an essay on the art of building and the nature of the universe, 20034) he has stressed the geometrical aspects of order in a quest for a computational approach to "wholenessextending" design (HarmonySeeking Computations: a science of nonclassical dynamics based on the progressive evolution of the larger whole, International Journal for Unconventional Computing (IJUC), 2009). Significantly he concludes that paper with the statement:
This approach is discussed separately (HarmonyComprehension and WholenessEngendering: eliciting psychosocial transformational principles from design, 2010). The question in relation to the design of any animation is what is the design which most fruitfully integrates disparate elements, whether visually or in terms of the rhythm and pace of its "story". The suggestion here is that a "design platform", such as offered by SVG, would enable many to experiment with different stories and to focus dispute on the implications of those stories as designs which could potentially embody greater elegance  or recognize diff rent preferences in that respect. As implied by the quest of Alexander, are there designs in threedimensions which might embody a greater sense of harmony? Dialogue implications of design and geometryOne approach to the pentagonal and hexagonal spaces is to see them as different dialogue arenas, whether in terms of theme or style. Common sides in any configuration of those spaces  pentagon meets hexagon  could be interpreted as a common discussion thread, or as an issue confronted from two sides. In complex situations like the Middle East there are many fora and many discussion threads. There is little attention to how these constitute any kind of system  or to how different dialogue arenas might be related. The issue has been partly explored separately (Interweaving Thematic Threads and Learning Pathways: noonautics, magic carpets and wizdomes, 2010), notably in relation to configuration through the web in threedimensions (Spherical Configuration of Interlocking Roundtables: Internet enhancement of global selforganization through patterns of dialogue, 1998), and notably in relation to governance (Enabling a 12fold Pattern of Systemic Dialogue for Governance, 2011). With respect to a pattern of dialogue arenas envisaged as "twodimensional", the question is how to envisage the "hooking capacity" which would enable the pattern as a whole to be "folded" into "threedimensions"  whatever that may be understood in terms of greater integration, "clicking into place", or enabling emergence and recognition of understandings of a subtler order. Such understandings are of course acclaimed in principle in the case of faiths obliged to offer twodimensional reflections of an understanding which cannot be fully embodied within such constraints. A pattern of dialogue arenas in "twodimensions" may simply not be viable, however they are shuffled into different configurations  as suggested by the animation. To what degree it may "work" in "threedimensions" remains to be determined, and therefore merits attention. Reference is made to the football design, as an illustration of the truncated icosahedron when describing carbon buckyballs or the root structure of geodesic domes, as separately discussed (Understanding Sustainable Dialogue: the secret within Bucky's Ball? 1996). Systems perspective on threedimensional cognitive configuration and communicationThere is widespread familiarity with systems diagrams, wiring diagrams and circuit boards as represented in two dimensions. They embody insights into connectivity vital to the operation of many systems. The twodimensionality of the representations may of course obscure the fact that crossover points do not necessarily imply connectivity. The key question raised by the Middle East dynamic is how much connectivity is required to channel dynamics which otherwise take chaotic form? What form does this connectivity need to take? How complex does the pattern need to be to hold the complexity of the situation  specifically to reconcile patterns of organization based on 6fold and 5fold (cognitive) organization? It is appropriate to note that the effectiveness of memory operation in supercomputers is dependent on wiring best understood in terms of three dimensional configurations  notably patterned on polyhedral forms. Why should it be so readily assumed that the complex dynamics of the Middle East do not require consideration of corresponding complexity? In the light of a biological systems perspective, requisite integration involves a pattern of metabolic pathways. It is possible to see the chaos of the Middle East in terms of subsystems of such a pattern  between which adequate connectivity has not been provided to ensure viability. "Viability" means "life"  as enabled by appropriate feedback loops through which the system is cybernetically controlled.
