
9th June 2008
 Draft
Polyhedral Empowerment of Networks through SymmetryPsychosocial implications for organization and global governance /  Associated with Towards Polyhedral Global Governance: complexifying oversimplistic strategic metaphors (2008), Polyhedral Pattern Language: software facilitation of emergence, representation and transformation of psychosocial organization (2008), Configuring Global Governance Groups: experimental visualization of possible integrative relationships (2008) and Configuring Global Governance Groups: experimental animations and video sequences (2008). Context Challenge of network connectivity and networking efficiencies Polyhedral approaches to social network analysis Polyhedral networks: designing for robustness and survivability Epistemic networks, simplicial complexes and polyhedra Polyhedral dynamics and Qanalysis Polyhedral theory Polyhedral computing: optimizing responses to complexity Polyhedral methods of conjoint analysis Polyhedral design of computer memory utilization processes Polyhedral databases: operational significance Polyhedral patterns: representation of complex numerical abstractions Polyhedral networks: strategic significance Polyhedral relationship networks? Polyhedrabased sense of identity? Missing link: selfreflexive closure? Symmetry: competition vs complementarity? Polyhedral empowerment: "eliciting the sparkle from networks"? Recognition of curvature as fundamental to a polyhedral psychosocial universe? References ContextIn the further exploration of the possibilities highlighted in the abovementioned papers, reference is made here to the current applications of "polyhedral approaches" to networks, their operation, and to their significance for information organization in situations calling for higher orders of efficiency and robustness  and to the insights offered for new approaches to psychosocial organization. The main emphasis below has been to indicate fields of study and application which may not necessarily be wellconnected, however relevant they may be with respect to any potential psychosocial implications. Given that the concept of "network" has proven over past decades to be as significant as a metaphor for social organization as it has as an analytical framework for the development of such organization, it is possible that the potential of "polyhedra" should be similarly understood. The basic argument is that in psychosocial usage "network" is relatively unstructured and has not achieved much of what was hoped initially in contrasting it with hierarchical modes of organization. Whilst seemingly quite unrelated, the faces and edges linking vertices of any 3dimensional polyhedron can be mapped in 2dimensions as a network (a polyhedral net)  by "unwrapping" the polyhedron. Studies of social networks show that desirable properties such as robustness and information transfer efficiencies can be achieved with networks that take the form of polyhedra having properties such as symmetry. Symmetrical polyhedra in 3dimensions are those of greatest aesthetic appeal and are intuitively comprehensible as a whole, despite their possible complexity. There is therefore a case for considering how, in the light of the various polyhedral approaches (considered below), networks might be "polyhedrally empowered" by using polyhedra as structural (or dynamic) templates. Of particular related interest is the degree to which complex multicriteria decisionmaking is now dependent on such approaches  suggesting that the comprehensibility and communicability of a solution to any strategic dilemma might be associated with a polyhedral form reflecting its "goodness of fit" as a pattern in a design sense. This would then have important implications for governance. Challenge of network connectivity and networking efficiencies"Tensing networks": In response to early optimism regarding the merits of social "networking", in contrast to the problematic aspects of hierarchical social organization, attention was focused on the inefficiencies of untensed networks and associtated "networking diseases" (Tensing Associative Networks to contain the Fragmentation and Erosion of Collective Memory, 1980; Implementing Principles by Balancing Configurations of Functions: a tensegrity organization approach, 1979; Tensed Networks: balancing and focusing network dynamics in response to networking diseases, 1978). These concerns resulted in a continuing preoccupation with tensegrity organization, namely ensuring a degree of tensional integrity within networks (From Networking to Tensegrity Organization, 1984; Documents relating to Networking, Tensegrity, Virtual Organization). Recent developments with respect to tensegrity as an extension of the focus on polyhedra, notably relevant software, are discussed elsewhere (Psychosocial operationalization of polyhedra through tensegrity representation, 2008). Network graphs as polyhedra: Just as threedimensional polyhedra may be represented in two dimensions as a network  a polyhedral net  so there have been explorations of the value of representing networks by polyhedra (Branko Grünbauma, Graphs of Polyhedra; Polyhedra as Graphs. Discrete Mathematics, 2007). Of particular interest has been the generic challenge of network connectivity (M. Grötschel, C.L. Monma and M. Stoer, A Polyhedral Approach to Network Connectivity Problems, 1992). Optimal networks  classification of polyhedra: As part of the quest for more useful networks, efforts have been made to classify polyhedra.
