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28th May 2008 | Draft

Polyhedral Pattern Language

Software facilitation of emergence, representation and transformation
of psycho-social organization

-- / --

Also available in a PDF version for printing convenience. Sequel to Towards Polyhedral Global Governance: complexifying oversimplistic strategic metaphors (2008). Associated with: Polyhedral Empowerment of Networks through Symmetry: psycho-social implications for organization and global governance (2008), Configuring Global Governance Groups: experimental visualization of possible integrative relationships (2008) and Configuring Global Governance Groups: experimental animations and video sequences (2008)

Context
PART A: Polyhedral mapping experiments and use of Stella Navigator
-- Focus and psycho-social relevance
-- Currently relevant features of Stella
-- Possible additional features
-- Possible applications relevant to psycho-social organization
-- Pattern language
-- Test applications
-- Mapping possibilities with regular polyhedra
PART B: Future extension to tensegrity representation
-- Psycho-social operationalization of polyhedra through tensegrity representation
-- Process of use
-- Multi-dimensional heuristic work space
-- "Cognitive circuits"
PART C: Meta-patterning considerations from other cultural perspectives
-- Patterns as enabling emergence of a "quality without a name"
-- Quality of space as a central challenge of governance
-- "Spirit of team": vibration and resonance
-- Insights from Chinese culture
-- Pattern dynamics: change, alternation, resonance
-- Possibilities of "variable geometry" in psycho-social organization
-- Comprehension of strategically appropriate patterns through the fourth dimension
-- Homeostatic equilibrium: necessary "human sacrifice" to the "gods"
CONCLUSION: "Globality", viability and comprehensibility
References

The significant problems we face can not be solved at the same level of thinking
we were at when we created them (Einstein)
Learning: Governance needs a new language?

Context

Part A is effectively a commentary on the polyhedral exploration software Stella Navigator, produced by Robert Webb (Stella: Polyhedron Navigator. Symmetry: Culture and Science, 2000); a helpful overview of the application is provided in Wikipedia. Its most recent version provides unique access to polyhedra in both three and four dimensions. Demo versions (3D or 4D) may be downloaded free of charge; an explanatory manual is available.

The commentary follows from a recent study of the relevance of polyhedra to global governance (Towards Polyhedral Global Governance: complexifying oversimplistic strategic metaphors, 2008). Here "polyhedral" is specifically used to convey the need to explore "many-sided" forms of psycho-social organization. These constitute an integrative challenge of governance from the world to the local level -- if not to personal self-governance and identity. In that respect "global" is also to be understood in its integrative sense of a "whole" and not only in its geo-political sense of world-wide (cf Future Generation through Global Conversation: in quest of collective well-being through conversation in the present moment, 1997) .

The concern here with polyhedra dates from work within the context of the Union of International Associations in the period 1997-2000 on the online, multi-media, interactive representation of complex networks of international organizations, world problems, global strategies and human values, notably developed from 1972 within the Encyclopedia of World Problems and Human Potential. The later work formed part of a project funded by the European Commission (Ecolynx: an information context for biodiversity conservation, 1997-2000) and subsequently evaluated positively by the World Bank for a development focus (INTERCEPT: Interactive Contextual Environmental Planning Tool for developing countries, 1998). In that period various experiments were undertaken to enable online users to associate particular portions of such networks with polyhedra in virtual reality (VRML) in order to facilitate their comprehension and as a point of entry to text profiles of the entities in the networks. Those in Figures 1 and 2 below were generated online directly from the databases, for example. Although the databases are still online -- some being freely accessible -- the transfer to a new platform has meant that these experiments are no longer accessible.

A range of these facilities is described elsewhere (Information Visualization and Sonification: displaying complexes of problems, strategies, values and organizations, 2001). The concern here however is to identify concrete possibilities and applications for the future.

Part B considers the possible future extension of Stella to tensegrity representation. Part C evokes a range of meta-patterning considerations, notably from other cultural perspectives.

PART A: Polyhedral mapping experiments and use of Stella Navigator

Focus and psycho-social relevance

It should be stressed that the Stella software has been primarily, if not uniquely, developed for exploration of the geometry of polyhedra. Any associated possible uses in the above context are therefore incidental. The purpose of this commentary is to determine how it may already be used for the exploration of psycho-social organization and how it might possibly be extended to facilitate further explorations of that kind.

The argument for the use of polyhedra has been developed in the earlier paper (Towards Polyhedral Global Governance: complexifying oversimplistic strategic metaphors, 2008). Basically the argument is that governance of any kind is cognitively challenged by the complexity of the psycho-social oranization that needs to be ordered and comprehended. Use of hierarchical organization charts, bullet points and simple mind maps is not adequate to the complexity and need for coherence.

Polyhedra, especially those of higher complexity, offer representational devices onto which complexity can be projected in such a manner as to enhance mnemonic possibilities and a sense of transformational potential -- especially when these are reinforced by properties of regularity, symmetry and appropriately variegated colouring. The concern in what follows is therefore with the use of polyhedra as "conceptual coat-hangers" (integrative memory aids or "cognitive prosthetics") and how the features of the Stella applications might be used for such purposes -- now or in the future..