What gets treated as an irrelevant externality by this approach, as discussed separately (Reintegration of a Remaindered World: cognitive recycling of objects of systemic neglect, 2011)? What gets dangerously excluded from any such pattern of checks and balances? Given the concern with change in relation to globalization, it is interesting to note that "metabolism" derives etymologically from "change" and "throw" (in the sense of ballistics, but also of ball in its various senses). Especially fascinating from this "metabolic" perspective is the sense of a requisite "metapattern", as notably articulated by Gregory Bateson (Mind and Nature: a necessary unity, 1979):
And it is from this perspective that he warned in a muchcited phrase: Break the pattern which connects the items of learning and you necessarily destroy all quality. How is this to be understood in relation to the symmetry breaking mentioned above  where it is the higher degrees of symmetry which are essential to the pattern, its memorability, and its communicability over time? How does this relate to governance of sustainability? Of some relevance, from a systems perspective, is the increasing recognizing of the role of trillions of bacteria in the metabolism and viability of the human body  outnumbering human cells by 10 to 1 (Jennifer Ackerman, How Bacteria in Our Bodies Protect Our Health, Scientific American, June 2012). Otherwise titled "The Ultimate Social Network", the article asks with respect to "Your Inner Ecosystem", who's in control? The question of what patterns need to be connected to enable integrative emergence can be discussed in terms of symbols, sets of principles or values, or other cognitive fundamentals. In the case of faiths, this may be understood in terms of theology  narrowly understood. However "theology" may be generalized to include any beliefs in fundamental, central subtleties, as separately discussed (Mathematical Theology: future science of confidence in belief, 2011; Patterns of Nfoldness: Comparison of integrated multiset concept schemes as forms of presentation, 1980). Also of relevance to the contrasting "orientations" which become evident in three dimensions, is the variety of patterns required to constitute an integrative whole  an alternative perspective on the cybernetic Law of Requisite Variety  as it might apply to knowing within human psychosocial systems. Whereas much has been made metaphorically for the "lateral thinking" which is strangely consistent with "twodimensional" creativity, and to that extent comprehensible to a degree, more mysterious is what might be implied by "voluminous thinking" consequent on eliciting "threedimensional" insight (From Lateral Thinking to Voluminous Thinking: unexplored options for subterranean habitats in dense urban areas, 2007). Triangulation, connectivity and "stitching": enabling coherent global system dynamicsThe role of triangulation has been reviewed separately with respect to triadic logic, triadic dialectics, triadic strategic applications,
triadic conceptualization and That discussion notably referred to the focus of R. Buckminster Fuller who argued extensively for the fundamental importance of triangulation as the basis for the stability of structures, notably with respect to his application of spherical triangulation to geodesic domes (Synergetics: explorations in the geometry of thinking, 1975). He demonstrates the need for omnitriangulation as a fundamental requirement of system integrity:
It is possible therefore that the integrity of psychosocial systems, and the connectivity of the "patterns which connect" of Gregory Bateson, involve an "omnitriangulated" emotional engagement. Fuller's insights have been applied separately in the explorations of the structural requirements for the possible polyhedral organization of governance (Towards Polyhedral Global Governance: complexifying oversimplistic strategic metaphors, 2008; Configuring Global Governance Groups: experimental visualization of possible integrative relationships, 2008; Geometry of Thinking for Sustainable Global Governance: cognitive Implication of Synergetics, 2009). The argument has also been developed in relation to a degree of intuitive understanding of geometry upon which strategic discourse relies through metaphor (Experience of Cognitive Implication in Fundamental Geometry: unexamined metaphoric framing of strategic discourse, 2012). This noted the case made from the perspective of cognitive psychology by George Lakoff and Rafael Nuñez (Where Mathematics Comes From: how the embodied mind brings mathematics into being, 2001).
The following images show how the lines defining the stars connect together in the truncated icosahedral pattern  also indicating where necessary connectivity is not (quite) possible in twodimensions. Note that the function of the external polygons as "vehicles" or templates is now of secondary significance  although those forms are now engendered within each star form, and coloured accordingly..