Such work focises the question of whether the range of polyhedra, especially with some degree of symmetry, constitute a rich repertoire for psychosocial organization appropriate to various conditions, as previously argued (Polyhedral Pattern Language: software facilitation of emergence, representation and transformation of psychosocial organization, 2008). Optimal networks  information traffic: There has naturally been a major interest in optimizing the flow of information through networks, notably in telecommunications. Raul J. Mondragon (Optimal Networks, Congestion and Braess' Paradox, 2007), for example, explores how to deliver information efficiently in a communications network and how to build networks to perform this function  notably by rewiring them. Following Dekker and Colbert's work, they seek a compromise between robust networks and optimal networks:
which leads to a focus on regular and symmetric graphs in which the nodes are all similarly linked. They cite the work of L. Donetti et al. (Optimal network topologies: Expanders, Cages, Ramanujan graphs, Entangled networks and all that, 2006; Entangled Networks, Synchronization, and Optimal Network Topology, 2005) in discussing the network congestion under load in various configurations. Such considerations are especially relevant to consideration of information (if not "knowledge" or "wisdom") flows in psychosocial networks. Given the increasing degree of enablement offered by the internet and the web, to what extent could the challenges of information overload and information underuse be circumvented by a polyhedral approach to optimization of such networks? In principle this is highly relevant to the challenges of a learning society and the threats to collective memory (Societal Learning and the Erosion of Collective Memory a critique of the Club of Rome Report: No Limits to Learning, 1980). Optimal networks  commodity distribution: Gábor Rétvári, József J. Bíró and Tibor Cinkler (Fairness in Capacitated Networks: a Polyhedral Approach) address the problem of allocating scarce resources in a network so that every user gets a fair share, for some reasonable definition of fairness. For example, a fair allocation would be such that every user gets the same share, and the allocation is maximal in the sense that there does not exist any larger, even and feasible allocation. We shall focus on the fair allocation problem that arises most often in networking: compute a fair rate at which users can send data in a telecommunications network, whose links are of limited capacity. The authors show that We show that the set of throughput configurations realizable in a capacitated network makes up a polyhedron, which gives rise to a maxmin fair allocation completely analogous to the conventional one. An algorithm to compute this polyhedron is also presented, whose viability is demonstrated by comprehensive evaluation studies. Gabriella Muratore (Polyhedral approaches to survivable network design, 1999) studies the problem of designing a costefficient multicommodity flow network with survivability features and the geometrical structure of several polyhedra arising in this context. For some of these polyhedra it proved possible to give a complete description by extreme points and by facets, while for others the complete description was given by extreme points. Several classes of facetdefining inequalities were identified. It might be inferred that a more sophisticated approach to "fairness" is what is required in response to the variety of forms of "unfairness" that drive social unrest and cycles of violence  whether collectively or within the interpersonal networks. Polyhedral approaches to social network analysisThe interest in the network structure of organizations, especially those of a potentially criminal or terrorist nature, has increased manifold as a result of the information revolution, as noted by B. Balasundaram, et al (Clique Relaxations in Social Network Analysis: The Maximum kplex Problem, 2006) . Their paper introduces and studies the maximum kplex problem. This arises in analysis of cohesion in social networks and is often used to explain and develop sociological theories. Members of a cohesive subgroup tend to share information, have homogeneity of thought, identity, beliefs, behavior, even food habits and illnesses. It is also believed to influence emergence of consensus among group members. The approach is relevant to the study of degrees of connectivity amongst sets of websites. It may be used in organizational management to study organizational structure to suggest better work practices and improve communication and work flow. Their study helpfully summarizes the distinction in the literature between three important structural properties expected of a cohesive subgroup that are idealized by models of clique models idealize:
Different models may relax relax different aspects of a cohesive subgroup:
This model relaxes familiarity within a cohesive subgroup and implicitly provides reachability and robustness. As the authors note:
It is curious that there are few memorable instances where such sophisticated tools have enabled better networks to form. The current explosion of "social networking" over the web, for example, does not appear to have benefitted from such insights. The relevant Wikipedia entry exemplifies the point with the statement: "Not to be confused with social network analysis, a type of social scientific model". It is however clear that the insights have been considered highly relevant in tracking criminal networks and terrorist suspects. Whereas they enable detection of problematic nodes against which security measures can be taken, they have not has yet enabled communitybuilding in practice. Polyhedral networks: designing for robustness and survivabilityCuriously it would appear that the array of network analysis skills has been most significantly applied "defensively" in response to possible vulnerabilities of vital networks  whether to protect against them or to exploit them. For example, the study by Jonathan T. Hamill (Analysis of Layered Social Networks, Air Force Institute of Technology, 2006) is concerned with prevention of nearterm terrorist attacks:
As noted by Anthony H. Dekker and Bernard D. Colbert (Network Robustness and Graph Topology, 2004; Network Robustness for Critical Infrastructure Networks, 2008):
After reviewing a range of polyhedra, the authors conclude:
The problem of robustness is vital to telecommunications networks as explored by Bernard Fortz (Design of Survivable Networks with Bounded Rings, 2000) using polyhedral analysis. The results obtained demonstrate how to use polyhedral theory for practical network design problems. From a military perspective, a recurring theme in the literature is the "design of surivable communication networks" (M. Grötschel, C.L. Monma, M. Stoer, Polyhedral and Computational Investigations for Designing Communication Networks with High Survivability Requirements, 1992). This perspective is notably relevant to telecommunications networks (Arie Koster, Polyhedral Combinatorics to Solve Network Design Problems, Paper for 9th INFORMS Telecommunications Conference, 2008). In the increasing concerns with sustainability, and the longerterm viability of catalytic social projects, the issue of what makes for robustness would appear to be vital  faced with the tendency of projects to collapse once seed funding ceases. Such robustness is clearly also of importance faced with the prospect of partial or complete social collapse, if only in the event of disasters. Recent disasters have indicated the vulnerability of food and utility supply networks, for example. Epistemic networks, simplicial complexes and polyhedraIn a society increasingly recognized to be significantly knowledgebased, social networks, concept networks and the manner of their apprehension are necessarily intimately intertwined  beyond the challenge of the description of networks in the abstract and the flows of information through them. This interweaving has been remarkably explored by Camille Roth (Coevolution in Epistemic Networks: reconstructing social complex systems, 2005). The author frames his discussion as folows:
In contrast to the graph theory tools typically deployed for social network analysis, Roth bases her analyses on the use of Galois lattices that may be termed 'concept lattices' in other contexts (Wille, 1992; Stumme, 2002). With regard to the above objective, Roth notes:
With respect to polyhedra, Roth notes that the principles underlying use of Galois lattices (GLs) strongly relate to Qanalysis, notably that of Ron Atkin on simplicial complexes:
With respect to the structure of any knowledge community, Roth introduces a formal framework based on Galois lattices that categorizes epistemic communities automatically and hierarchically, rebuilding a whole community taxonomy in the form of a hypergraph of significant subcommunities. She argues that modeling social complex systems tends to require the introduction of coevolutive frameworks.
Roth notes that beyond the profusion of communityfinding methods, often leaning towards AIoriented clustering, an interesting issue concerns the representation of communities in an ordered fashion. She cites as examples of different techniques for producing and representing categorical structures: hierarchical clustering, Qanalysis, formal concept analysis, information theory, blockmodeling graph theorybased techniques, neural networks, association mining, and dynamic exploration of taxonomies. In later work relevant to the coevolution of social and knowledge networks, Camille Roth (Patterns and Processes in Sociosemantic Networks, 2007) notes:
Also of note is R. Cowan, et al. (The Joint Dynamics of Networks and Knowledge, 2002), as well as the literature on the diffusion of innovation through networks. It is however intriguing that Roth's central focus on "reconstructing social complex systems" might be understood purely as one of construcitng a more adequate model of such a system rather than responsing to the challenge of designing and facilitating the operation of more appropriate systems. This has long been a weakness of social network "analysis" which (as noted above) has not yet proven to be significant in designing better networks, whatever that might mean. Polyhedral dynamics and QanalysisPolyhedral dynamics is a tool for representing network structure and behaviour. In introducing this field and its relevance to IIASA (International Institute for Applied Systems Analysis), J. Casti (Polyhedral Dynamics: the relevance of algebraic topology to human affairs, 1975) notes:
Casti notes that the approach was introduced by Ron Atkin (Mathematical Structure in Human Affairs, 1974) using ideas of algebraic topology.