The focus here is therefore both on how the application can be currently used for that purpose and how it might be extended in support of other uses. Additionally there is the question of whether such an application suggests unforeseen approaches to facilitating the emergence, representation and transformation of psycho-social organization (whether groups, organizations, networks, problems, strategies, concepts or values).

This focus on facilitating psycho-social organization, its emergence and representation (especially through symmetrical polyhedral forms of tensegity), depends on the confluence of several essentially independent lines of investigation -- some of which have been unfortunately inhibited by intellectual property rights. These are:

The relevance is foreseen in relation to:

Figure 1: Mapping of problems and organizations onto selected polyhedra in virtual reality
(extract from Exploring Intelligible Associations: integrative modes and metaphors, 2006
as presented to the German Research Centre for Artificial Intelligence)
Selected issues relating to "discrimation against women" Selected components of the "EU system"
Problem polyhedron Organizatioal polyhedron

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Figure 2: Selected screen shots of previous experimental visualizations (1997-2001)
(click on each for navigable version in virtual reality; possibly with the free browser Cortona plugin
-- available for PC, Mac OS X, and Pocket PC platforms --
mouseover nodes or faces for entity names; many hotlinked to text profiles)
Experimental visualization of a problem Experimental visualization of a problem complex Experimental visualization of an organization complex
Interlocking world problem loops displayed as a "polyhedron" of intersecting planes Clusters of interlocking problem loops with links between them International organizations mapped onto faces of an (incomplete) spherical polyhedron (larger file, slow to mani9pulate)

Currently relevant features of Stella

The general features are detailed in the original paper by Robert Webb (Stella: Polyhedron Navigator. Symmetry: Culture and Science, 2000). They are of course presented in detail in the current help manuals associated with the applications.

The following is a commentary on features of special interest that enhance the mnemonic function that might prove significant in various institutional and strategic design situations:

Figure 3: Transformation of representations of issue arenas on an icosidodecahedral net (using Stella)
Derived from one original mapping of a set of global strategic dilemmas (as explained in
Configuring Globally and Contending Locally: shaping the global network of local bargains by decoding and mapping Earth Summit inter-sectoral issues, 1992)
The succession of images below shows stages in the unfolding of an icosidodecahedral mapping from 3D to 2D; a wide variety of other manipulations may be used in Stella to explore this pattern. Of particular interest is the relationship between the parts and the implications for any specific "roundtables" in relation to any global bargain (see commentary)
The 20 labelled triangles (paired) correspond to 3-fold issue complexes, whereas the 12 unlabelled pentagons correspond to 5-fold issue complexes, each necessitating more complex dialogue; the structure is integrated by 5 strategic themes. An unlabelled version of the structure, exported from Stella, can be viewed in virtual reality, possibly with the free browser Cortona plugin (for PC, Mac OS X, and Pocket PC platforms)
[click on each image below for enlarged version]
Transformation of issue arenas on an icosidodecahedral net Transformation of issue arenas on an icosidodecahedral net Transformation of issue arenas on an icosidodecahedral net
Transformation of issue arenas on an icosidodecahedral net Transformation of issue arenas on an icosidodecahedral net Transformation of issue arenas on an icosidodecahedral net

The above representation suggests a distinct between:

Framed in this way, it is how the configuration of elements offer a degree of integrity to the variety of otherwise fragmented elements in the global debates (cf Spherical Configuration of Interlocking Roundtables: Internet enhancement of global self-organization through patterns of dialogue, 1998).

Possible additional features

There are clearly several options in this respect:

It is important to recognize the prime function of Stella in considering such options. Such changes might simply complicate unnecessarily a well-honed application. One alternative is of course to allow users to select additional tool bars for facilities unrelated to the primary function -- such as some discussed below.

Labelling and images: As it currently functions, Stella ensures that colours, labels and images "stick" to surfaces to a very useful degree when rotated, etc. However, colours seem to stick over a wider variety of transformations -- including morphing -- than labels or images. Colours are also appropriately reflected in both 2D and 3D versions?

It would be convenient if labels and images applied to a face in 2D net representation could be preserved rather than lost when the image is folded into 3D. To what degree can labels and images be preserved through a greater variety of transformations

External links: These could be of various types:

All such links would currently require significant effort to export from Stella to VRML and might therefore be more appropriate in a separate application.

Identification of usable representations: With respect to on-the-fly input, one interesting feature would be to be able to call up (and/or select) a list of polyhedra with user-specified numeric properties, based on numbers of faces, edges, and/or vertices. Whilst polyhedra may be selected by type through a drop down menu, and many properties of any selected polyhedron are listed, there is no apparent means to search on such properties.

As an alternative, there is of course the Wikipedia List of uniform polyhedra, presented in tabular form, with images (in fact created using Stella) and with links to entries in Wikipedia on each. The table permits polyhedra to be identified by numbers or faces, edges or vertices. However the tabulation only includes a small subset of the polyhedra available through Stella. In Wikipedia the typology of uniform polyhedra is usefully described separately.

Thus if one had an organization with N major divisions, what polyhedra might one project (map) it on to -- in terms of faces, edges or vertices? Could the user run through a proposed list of possible "good fits" and make a selection?