The "lines" in the image above also call for reflection on how they might function in systemic terms, especially with respect to the emergent dynamics of any such system:
Other suggestive metaphors, readily recognized as having greater cognitive relevance, include:
Potentially more intriguing are the implications suggested by:
The pentagon and the hexagon, as shown above, are intimately related in geometrical terms to a form of triangulation from which the 5fold and 6fold stars emerge. The question which might fruitfully be asked in relation to the earlier arguments concerning the cognitive significance of polyhedra and triangulation, is whether there are systemic implications suggested by the metaphors mentioned immediately above. The patterns suggest the significance of connectivity being enabled or disenabled in dynamic systemic terms  readily understood in the operation of a model train set in which traffic is allowed or disallowed. If such connectivity were to be understood as sequentially phased, as is characteristic of electrical systems, is there then a sense of effects analogous to those of the rotation operations of motors and dynamos? How might cognitive "cycles" be identified as characteristic of the global organization of the pentagonhexagon patterns in a truncated icosahedron (cf. Cognitive Cycles Vital to Sustainable SelfGovernance, 2009)? Resonance, fullerenes and the Middle East?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 cagelike polyhedra molecular structures is now fullerenes of which the most important is C_{60}  as with the number of vertices in a truncated icosahedron, to whose polyhedral structure it corresponds (see especially (cf Fullerenes, KirkOthmer 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. Catoms), 3n/2 edges, 12 pentagonal and (n20)/2 hexagonal faces. Other forms include C_{70}, C_{76}, C_{82} and C_{84}  with predictions extending to C_{3996}. The smallest "fullerene" is said to be C_{20}, 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 C_{60}. 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 C_{60} 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 C_{60}. 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 Edgeconnectivity 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 edgeconnectivity", 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 Edgebound Comprehension and Modal Impotence: combining qholes 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 C_{60}. Such diagrams provide a tool for studying combinatorial and topological properties of polytopes. For threedimensional polyhedra, they offer a projection into a plane. In the case of a fourdimensional polychoron, this is projected into 3space, and is therefore commonly used as a means of visualizing fourdimensional 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.
Experimentally the 5star and 6star star images can be presented from the hyperplane perspective (above) as indicated below  with particular lighting effects.
The standard Schlegel diagram can be used to make evident a unique property of the resonance structure of C_{60}  of potential relevance to the above argument  namely that it avoids double bonds within pentagons and maximizes the number of alternating singledouble 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).
Also of great potential interest is recent recognition of what might be understood as the stages in a process of "selforganization" within fullerene polyhedral cages (Stan Schein and Michelle SandsKidner, A Geometric Principle May Guide SelfAssembly 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 fourdimensional 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 twodimensional 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 5fold 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 higherdimensional lattice. There are two known types of quasicrystal:
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.
Peace as a metapattern of resonance: psychosocial, implicit and emergent?There is of course a very extensive literature on peace  from a multitude of perspectives. In the case of the Middle East  as in other arenas of continuing violence  concern is repeatedly expressed with respect to the "peace process", its breakdown, and "getting it back on track". The argument above is merely a further development of a possibility previously envisaged (Computeraided Visualization of Psychosocial Structures: peace as an evolving balance of conceptual and organizational relationships, 1971). "Voluminous thinking"? The emphasis here is on the contrast between the peace so desperately sought within a twodimensional framework and that implied by a threedimensional framework. The feasibility of peace in the former case were caricatured by comparison with the "forcing" the pieces into a jigsaw puzzle  despite the fact that they do not interlock. Banging them in is not a viable solution  irrespective of the quality of the "lateral thinking" applied to that process. The question explored here is with regard to a threedimensional framework. Might the pieces fit together when understood in that way? The point was made above by reference to "voluminous thinking", illustrated by the example of housing shortage (From Lateral Thinking to Voluminous Thinking: unexplored options for subterranean habitats in dense urban areas, 2007). A major concern is that twodimensional peace is intuitively comprehensible and appealing, notably in the light of its implications regarding resolution of disputes over flat territory. However it is evident that this is perspective is inadequate  as when the jigsaw puzzle pieces do not "fit" together. Threedimensional