Elsewhere (Topological Methods for Social and Behavioural Systems, 1982) Casti notes:
Atkin introduced polyhedral dynamics in terms of Qanalysis whereby patterns of qconnectivity are analyzed rather than connectivity (Combinatorial Connectivities in Social Systems; an application of simplicial complex structures to the study of large organizations, 1977). The approach has been applied with varied success as discussed by Jacky Legrand (How far can Qanalysis go into Social Systems Understanding?, 2002) and S. B. Seidman (Relational Models for Social Systems, 1987). Examples include: P. Doreian (Polyhedral Dynamics and Conflict Mobilization in Social Networks, 1981) and L. C. Freeman (Qanalysis and the Structure of Friendship Networks, 1980). In a later work (Ron Atkin, Multidimensional Man: can man live in three dimensions? 1981) he develops a theory of polyhedral events and makes a convincing argument that structural events are related to clock time in a nonlinear way related to their dimensions. He gives a convincing explanation why higher dimensional events take a lot longer to occur in clock time than simple events. Jeffrey Johnson (Hypernetworks for Reconstructing the Dynamics of Multilevel Systems, 2006) develops this argument that polyhedral dynamics form trajectories in a nonlinear way in clock time. The formation of a simplex is a polyhedral event. Polyhedral events mark the passage of system time. Events occur at different levels on multilevel systems, and they have to be coordinated. Johnson's argument is that:
The relevance of Qanalysis to psychosocial issues is discussed in a commentary on the Global Strategies Project (Comprehension: social organization determined by incommunicability of insights). It is also discussed in Beyond Edgebound Comprehension and Modal Impotence: combining qholes through a pattern language (1981). Polyhedral theoryPolyhedra, linear inequalities and linear programming can be seen as three views of the same concept. Polyhedra represent a geometrical point of view, linear inequalities represent the algebraic point of view, and linear programming the optimization point of view. As noted by Geir Dahl (An Introduction to Convexity, Polyhedral Theory and Combinatorial Optimization. 1997)
It is the identification of the closure associated with appropriate "convex polyhedra" that is closely associated with identification of an "optimal subset". Most problems studied in combinatorial optimization involve looking for certain structures in graphs. The simplest symmetrical polyhedra (ocathedron, docdecahedron, etc) may therefore be understood as representing or "containing" the solutions to challenges of optimization in many fields. As Dahl further notes:
Polyhedral computing: optimizing responses to complexityThe set of feasible solutions of a linear optimization problem is a convex polyhedron. Specially structured variants of these problems define polytopes with special structures. In this way the theory and algorithms of linear optimization are inherently linked to polyhedral theory and properties of convex bodies. The generic issues posed by the above concerns were brought to a focus at the Centre de recherches mathématiques (University of Montreal) in the form of a semester on Polyhedral Computation Combinatorial Optimization (2006) presented as follows:
Polyhedral computation addresses the computational complexity of solving problems associated with convex polyhedra and search for efficient algorithms. One of the most fundamental problems is the vertex enumeration problem that is to list all vertices of a convex polytope given as the solution set to a system of m linear inequalities in dvariables  as succinctly presented by Jakub Marecek (Polyhedral Approach to Multicriterial Optimization, 2006). It is in this sense that polyhedral approaches are fundamental to decisionmaking under uncertainty, if only in the field of economics (Andreas Eicchorn, et al. Polyhedral Risk Measures and Langrangian Relaxation in Electricity Portfoloio Optimization, 2005). More generally it is worth considering the possibility that an appropriate polyhedral configuration of criteria, opportunities and constraints constitutes a comprehensible, credible "solution" to any strategic dilemma. The argument here, notably in the light of the "epistemic" framing provided by Roth (above), is that it is precisely the polyhedral coherence (through symmetry effects) that renders a solution both understandable and memorable as a gestalt that "works". Thus what is otherwise understood through the intuitionist school of mathematics is echoed in how a viable solution is apprehended beyond that discipline. This possibility is clearly of significance to governance and to that to which the governed are asked to subscribe  and to the manner in which a complex solution can be communicated (notably in the light of Atkin's arguments regarding the communicability of insights). How should governance envisage optimizing the multicriteria decisionmaking challenges of the future  given the computing resources on which it can call  in a manner to render the "solutions" comprehensible and credible to all concerned? In purely material terms, such arguments are of course consistent with the aesthetic appeal of sacred geometry as a response to architectural and design challenges. The question is how such polyhedral configuration might prove significant in psychosocial organization and knowledge structure design, notably when it takes virtual form on the web (cf Sacralization of Hyperlink Geometry, 1997; Spherical Configuration of Categories  to reflect systemic patterns of environmental checks and balances, 1994; Spherical Configuration of Interlocking Roundtables: internet enhancement of global selforganization through patterns of dialogue, 1998). Polyhedral methods of conjoint analysisConjoint analysis, also called multiattribute compositional models or stated preference analysis, is a statistical technique that originated in mathematical psychology. Today it is used in many of the social sciences and applied sciences including marketing, product management, and operations research. Olivier Toubia (New Approaches to Idea Generation and Consumer Input in the Product Development Process, 2001) describes a method, dependent on computer support, for an adaptive question design method that attempts to reduce respondent burden while simultaneously improving accuracy. For each respondent the question design method dynamically adapts the design of the next question using that respondent's answers to previous questions. The adaptive method interprets question design as a mathematical program and estimates the solution to the program using recent developments based on the interior points of polyhedra. Toubia begins with a conceptual description that highlights the geometry of the conjointanalysis parameter space, permitting analyses of decision challenges involving many "dimensions"  understood as distinct features of a product (3 to 100, say) which respondents are called upon to evaluate. The polyhedral method is designed to "shrink" the feasible set of features  reducing its dimensionality  determining the key features of the product design. The respondent's answers to the first q questions define a (pq)dimensional hyperplane which intersects the initial pdimensional polyhedron to give a (pq)dimensional polyhedron, namely one of lower dimensionality. The challenge is to select questions that reduce the dimensionality of the polyhedron as fast as possible. Conjoint analysis is one of a set of techniques of multivariate analysis in which, for example, potential customers are asked to compare pairs of products and make judgements about their similarity (or their dissimilarity in the case of ordination statistics). By contrast, whereas techniques such as factor analysis, discriminant analysis, and conjoint analysis obtain underlying dimensions from responses to product attributes identified by the researcher, multidimensional scaling obtains the underlying dimensions from respondents' judgements about the similarity of products rather than being dependent on researchers' judgments (in furnishing a list of attributes to be shown to the respondents). The underlying dimensions then emerge from respondents' judgements about pairs of products making it the most common approach to perceptual mapping. Polyhedral design of computer memory utilization processesThe power of supercomputers is partly due to their use of a design based on a hypercube configuration of distributed memory parallel computers (see Ndimensional modified hypercube). The plurality of nodes or cells is interconnected to provide a shared memory with processors of the network and their memory providing the network routing and shared memory. Distributed memory parallel computers offer both the potential for a dramatic improvement in cost/performance over conventional supercomputers. Of relevance here is the application to resource intensive computations in fluid dyanmics required by applications, for example:
Combinatorial computational geometry: The primary goal of research in combinatorial computational geometry is to develop efficient algorithms and data structures for solving problems stated in terms of basic geometrical objects: points, line segments, polygons, polyhedra, etc. There is tremendous pressure to develop computers of higher performance in response to certain modelling challenges, notably those related to climate and ocean currents and to challenges of astrophysics and fundamental physics. But it is curious that the applications of relevance to psychosocial challenges have been limited to relatively simplistic forms of "global modelling". There has been no implication that there might be equally demanding applications, of equivalent significance for humanity, associated with building richer and more complex psychosocial systems. Polyhedral databases: operational significanceThe efficiencies associated with memory process organization in computers have given rise, as an example, to databases that reflect these efficiencies. On of these is Polyhedra which is used as a key component in the infrastructure of an embedded application where data management is a considerable factor in the design. The application brings together the benefits of SQL database technology with a powerful set of high performance features designed specifically for the embedded market. The product is used in mission critical military systems (Lockheed Martin UK Selects Enea's Database Management Systems for Merlin Helicopter. Military Embedded Systems, 2007). Polyhedra uses a memoryresident design that boosts performance by up to an order of magnitude relative to conventional disk and flashbased RDBMSs. The active, eventdriven technology makes databases more robust, simplifies applications and en¬hances performance. The active query mechanism allows applications to be kept up to date without the need for applications to poll the server (to detect changes). Because the client application is told precisely what has changed, the application need not refresh its queries. This all leads to a responsive system with low latency data distribution, and good scalability with graceful degradation in times of peak load. Presumably such techniques will shortly prove to be significant to handheld computer access to the next generation of the web  driven by commercial innovations  irrespective of any consideration of the potential psychosocial significance of such developments. Polyhedral patterns: representation of complex numerical abstractionsAs a consequence of the computerrelated developments above, a polyhedral pattern library has been instigated by Roberto Bagnara at the University of Parma (Roberto Bagnara, Convex Polyhedra for the Analysis and Verification of Hardware and Software Systems: the "Parma Polyhedra Library", 2003; Roberto Bagnara et al., The PPL: A Library for Representing Numerical Abstractions: Current and Future Plans, 2004). The Parma Polyhedra Library (PPL) is a modern and reasonably complete library providing numerical abstractions especially targeted at applications in the field of analysis and verification of complex systems. The PPL can handle all the convex polyhedra that can be defined as the intersection of a finite number of (open or closed) hyperspaces, each described by an equality or inequality (strict or nonstrict) with rational coefficients. The PPL also handles restricted classes of polyhedra that offer interesting complexity/precision tradeoffs. The library also supports finite powersets of (any kind of) polyhedra and linear programming problems solved with an exactarithmetic version of the simplex algorithm. As highlighted by the PPL (The Parma Polyhedra Library), justifications for the creation of such a library include:
Again the question is on what range of polyhedral patterns is psychosocial organization currently based and whether much of relevance could be enabled by extending that range. Does the restriction of the range considered credible result in what Magoroh Maruyama has termed "subunderstanding" (Polyocular Vision or Subunderstanding, 2004)? In his terms is "polyocular" ensured by "polyhedral"? Polyhedral networks: strategic significanceAs noted above, network robustness (Anthony H. Dekker and Bernard D. Colbert, Network Robustness and Graph Topology. 2004) is a significant concern in the increasing focus on netcentric warfare (D. Alberts, J. Garstka and F. Stein. Network Centric Warfare: developing and leveraging information superiority, 1999; John Arquilla and David Ronfeldt (Eds.). Networks and Netwars: the future of terror, crime, and militancy. RAND Corporation, 2003). Dekker and Colbert make very clear the strategic advantages of particular polyhedral configurations of networks. Dekker has also explored related concerns in military performance (Network Topology and Military Performance, 2005; Agility in Networked Military Systems: a simulation experiment, 2006). The earlier paper (Towards Polyhedral Global Governance: complexifying oversimplistic strategic metaphors, 2008) raised the question of whether a polyhedral approach would help to reframe the essentially "stuck" and obsolete approaches to governance of recent decades. It is clear that such approaches are considered vital to more complex forms of decisionmaking, notably as faced by business and the security services. Presumably the challenges and opportunities of governance are not otherwise understood to be complex, nor to offer windows of opportunity which more complex approaches could detect  as they are used to detect marketing opportunities for consumer products. It is curious that no consideration has apparently been given to their application to the supposedly critical challenges of global governance. The question was raised elsewhere as to why significant segments of leadership seemingly subscribed to what amounts to a "flatearth" approach to the strategic challenges of the future (Irresponsible Dependence on a Flat Earth Mentality  in response to global governance challenges, 2008). The situation does however result in a multiplicity of governance initiatives wandering the plains of such a flat Earth and competing for resources  a metaphor of the challenge that the peoples of the Earth collectively face. It is for this reason that there is a case for experimenting with the configuration of the set of such initiatives together as a polyhedron in order to suggest more global patterns of signifiance from their complementarity (see Configuring Global Governance Groups: experimental visualization of possible integrative relationships, 2008). Polyhedral relationship