peace indicates that it might imply counterintuitive comprehension  as suggested in some measure by the Biblical peace which "passeth all understanding". Hence the value of animations. The symmetry group mathematics and its visual renderings suggest that there may forms of peace which are an even greater challenge to comprehension (Dynamics of Symmetry Group Theorizing comprehension of psychosocial implication, 2008). Ironically the threedimensional approach lends itself more readily to comprehensible visualization as illustrated by the association football with which people interact worldwide. A "peace football" could be made simply by appropriately applying the symbols of the two conflicting Middle Eastern cultures to each of its 32 facets. This could well enable reflection on Transcending Simplistic Binary Contractual Relationships: what is hindering their exploration? (2012). Peace as a metapattern: The nature of cognitive incommensurables, as reflected in geometrical patterns, may require the exploration of far more paradoxical forms. For example, the Stella Polyhedron Navigator facilitates visual exploration of patterns in fourdimensions to some degree. The epistemological implications were explored to a degree in the later work of Gregory Bateson (Angels Fear: towards an epistemology of the sacred, 1987). In this sense peace is best understood, not so much as a pattern in three dimensions, but as a metapattern to which Bateson refers (above). Such consideration of metapattern is evoked in Quaker reflection (Approaches to Peace and NonViolence for Modern Quakers, 2005). Emergent insight: Curiously the process of constructing a football from flat pieces results in a form of "emergence" of a new pattern when these are folded and stitched into spherical form. The stepbystep logic of this process can be readily followed, although it is only through the making process (understood as poieisis) that its significance is best realized  as the emergence of a new form of order. It might be said to defy explanation and justification within a twodimensional framework. Relevant arguments regarding recognition of the bigger picture implied by peace are offered by Silvia Hartmann (Meta Patterns and Genius: a simple game to increase a child's intelligence, 2005; The Problem With Peace: why peace is NOT the answer, 2009). Does the surprising recent recognition of a new form of carbon  previously assumed to be one of the commonest and best known elements  imply the possible emergence of a new form of peace? Indications of counterintuitive comprehension: It is useful to note the variety of metaphorical and other pointers to such a counterintuitive form  of which there is evidently a degree of intuitive comprehension:
With respect to the "weaving" metaphor of connectivity, it is appropriate to note the beading diagrams for the fullerenes as explored by BihYaw Jin (C60 beading procedure. The Beaded Molecules: the wonderful world of beaded fullerenes, October 2011). It is also appropriate to note the importance of carpet design to the principles of order identified in the work of Christopher Alexander (A Foreshadowing of 21st Century Art: the color and geometry of very early Turkish carpets, 1993). Design as its own metaphor: It is intriguing that the challenge of "peace", especially in the Middle East, could be framed metaphorically in terms of design and the tools which currently enable it. The production of the above animation, using one of the common design applications, enables a twodimensional pattern to be manipulated into three dimensions. The key to this process is the common instruction to "select all", thereby grouping all the elements on the drawing board. With respect to "peace", the sense of how to "select all" remains elusive, usefully understood as the challenge of "seeing things whole" or the ability to distinguish the "wood from the trees". The tendency is simply to select a subset of elements, treating the remainder as externalities, as separately discussed (Reintegration of a Remaindered World: cognitive recycling of objects of systemic neglect, 2011). Echoing the quest of physicists for a Theory of Everything in a universe characterized by subtle understanding of curvature, peace might then be understood in terms of enabling curvature into a "global configuration of everything". The challenge might be expressed metaphorically in terms of (en)activating the "great circle" pathways. Incommensurable cognitive patterns and their symbolismThe (questionable) assumption of this argument is that effective choice by a group of a primary symbol  as on its flag  reflects the distinctive identity of that group and its engagement with life. The focus here is on the 5fold in relation to the 6fold. Similar arguments  and analogous animations  could be explored with respect to other psychosocial relations that are intractable and incommensurable in "twodimensions", perhaps 3 vs 4, 4 vs 5, 6 vs 8, etc. Recognizing contrasting systems of belief: The 5 and 6pointed stars on which this exploration is based are instances of a set of star polygons which have not as yet been formally defined. Only the most regular ones have been studied in any depth. They can be represented as a sequence of stellations of a convex regular core polygon. Wikipedia offers convenient images of stellation models of associated polyhedra (see stellations of: octahedron, dodecahedron, icosahedron, cuboctahedron and icosidodecahedron). The image below makes clear that the above argument relates only to first two instances in the second diagonal line. Other stars in the diagram are potentially variously significant to other belief systems.