networks?Whilst emphasis may be appropriately placed on the larger challenge of global governance and the new approaches seemingly required, it is useful to be aware of the many challenges faced by family and interpersonal relationships at this time. The incidence of family violence and divorce are but indicators of this. It could however be argued that the inadequacies of personal relationships are reflected in the inadequacies in governance relationships. As remarked by Gregory Bateson: We are our own metaphor. As noted above, there is a blossoming of awareness of social network relationships  especially beyond the family. Extended family networks, and kinship networks, remain of significance to many. Beyond the partners often desperately sought, individuals tend to be very focused on supportive networks of friends. The question might be asked as to whether, other than the somewhat arbitrary structure of such "networks", are there other ways of understanding those relationships that might enhance a sense of wellbeing and identity to a greater degree. Is the coherence and quality of a set of relationships enhanced by what is effectively its configuration as a polyhedron of some form  which may evolve or transform in response to circumstances, as with any sense of team? Given the complexities of the relationships within which many are embedded, would such configuration enhance the kind of desirable outcome to which family therapists may aspire? Alternatively, what is it that gives significance and a sense of wellbeing to an ordered pattern of relationships  beyond the hierarchical, patriarchal or matriachal models? Viable projects and businesses make much of the elaboration of an appropriate "business model". Is it possible that the set of polyhedra might highlight unforeseen patterns of relationship  as "psychosocial models"  that would enhance the meaning of those so engaged and empowered? One concrete approach is that explored by Stafford Beer (Beyond Dispute: the invention of team syntegrity, 1994) based on the icosahedron  and subsequently subject to restrictive licensing. Inspired by Fuller, the concept of a selforganizing geodesic democracy has, for example, been developed in a series of documents by Roan Carratu (The Geodesic Direct Democratic Network; Structure; Process; Modes; Finances; General Archives; Projects; Growth; Details of Specific Procedures, 2005) and on an associated website on geodemocracy. Like syntegrity, it focuses on small group processes icosahedrally organized  but set here in a larger context (see explanation of Geonet, 2000). Is there a much larger range of possibilities to be explored from which many might benefit  if only in the learnings to be derived from their explorations of them? The explosion of social networking offers an ideal context for such exploration  with an early proposal providing an interesting example (Group Questing or Twelving: proposal for a largescale smallgroup development process, 1976). Given that understandings of transparency and its desirability in configurations of relationships, there is a case for exploring this metaphorically through new understandings of glass  especially given the widespread strategic use of optical metaphors (perspectiove, focus, vision, etc). Glass may be understood as an inorganic product of fusion, cooled to a rigid condition without crystallizing. Whereas the solid state of all known metals and metallic alloys consists of regular, periodic arrangements of the atoms, a metallic glass consists of metallic atoms arranged in a random manner with no obvious longrange correlation in the atomic positions. These new have proved to be of considerable technological importance for their unique magnetic, mechanical and corrosionresistance properties. It might be asked whether this suggests a way of thinking about potentially valuable new configurations of psychosocial relationships. The work of Subir Sachdev (Icosahedral Ordering in Supercooled Liquids and Metallic Glasses, 1992) focuses on the structural properties of dense and supercooled systems of atoms interacting with each other through spherically symmetric forces. He finds that there are significant shortrange orientational correlations between the atomic arrangements applicable to any dense, supercooled liquid of spheres interacting with a pairpotential which has a repulsive hardcore and a weak longrange attraction  readily modelled in computer. Perhaps ironically a key property of the systems considered is that they are "frustrated". Sachdev remarks:
Sachdev's discussion of how the the "frustration" of flat three dimensional space is relieved, by appropriate minimumenergy icosahedral packing of polyhedra, points to possibilities of ordering relationships in new ways. Polyhedrabased sense of identity?Elsewhere (Emergence of Cyclical Psychosocial Identity: sustainability as "psyclically" defined, 2007) in a discussion of Interlocking cycles as the key to identity it was suggested that:
In this sense one or more polyhedra may prove to be a suggestive template or scaffolding onto which to project identity or with which to associate or order it. The sense of identity is then carried to a significant degree by the symmetry of the polyhedron and the manner in which the facets, edges and vertices complement each other  whatever content is mapped onto them. Speculatively, the individual facets might then be understood as distinct "windows" on the world  through which the world is engaged and through which identity is reinforced. Metaphorically the facets might be understood as having lenslike optical characteristics, serving a "polyocular" function in Magoroh Maruyama's terms, in bringing reality into focus. Each polyhedron then constitutes an array of such facets through which the world may be engaged (Spherical Configuration of Categories  to reflect systemic patterns of environmental checks and balances, 1994). Identity understood dynamically in this way might then also have its correspondence at the collective level. Missing link: selfreflexive closure?The polyhedral approaches presented above do indeed point to possibilities in relation to psychosocial organization. Missing however is any extensive exploration of how the polyhedral form can be cognitively decoded as a meaningful mnemonic. The work on epistemic networks is only suggestive of such possibilities. The work at the Cognitive Engineering Lab (University of Western Ontario) by Jim Morey and Kamran Sedig on an Archimedean Kaleidoscope as a "a cognitive tool to support thinking and reasoning about geometric solids" (Interactive Metamorphic Visuals: exploring polyhedral relationships, 2001) indicates other possibilities [explore their applet]. The focus is however on rendering polyhedra comprehensible and not on the cognitive processes of which they may constitute a formal representation. How is it that "network" proved so intuitively appealing as a metaphor? What is it that makes forms like yantras, mandalas or rose windows so appealing (to some) as patterns or indicators of significance? What is it that elicits such appreciation within analogous 3dimensional architectural spaces? This was the specific focus of Christopher Alexander and his team (A Pattern Language, 1977; The Timeless Way of Building, 1979) in identifying patterns that offered a subtle sense of a desirable "place to be" or a "sense of place"  of feeling "at home", as discussed separately (Patterns as enabling emergence of a "quality without a name"). From a design perspective this is a sense of "goodness of fit"  a concept shared with decisions or solutions meeting complex criteria in several relevant polyhedral approaches reviewed above. The question is then the nature of the psychological engagement  or identification  with the configuration of criteria that constitutes closure