Embodiment of vital insights by the "other": In an intractable conflict it is interesting to explore how their "reconciliation" in a threedimensional configuration would seem to highlight other numberbased patterns in an unexpected manner. In the truncated icosahedron, it is the "Islamic" 5fold which effectively represents the pattern of "12 tribes" so important to the traditions of Israel. There are 12 pentagons in which the Islamic Star has been embedded. Is this characteristic of the sense of problematic "otherness"  as psychologists would tend to claim? What might the pattern of 20 hexagonal  with the embedded Star of David  be recognized as representing as vital within Islamic tradition? Rather than explore this possibility, the question it raises is how the numbers associated with truncated icosahedron (faces, edges, vertices, etc) might interweave elements of both traditions. Failure of systemic connectivity: The widespread importance of 12fold pattern, notably characteristic of jury systems and the like, is noted separately (Checklist of 12fold Principles, Plans, Symbols and Concepts, 2011). Their operational implications have also been explored (Enabling a 12fold Pattern of Systemic Dialogue for Governance, 2011; Eliciting a 12fold Pattern of Generic Operational Insights: Recognition of memory constraints on collective strategic comprehension, 2011). The disconnect between the 5fold and 6fold in the Middle East is an indication of a failure of systemic connectivity and its requisite underlying conceptual organization. Polyocular strategic vision: Whether in that particular case, or in the case of other concept schemes which could be mapped into a set of polygons in two dimensions, each polygon can be understood as having a particular systemic function. In effect each operates as a distinctive "eye"  an organ of strategic "vision". Configured in three dimensions, it is more understandable that each relates to a subsystem with which a form of "subunderstanding" is associated  as with dependence on the right or left eye when both are required for depth perception through stereoscopic vision. A 12fold vision, implicit in the justification of jury size, is then to be understood as enabling "polyocurlar vision" as articulated by Magoroh Maruyama (Polyocular Vision or Subunderstanding? Organization Studies, 2004). Assumptions relating to a singular vision, or a single strategic sense, can then be usefully challenged (Cyclopean Vision vs Polysensual Engagement, 2006). How does each of the 12 both understand and not understand  as implied by "blindspots" and "horizon effects" within the vision metaphor? Polyhedra as systems  of belief: Also intriguing is the sense in which configurations of pentagons around hexagons, or hexagons around pentagons (as presented above)  implying extended zones of discourse and systemic preoccupation  may overlap or interlock within the threedimensional structure. It is appropriate to note the arguments of R. Buckminster Fuller by which polyhedra, especially the regular polyhedra, are to be understood as systems. This reinforces the subsystemic status of individual polygons, or clusters of polygons, through which any threedimensional structure is configured. The case of the Middle East offers a further possibility for exploration  given the role of Christianity, most notably in relation to Jerusalem. This suggests the need to identify polyhedra which combine polygons representative of Islam and Israel with those distinctively representative of Christianity  perhaps a triangle. It follows that there is the intriguing possibility of not only associating the expression of faith with a polygon  ironically to be caricatured as "putting a face" on each. Going further, there is the possibility of associating the integrity of that faith as a "closed system" with a regular polyhedron, potentially as follows::
These could then each be understood as the vehicle or container for the respective faith  when unchallenged by any "other". The mutual challenge of the main faiths is then a form of "geometrical challenge" between the regular polyhedra. In the case of the Middle East, primarily between the dodecahedron and the icosahedron, say. This symbolic mapping could be readily deprecated as simplistic but this would preclude more complex understandings arising from the possibilities in which polyhedra morph into each other dynamically to express particular insights under certain conditions. If polygons and polyhedra can be usefully understood as representative of faiths  if only as mnemonic mappings  this then raises the possibility of a useful relationship between any periodic table of polytopes (polygons and polyhedra) and any periodic table of religions, as separately discussed (Tuning a Periodic Table of Religions, Epistemologies and Spirituality  including the sciences and other belief systems, 2007). Within such a framework an atheistic belief system might be appropriately, or provocatively associated with the nobles gases. "Resonance"? Given the role of resonance in enabling the biology of living structures, is there the intriguing possibility that a process akin to resonance may play a vital role with respect to the viability and sustainability of psychosocial systems  and to "belief"? Is there some intuition that the currently muchchallenged "bond" vital to "confidence" in any psychosocial system  "my word is my bond"  has characteristics which could benefit from the insightful consideration of the dynamics of bonds in chemical molecules? How might "confidencebuilding" then be understood? Of relevance to this possibility is the extensively articulated description of one process of the Arab Spring by Gaston Gordillo (Resonance and the Egyptian Revolution, Space and Politics, 6 February 2011), variously indicating:
ConclusionThe irony of the presentation of this argument, on the occasion of worldwide media focus on the patterned movement of a truncated icosahedron in Kiev, was noted above  as with the United Nations demonstration of its longstanding inability to elicit relevant intelligence in more creative response to ongoing massacres, most notably in the Middle East. The priority given to the application of the insights of mathematicians of different faiths to the development of ballistics was also noted, in contrast to the possibilities of taking more seriously the challenges of violent difference of belief (Mathematical Theology: future science of confidence in belief, 2011). This priority is of course also evident in relation to other securityrelated opportunities, most notably with respect to cyberwarfare. The irony is all the greater in that technology has advanced to the point at which footballs (and the boots of players) are in process of being enhanced with computer chips to facilitate game pattern analysis. This is to be compared with the total confusion within the international community regarding the emergence of the global financial crisis and how it might best be managed  seemingly in the absence of any comprehensible simulation of both remedial strategic possibilities and the implications of global strategic incompetence of a very high order. Patternlearning via gameplaying: There is a case for recognizing the extent to which games are played as an unconscious psychosocial device for learning the patterns which are yet to be appropriately acquired in the handling of psychosocial relations. This point has been made with respect to the organization of popularly preferred styles of music, by contrasting the classical patterns implicit in current institutional organization with the modern music yet to acquire relevant organizational expression (Jacques Attali, Noise: the political economy of music, 1977). This suggests that the acquisition of insights from biomimicry and technomimicry could be extended to "ludomimicry"  following the critical arguments of Roger Caillois (The Definition of Play and the Classification of Games, 2006) regarding those of Johan Huizinga (Homo Ludens: a study of the playelement in culture, 1950), as discussed by Andrew Brown (Agon, Alea, Mimicry and Ilinx, Embodied Knowers). What is a global civilization endeavouring to learn by kicking around a truncated icosahedron  whilst others are being kicked to death? How does such a game relate to the possibility outlined by James P. Carse (Finite and Infinite Games: a vision of life as play and possibility, 1987)? Curiously, although it has long been known that there can be no hexagonpentagon polyhedron with less than 20 vertices, a classical theorem exists to show that polyhedra of any size can be created  composed solely of hexagons and pentagons  provided that there are exactly 12 pentagons (Thayer Watkins, Euler's Theorem Concerning Polyhedra Composed of Pentagons and Hexagons: there must be exactly 12 pentagons). What pattern dynamics is society exploring, especially in the Middle East, given that the relevant equation from that theorem excludes the possibility of a polyhedron with 22 vertices  namely the number of players in association football, acclaimed as the world's most popular sport? If the argument above relates the Islamic faith with the 12 pentagons, what then is the implication of the relationship of Israel to the 20 hexagons in the truncated icosahedron? Why then should the number of hexagons in a hexagonpentagon polyhedron be constrained to half the number of vertices minus 10 (namely with the number of vertices equal to twice the number of hexagons plus 20)? Mathematical theology: What might the neglected discipline of mathematical theology contribute to reframing the relation of Judaism to Islam? Given the identification of Christianity with Jerusalem, how might a "third polygon" pattern be incorporated into a spherical geometry reframing of the cognitive challenge? Especially intriguing is the sense in which the "holiness" so powerfully associated with Jerusalem by the three faiths may imply an exceptionally elegant geometrical representation in which the three are configured together. This is consistent with the more general argument above that the distinctions between the faiths so important to the peoples of the world (including "science") may be strangely related to distinct cognitive patterning preferences  represented to a degree by polygonal and polyhedral geometry and the possibilities of its configuration, possible in higher dimensions. Global