on an acceptable solution. What in fact is the psychological appeal of a "solution" as a kind of strange attractor  akin to "coming home" (Human Values as Strange Attractors: coevolution of classes of governance principles, 1993)? How is that cognitively embodied (cf George Lakoff and Mark Johnson, Philosophy in the Flesh: the embodied mind and its challenge to western thought, 1999; E. Thompson and F. J. Varela, Radical Embodiment: neural dynamics and consciousness, 2001)? Curiously it may be that aesthetics, notably poetry, offers pointers to pattern embodiment  especially in relation to governance (Poetrymaking and Policymaking: arranging a marriage between Beauty and the Beast, 1993). George Hart (The Pavilion of Polyhedreality) associates distinct pieces from J. S. Bach with polyhedral images  recalling Johannes Kepler's Harmonice Mundi (1619). The missing link might then be understood as the cognitive processes implicitly mapped in the steps of polyhedral approaches to decisionmaking. The challenge might be framed in terms of the "cognitive decoding" of those steps  as so clearly identified by Jakub Marecek (Polyhedral Approach to Multicriterial Optimization, 2006). It is somewhat ironic that the most comprehensible articulation of the process (found during this literature review) is that of Olivier Toubia (New Approaches to Idea Generation and Consumer Input in the Product Development Process, 2004) with respect to polyhedral methods for adaptive choicebased conjoint analysis in market research  especially the discussion of polyhedral question design and estimation. Whilst links and nodes are widely understood as metaphors  notably in relation to the web  what about faces, planes, cuts, cones and point location? What of the processes of truncation and stellation  and the implications of complementary duals? What form do these decisionmaking steps intuitively take in real life, independently of the mathematical formalization with which Marecek and others are so familiar? Where is the evocative wording that would build the vital bridge between the riches of formalization and the intimacy of the experiential encounter with decisionmaking? What of the closure that highlights the importance of convex polyhedra? Is it characteristic of a degree of selfreflexivity? How might such closure on a solution relate to the intuitive understanding of a "settlement" achieved by an appropriate configuration of "stakeholders"  exploiting the ambiguity of these terms in relation to both decisionmaking and architectural construction of a "safe and sheltered communal space"? Any exploration of the architectural metaphor could usefully take account of the role of "compression" and "tension" elements that enable material construction and their implications for construction of cognitive spaces, existential "places to be" and even "comfort zones" (Groupware Configurations of Challenge and Harmony  an alternative approach to "alternative organization'', 1979). These are issues at the interface between tensegrity structures and polyhedra (as mentioned above). The challenge might be framed in terms of the question: what sort of network feels like home or like a community? Agaikn Christopher Alexander's design pattern language offers a suggestive template for any cognitive analogue (see 5fold Pattern Language, 1984; Governance through Patterning Language: creative cognitive engagement contrasted with abdication of responsibility, 2006). Does intuition offer access to decisions arising from polyhedra in more than 3 dimensions  as suggested by Ron Atkin (1981)? To what degree are solutions associated with ndimensional polyhedra meaningful? Is it polyhedra of this kind that offer coherent, comprehensible solutions to the dilemmas of sustainable development? It is this that would give a sense to what might be experienced in a "polyhedrally empowered" network  how it "hangs together" comprehensibly as a viable, communicable solution. Also of potential significance is the implication that it is crystals, as natural forms of polyhedra, that are capable of reflecting and refracting light  not networks. Whilst "crystalization" is considered inherently problematic in psychosocial organization, it may prove to be of unsuspected significance in a "supersaturated" knowledge society (see separate discussion Patterning Archetypal Templates of Emergent Order: implications of diamond faceting for enlightening dialogue, 2002). Beyond the use of "Windows" as an operating system with an associated interface to the web, there is a certain irony to the parallel between the array of wallmounted rectangular screens characteristic of:
The evolution of this arrays into wraparound (effectively polyhedral) cognitive experiences suggests a strong link to the design challenges of "cognitive fusion" in which disparate information is brought into focus (Enactivating a Cognitive Fusion Reactor Imaginal Transformation of Energy Resourcing (ITER8), 2006). The screens then function as cognitive lenses justifying exploration of optical metaphors. Such explorations might then offer a new approach to strategic dilemmas, using polyhedral forms as a guide to strategic "goodness of fit" as a pattern in a design sense. This would then have important implications for governance. In building this bridge, it is worth recalling the early words of Harold Lasswell (The Transition toward more Sophisticated Procedures, 1968):
Symmetry: competition vs complementarity?The argument above points to the value of symmetry both in ensuring network robutness and comprehensibility of complexity. However, from a complexity science perspective as argued by Chris Lucas (personal communication):
This framing highlights the question of how nodes are distinguished in a network and how vertices are distinguished in a polyhedron. In the case of a network, the strategic arguments for robustness (notably from a military perspective) stress a degree of uniformity and replaceability of nodes  no node being significantly distinct from another. However, from another perspective, it might be argued that if all the nodes were indistinguishable there would be no significance to relationships between them. Missing from the complexity argument is the sense in which a variety of nodes  if only of different colour  may be configured in a symmetrical array. Good examples of this are teams. The above argument clearly does not apply to a football team or to two opposing teams. There is no question of arguing for a different array of players  10 or 15  for that game. The same is true of a card game like bridge. Of great management significance is the desirable range of roles in a management team as identified by Meredith Belbin (Management Teams, 1981). He distinguished nine key roles: plant, resource investigator, coordinator, shaper, monitor/evaluator, team worker, implementer, completer finisher, and specialist. Other such arrays have been suggested. How many distinct roles and functions are desirable for its sustainable development in any system  whether an ecosystem or a psychosocial system? This points to the complementarity implied by nodes in a symmetrical array. Significantly this is seldom explicit in the design of social networks, or the enthusiasm for that mode. People may have complementary roles but this tends to be recognized, if at all, through the dynamics of the network. Indeed, in practice, the roles identified by Belbin may be emergent rather than designed in. The problem for such a management team may arise if particular roles are missing or overrepresented. This would suggest that in this sense the nodes in a "polyhedrally empowered network" are empowered precisely because they constitute an ordered diversity of skills or perspectives  a viable system in the sense advocated by R Buckminster Fuller (Synergetics: explorations in the geometry of thinking, 1975/1979) and by Stafford Beer (Beyond Dispute: the invention of team syntegrity, 1994). Intriguing in the case of the spherical symmetry explored above is the sense that some nodes, often half, are not visible from any perspective when the polyhedron is viewed. They may be understood to be in a "shadow" zone. The sphere has to be rotated to bring them successively into view. This suggests a way of thinking about "otherness", namely "them" rather than "us"  typically a challenge to be met, possibly through the dynamics of competition. The interplay between complementarity and competition is of course most evident in the archetypal symmetry of the relationship between man and woman. The challenge, exemplified in that context (and by the challenges of comprehending spherical symmetry), is how to internalize the "other". Another metaphor, using the dual form of polyhedra, is the implication of the possible need to alternate into the dual form, or to allow for its expression  a transformation wellillustrated by morphing (Carl Erikson, Morphing Three Dimensional Polyhedral Objects, 1994; Wayne Carlson, et. al. Shape Transformation for Polyhedral Objects, 1992). With respect to dynamics, Chris Lucas again argues:
This argument is obviously valid in those terms. However it does not take account of classic examples such as the resonance hybrid dynamics of the benzene molecule so fundamental to the organic world. Again it does not take account of how symmetrical arrays of team players with specific roles engage with one another in football. Nor does it take account of the archetypal dance between man and women. Of great interest in relation to the above references to Galois lattices, is the lattice focus of the work of Patrick Heelan (Logic of Changing Classificatory Framework, 1974). Of particular interest are the dynamics of possible relationships between polyhedral forms. As with a strategic play in footbll, a polyhedrally empowered network might may morph into another array, or between several arrays, as indicated in the images in the related paper (Configuring Global Governance Groups: experimental visualization of possible integrative relationships, 2008). Elsewhere it was suggested that sustainable development was perhaps to be understood as based on alternation (Policy Alternation for Development, 1984). Polyhedral empowerment: "eliciting the sparkle from networks"?There is a fruitful play on the relationship between "dual" and "jewel" in that the wealth of civilization  its highest values  may be epitomized not so much by its precious stones (often set in symbolic "crowns", rings, or pendants) as by the dynamics of duality, so well mapped by morphing polyhedra. Hence also the association with sacred geometry as emblematic of the highest forms of integration and order. It is also interesting to recognize that networks as such do not "sparkle"; it is the crystals, associated with the polyhedral forms into which networks may be folded, that enable the reflection and refraction of light for which they are so valued. Eliciting this capacity, in its most enhanced forms in gems and jewels, is dependent on symmetry (Steven Dutch, Symmetry, Crystals and Polyhedra, 1999). More specifically gems are most notably associated with the set of polyhedral forms that exhibit a degree of selfduality amongst themselves, namely the 5 Platonic forms: tetrahedron, cube/octahedron, icosahedron/dodecahedron. Enhancing their capacity to sparkle is achieved by appropriate cutting, facetting and polishing  themselves offering interesting metaphors with respect to networks. Crystals are:
As noted above, gems offer an interesting metaphor regarding processes of relevance to governance (Patterning Archetypal Templates of Emergent Order: implications of diamond faceting for enlightening dialogue, 2002). In the above context, where polyhedral nets embody the necessary symmetry to enable them to "come alive" and "sparkle", what is to be said of the portions of the network that are appropriately configured to ensure that this process occurs?
Given the way in which reflection and refraction work in gems, the existence of gemlike forms in which light is only reflected from the outer surface offers interesting metaphors regarding those in which light may only be reflected between the inner surfaces, not allowing it to pass out  what might be termed "dark crystals". The issue might be related to the distinction between bonding ("within crystal") and bridging ("between crystals") (cf Dynamically Gated Conceptual Communities: emergent patterns of isolation within knowledge society, 2004; Y. Connie Yuan Geri Gay, Homophily of Network Ties and Bonding and Bridging Social Capital in ComputerMediated Distributed Teams, 2006). This is partially played out, notably in academic circles, through what have been termed "mutual citation pacts". Presumably such issues of reflection and refraction could be understood as being intimately related to the sense of image  especially selfimage  and the importance attached to it by a person or a group. Given the features of a network, people (or groups) may be variously identified with (and hold responsibility for):
Recognition of curvature as fundamental to a polyhedral psychosocial universe?Networks and Sponges: In dealing with the frustration of undertaking strategic initiatives, the omnipresence of networks is frequently acknowledged through exasperated phrases such as "everything is connected to everything". It is therefore appropriate to note that sponge structures are the most abundant forms in nature, on all possible scales of material existence. But, as noted by Michael Burt (Periodic Sponge Polyhedra: expanding the domain, 2007), the amount of morphological insight into the phenomenon is meager. It may therefore be appropriate to see unforeseen aspects of polyhedrally empowered networks in terms of insights into the ordering of sponges  as a generalization of the order conventionally associated with polyhedra based on planar surfaces. This is achieved by Burt through the introduction of the curvature of faces and edges  an intuitively reasonable expansion of perspective. "Curved" psychosocial relationships? It might be considered curious that psychosocial relationships, bonds and links are commonly conceived as "linear"  whether or not they are considered multivalent or as polyhedral nets. They may indeed have a quality of "curvature" as is often implicit in any understanding of their dynamics. Burt notes:
He does not mention their possible relevance to psychosocial order, but presumably would not exclude it, as implied elsewhere (Michael Burt, Periodic Sponge Surfaces and Uniform Sponge Polyhedra in Nature and in the Realm of the Theoretical Imaginable) in stating:
Spherically symmetrical radiolaria have long offered an appeal to the imagination as organic patterns with which psychosocial organization might be in some way isomorphous. The quality of sponginess, as noted by Burt, has been recognized in many languages as the description of a physical phenomenon characterized by porosity and visual permeability whose equivalences are recognized in "porous" psychosocial organization  and, pejoratively, in the dynamics between some people. Given the different degrees of appoximation of the regular polyhedra to the curvature of the circumsphere on which their vertices are located  and the intuitive significance of a sphere as emblematic of desirable coherence and wholeness  there is a case for envisaging the concave spaces within spongelike forms as incomplete "reflections" of that encompassing form. Symmetry, topology and combinatorics: Burt describes his approach to the spongifom domain as follows:
Primary Parameters of the Polyhedral Universe. The parameters found to be most suitable by Burt to describe the polyhedral phenomenology in 3D space are the following:
Periodic Table of the Polyhedral Universe (Michael Burt, 1996): This remarkable synthesis has been subsequently described by Burt (Periodic Sponge Surfaces and Uniform Sponge Polyhedra in Nature and in the Realm of the Theoretical Imaginable) as to some extent a visual animation of Euler's formula of VE+F = 2(1g), producing a medium through which all its apparent and hidden meanings are revealed. Burt's synthesis is constructed on the basis of thorough selection of the above primary parameters. These:
Of critical importance therefore, when dealing with the polyhedral universe and especially with its "sponge domain", is the determination of these primary parameters on the basis of which, as coordinates of a Cartesian space, the Periodic Table of the Polyhedral Universe is constructed. This then provides a domain in which every conceivable polyhedron has a unique point representation and discloses patterns of polyhedra sharing various geometrictopological characteristics, related to these primary parameters. It constitutes a powerful research tool of the polyhedral universe. Burt further notes:
Burt (2007) concludes:
Isomorphism relevant to psychosocial organization?: It is curious that Burt's work on the fundamentals of order has emerged from the designfocused context of a university faculty of architecture and town planning  just as has the past and recent work on a pattern language of Christopher Alexander and his team (A Pattern Language, 1977; The Timeless Way of Building, 1979; The Nature of Order, 20032004). In the absence of analogous insights into the fundamentals of order specific to psychosocial organization in a knowledge society  rather than for its "shelters" and "settlements"  these approaches would seem to constitute a strong justification for the exploration of their implications for new understanding of more appropriate patterns of psychosocial and cognitive organization. The focus on sponges, given their essentially organic and biological nature (despite their interest for architects), highlights in a remarkable manner the inappropriateness of an essentially inorganic and abstract assumption regarding networks. Discussion of networks of all kinds typically focuses on their abstraction as linear relationships between nodes  themselves treated as abstractions. This is necessarily true of polyhedral nets and of the polyhedra they may form in three dimensions (despite their significance for psychosocial organization as argued above). Absent from any such discussion is the existential participatory experience (in the moment) within any such network. Such networks are at best treated like telecommunication networks of wires between devices  of which the organic equivalent is the nervous system. This is far from the experience of such a network  to which Atkin's work so notably points. Embodying an experiential dimension: There is a seemingly unexplored approach to representation of networks, and their configuration into polyhedra, which relates it to Burt's work on sponges. It may be recognized that:
In this way the tubes of the network link experiential spaces in a manner somewhat similar to the manner in which any sponge is organized. There is a certain irony to any focus on the humble "sponge" (in all its variety, as identified by Burt) potentially offering a more organic insight into the experiential significance of networks, whether polyhedrally configured or not. Further support to imaginative reflection on the complexities of psychosocial organization is offered by the possibility of allowing the diameter of either the link (edge) or the node (vertex) to increase considerably, and even disproportionately, whilst retaining the geometry of the polyhedron. Possibilities include:
Perceived from a given vertex, the intersection with another vertex can be understood as a form of Venn diagram (or one of its variants), whether in 2 or 3 dimensions  suggestive of "cognitive sponges". Further possibilities include:
Decisionmaking as changing orientation: The introduction of curvature by Burt highlights interesting perspectives within a psychosocial context in which "horizon effects" are necessarily a feature of ignorance and denial  as effectively recognized by Atkin. Any "linear" progression over a convex curved surface eventually results in the starting place becoming "out of sight" as the effect of curvature comes into play  especially problematic if curvature is denied. To the extent that spherically symmetrical polyhedra approximate to the curvature of a sphere as they increase in complexity, there is a case for recognizing that decisionmaking at a nodal point (a vertex) is associated with a change of orientation  from that of one link (edge) to any others meeting at that node. A vertex is then a point at which the various decision options are presented  all of which will prove (after linear pursuit of their orientation) to be suboptimal with respect to the integrity associated with the circumsphere on which the vertex is located. [N.B. This postpones any discussion of the potential significance of stellation  precluding the association of some vertices with the circumsphere. It also postpones discussion of the contrasting integrity that might be associated with the insphere  the largest sphere that is contained wholly within the polyhedron, being dual to the dual polyhedron's circumsphere.] There will be preferences (vigorously) articulated for one orientation rather than for another. Typically the focus is on selecting the "right" decision for further "forward" progress. This may be understood as either:
However the polyhedral framework highlights the interrelationships amongst the set of disparate orientations (and the interlocking great circles) that effectively defines the system as a whole  whether to be understood as a psychosocial ecosystem, an "egosystem", or even as a dialogical "echosystem". It suggests a degree of fruitless inappropriateness in seeking to persuade all of the inherent superiority of any particular great circle pathway (Spherical Configuration of Interlocking Roundtables: internet enhancement of global selforganization through patterns of dialogue, 1998). This is especially the case where, as a learning pathway, each is best understood in temporal terms, as a temporal learning/action cycle (notably as explored by Arthur Young, The Geometry of Meaning, 1976, summarized in Characteristics of phases in 12phase learningaction cycle, 1998). The challenge of achieving significant consensus is then epitomized by that of ensuring fruitful multigenerational, multicultural (community) gatherings  given the experiential and orientational differences. Following the discussion above with respect to a polyhedrallybased sense of identity, and elsewhere (Emergence of Cyclical Psychosocial Identity: sustainability as "psyclically" defined, 2007), any challenge of consensus might then be best understood as one of reconciling an array of "voices", each developing a distinct theme from one vertex to another  in song. This is the musical challenge of eliciting harmony from polyphony and multipart singing in large choirs. It highlights the cultural and cognitive significance traditionally associated with song and rhythm, as notably explored from a philosophical perspective by Antonio de Nicolas (Meditations through the Rg Veda, 1978) and Ernest G McLain (The Myth of Invariance: the origins of the gods, mathematics and music from the Rg Veda to Plato, 1976; The Pythagorean Plato: prelude to the song itself, 1978). The underlying polyhedral structure, whether temporal or otherwise, may then be understood as a "metapattern" in terms of the central thesis of Gregory Bateson (Mind and Nature: a necessary unity, 1979):
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