connectivity: Global civilization is currently characterized by erosion of confidence in various forms of authority with concerns widely expressed regarding the need for "confidencebuilding"  but with little sense as to what that means. What is the nature of the connectivity required and how can its emergence be facilitated? It is in this sense that the failure of mathematicians is so extraordinary  given their insights into "connectivity" of the highest order, and their inability to "connect" with those concerned with "confidence" and "belief". The disconnect, an "empathy deficit", might be compared to the challenge for those on the autism spectrum  a recognized characteristic of mathematical excellence  in comprehending the emotional dimension on which faith is based and which ensures cognitive engagement with a form of "psychoactive curvature" (Zippora ArziGonczarowski, Self, Empathy, Manipulativity: Mathematical Connections between Higher Order Perception, Emotions, and Social Cognition,, 2001). It is this "empathy curvature" which is evoked by spherical geometry and its "global" implications in a faithbased society (cf Wikipedia entry on EmpathizingSystemizing theory). Quest for a bigger "tent": It is ironic that one of the most significant contributors to the development of spherical geometry, fundamental to the argument above, was Omar Khayyam  from a country whose bombardment is currently envisaged with the use of ballistic skills to whose development he could be said to have contributed. Curiously Islam and Judaism share in the fundamental metaphorical significance they attach to the protective role of a tent and to its construction  a metaphor inherited by Christianity to some degree. Omar Khayyam valued the inspiration of his tentmaking tradition, as discussed separately (Global Brane Comprehension Enabling a Higher Dimensional Big Tent? Strategic implication in encompassing nothing and coming to naught, 2011). Given the counterintuitive challenge of "stitching" a football, as noted above, it is appropriate to note his selfreflexive use of the metaphor: Khayyám, who stitched the tents of science. What "higher dimensionality", offering connectivity and coherence, would facilitate the evident incoherence and rivalry between faiths at a "lower dimensionality"? (cf. Stephen Prothero, God is Not One: the eight rival religions than run the world and why their differences matter, 2011; Thomas Sowell, A Conflict of visions: ideological origins of political struggles, 1987). How might such tentmaking now be understood (Towards higher dimensional "tentmaking"? 2011). Renowned for his skills in both mathematics and poetry, Omar Khayyam exemplifies the insight where characterized as "empathy curvature". Strangely the nature of the "nothing" which a tent usefully encompasses is echoed both in the football, so enthusiastically kicked around globally, and in enthusiasm for the spirit of the game and its outcome, as might be detected in one of the quatrains of his Rubáiyát: And if the Wine you drink, the Lip you press, At a time when the culture of Omar Khayyam is being associated with Nazism (Israeli foreign minister Lieberman compares Iran to Nazi Germany, The Guardian, 28 June 2012), the mysteriously dangerous relationship between "thoughtlessness" and "nothingness" merits further consideration, as can be variously argued (cf. Unthought as Cognitive Foundation of Global Civilization, 2012; Exploring the Hidden Mysteries of Oxfam's Doughnut, 2012; Swastika as Dynamic Pattern Underlying Psychosocial Power Processes, 2012). The poetic poignancy of "nothing" for the individual is now echoed in the collective experience of collapse of confidence, historic deficits, and the failure of decisionmaking  so recently exemplified by the Earth Summit. Reframing of remedies in terms of global missileenhanced "security" then merely conflates the emptiness of political rhetoric, and the annihilation it engenders, with the nothingness of that insight  as conventionally understood. It is the cognitive implication of the spherical geometry capable of encompassing and containing "nothing" that then merits attention. Football offers a good reminder. Facilitating imagination: The argument has been presented using relatively sophisticated computer applications  although readily accessible. In highlighting the need for a computerassisted pattern language, the point was made that the Stella Polyhedron Navigator offers numerous facilities vital to education of spherical imagination, as previously argued (Polyhedral Pattern Language: software facilitation of emergence, representation and transformation of psychosocial organization, 2008). With the rapid development of mobile applications on smartphones, it is not difficult to predict that such facilities will soon enhance social networking interactivity in surprising ways (Polyhedral Empowerment of Networks through Symmetry: psychosocial implications for organization and global governance, 2008). Whilst such software also enables paper construction of polyehdra for educational purposes, it was also stressed that many "counterintuitive" aspects of the argument regarding requisite connectivity can be developed using a marker pen and an ordinary football. Replicating restrictive territoriality through copyright: It is appropriate to conclude with a reference to the regrettable tendency for the most remarkable creativity with respect to relevant spherical innovation to be marked by a counterproductive preoccupation with intellectual copyright  suggesting a tendency to replicate in higher dimensions the problematic patterns so evident with regard to twodimensional territory. Noteworthy examples relate to the legacy of R. Buckminster Fuller and Stafford Beer (Beyond Dispute: the invention of team syntegrity, 1994) and the tragic legal dispute between Stan Tenen and Dan Winter (Future Coping Strategies: beyond the constraints of proprietary metaphors, 1992). The only merit this offers is the motivation to explore alternatives by ignoring those based on such misappropriation of the cultural heritage of a global knowledgebased civilization. ReferencesChristopher Alexander:
Zippora ArziGonczarowski. Self, Empathy, Manipulativity: Mathematical Connections between Higher Order Perception, Emotions, and Social Cognition. AAAI Technical Report FS0102, 2001 [text] Jacques Attali. Noise: the political economy of music. Presses Universitaires de France, 1977 [summary] Gregory Bateson. Mind and Nature: a necessary unity. Hampton Press, 1979 Gregory Bateson with Mary Catherine Bateson. Angels Fear: towards an epistemology of the sacred. Hampton Press, 1987 [summary] Stafford Beer. Beyond Dispute: the invention of team syntegrity. John Wiley, 1994 William Betz. Central Issues in the Program of Mathematics for a World at Peace. School Science and Mathematics, 46, 1946, 5, pp. 446447 [text] J. Bourgoin. Arabic Geometrical Pattern and Design. Dover Publications, 1973 Eric Broug. Islamic Geometric Patterns. Thames and Hudson, 2008 Roger Caillois:
James P. Carse. Finite and Infinite Games: a vision of life as play and possibility. Ballantine Books, 1987 Peter T. Coleman. The Mathematics of Middle East Conflict and Peace: the Israeli/Palestinian conflict requires new thinking  try mathematics. In: The Five Percent: finding solutions to seemingly impossible conflicts. PublicAffairs, 2011 [text] Alex Comfort. Reality and Empathy: physics, mind, and science in the 21st Century. SUNY Press, 1984 Keith Critchlow. Islamic Patterns: an analytical and cosmological approach. Inner Traditions, 1999 Ubiratan D'Ambrosio:
Mircea V. Diudea. Fantasmagoric Fulleroids Revisited. Leonardo Electronic Journal of Practices and Technologies [text] Issam ElSaid. Islamic Art and Architecture: the system of geometric design. Garnet Publishing, 2008 P. W. Fowler. How unusual is C60? Magic numbers for carbon clusters. Chemical Physics Letters, 131, 1986, pp. 444  450 P. W. Fowler and D. E. Manolopoulos, An Atlas of Fullerenes. Oxford, 1995 Harald Fripertinger. The Cycle Index of the Symmetry Group of the Fullerene C60. 1996 [text] Stafford Beer. Beyond Dispute: the invention of team syntegrity. Wiley, 1994 R. Buckminster Fuller with E. J. Applewhite:
Johan Galtung. Peace Mathematics: does it exist? Transcend Media Service, 12 March 2012 [text] Johan Galtung and Dietrich Fischer. Peace Mathematics. Transcend University Press, 2012 Johan Huizinga. Homo Ludens: a study of the playelement in culture. Beacon Press, 1950 BihYaw Jin. C60 beading procedure. The Beaded Molecules: the wonderful world of beaded fullerenes, October 2011 [text] Robert D. Johnson, Donald S. Bethune and Costantino S. Yannoni. Fullerene Structure and Dynamics: a magnetic resonance potpourri. Acc. Chem. Res., 1992, 25 (3), pp. 169175 [abstract] Tomas Kaiser, Matej Stehlokz and Riste Skrekovski. On the 2resonance of fullerenes. 2010 [text] Craig Kaplan. The Trouble with Five. Plus Magazine, December 2007, 45 [text] K. Kutnar and D. Marusic. On cyclic edgeconnectivity of fullerenes. Discrete Appl. Math. 156, 10, 2008, pp. 16611669. George Lakoff and Rafael Nuñez. Where Mathematics Comes From: how the embodied mind brings mathematics into being. Basic Books, 2001 Magoroh Maruyama:
Pepin Press. Arabian Geometric Patterns. Pepin Press, 2011 Stephen Prothero. God is Not One: the eight rival religions than run the world and why their differences matter. Black, 2010 Anna Rosa Scarafiotti and Annarosa Giannetti. Can Mathematics Educate for Peace? ZDM 98/3 [text] Stan Schein and Michelle SandsKidner. A Geometric Principle May Guide SelfAssembly of Fullerene Cages from Clathrin Triskelia and from Carbon Atoms. Biophys Journal, 94, 2008, 3, pp. 958976 [abstract] WaiChee Shiu, Peter Che Bor Lam, Heping Zhang. Clar and sextet polynomials of buckminsterfullerene. [text] Thomas Sowell. A Conflict of visions: ideological origins of political struggles. William Morrow, 1987 Bharath Sriraman. International Perspectives on Social Justice in Mathematics Education. IAP, 2007 Daud Sutton. Islamic Design: a genius for geometry. Walker, 2007 David Wagner. Another Look at Relevance. (submission to Mathematics Teacher) 2005 [text] Dong Ye, Zhongbin Qi and Heping Zhang. On kresonant fullerene graphs. 2009 [text] Heping Zhang and Dong Ye. Cyclical Edgeconnectivity and Resonance of Fullerenes. School of Mathematics and Statistics, Lanzhou University, 2007 [text]