How might one distinguish between primary features and secondary features, namely if each division had sections, how might these be attributed to smaller faces, etc? Or is the divisional level to be associated with groups of faces appropriately shaped and coloured?

Importing sets of linked entities: In the earlier online exploration, a set of linked entities was imported (effectively as output from a search process by keyword) and then fitted, according to a (crude) algorithm. The interface then allowed the user to intervene in the process (see image ***).

With an easy input of a nested hierarchy of an organization structure, or array of nested strategies, this would enable a quick flip into a really integrative polyhedral overview of the initiative. Presumably outputs in a corresponding format might be made. One example of these approaches is the application Decision Explorer.

Matching and "fitting": Implicit in the previous point is the issue of how best to "fit" the structure to an appropriate polyhedron, or possibly a set of alternatives amongst which the user can choose or switch.

Clearly solutions vary greatly from simple to very difficult depending on how good a match is required. Issues include:

Clearly links should not be assigned to faces that share an edge, which would make a solution impossible most of the time.

Concentric polyhedra: The possibility of having concentric polyhedra (that could be displayed in wire-frame mode or with relatively high transparency) differently rotated. Seemingly this is possible, although trickier to arrange with Stella as it is.

A related possibility is to play with a light source from that centre such that, according to rotation of the concentric polyhedra, the light only "got through" certain "aligned" facets and not otherwise -- as an indication of consistency between different kinds of psycho-social structure (values, strategies, institutions and problems, for example).

Pattern language

The large set of polyhedra of distinct types together constitute a repertoire of patterns of order. As such they are a resource on which to call for subtler forms of psycho-social organization.

The key initiative with respect to pattern language has been that of Christopher Alexander and his team (A Pattern Language, 1977; The Timeless Way of Building, 1979). The idea of a pattern language appears to apply to any complex engineering task, and has been especially influential in software engineering where patterns have been used to document collective knowledge in the field. In a subsequent study (The Nature of Order, 2003-2004) the mostly static patterns from A Pattern Language have been amended by more dynamic sequences, which describe how to work towards patterns.

Alexander's earlier pattern language insights have been adapted experimentally to explore a more generic 5-fold Pattern Language (1984) of relevance to psycho-social organization:

A question to be explored with a polyhedral software application is the extent to which such patterns can be suitably, and memorably, represented by families of polyhedra. This is the crucial question, determining the relevance of this approach.

Is the taxonomy of psycho-social organization now in what might be termed a pre-Linnaean stage, given the significance that came to be attached to Linnaean taxonomy? Of interest in this respect is the role first played in flowering plant classification of the number of stamens in the bloom. It might be asked whether conceptual models (including their strategic and organizational reflections) could not be usefully distinguished by the kinds of numbers used too distinguished polyhedra (cf Representation, Comprehension and Communication of Sets: the role of number, 1978).

It would be very interesting if any periodic table of polyhedra -- each understood as a classification -- could be understood as providing insight into a classification of classification systems (cf Birger Hjørland, Lifeboat for Knowledge Organization; Classification, 2008). What indeed are the Patterns of Conceptual Integration (1984) in the light of the range of challenging examples (cf Patterns of N-foldness: comparison of integrated multi-set concept schemes as forms of presentation, 1980)? How such a periodic table might be "tuned" is fundamental to any response to the divisive psycho-social initiatives that are a preoccupation of governance (Tuning a Periodic Table of Religions, Epistemologies and Spirituality -- including the sciences and other belief systems, 2007).

Also of interest in this context s is the possibility of considering each polyhedron as modelling a learning system. This would then relate to the literature on the classification of learning systems, currently of relevance to artificial intelligence research (Douglas J. Pearson and John E. Laird, Incremental Learning of Procedural Planning Knowledge in Challenging Environments, 2005). Of relevance to understanding through polyhedra is the investigation of adaptive learning of geometry (Harri Ketamo, An Adaptive Geometry Game for Handheld Devices, Educational Technology and Society, 6 (1) 2003).

Possible applications relevant to psycho-social organization

Team building and associated strategy:

Complementarity vs Imbalance: Especially important to team building, and the formation of coherent coalitions of stakeholders, is to ensure a requisite variety of complementary elements (whether people, organizations, strategies, values or concepts). The symmetry properties of the polyhedra can be used to distinguish such elements, notably by colour. Use of polyhedra in this way also serves to highlight possible imbalance.

Mapping and encoding psycho-social functions: The value of a polyhedral pattern language is clearly associated with the degree to which cognitive significance can be usefully mapped onto its features such as to highlight and hold complementarities and contrasts. This may be primarily a matter of exploration within the contexts for which the mapping is to be used, whether a small group or a large community of interest. It may be a communication device, symbolizing an initiative, irrespective of whether the mapping is universally acceptable. The range of geographical projections of the world points to the kind of variety that is possible and variously considered desirable

In its simplest form, the question is if a set of specific psycho-social functions is distinguished -- whether as principles, action programmes, values, qualities, etc -- is it helpful to map these onto a polyhedral form rather than present them solely as a checklist? Many such sets have been elaborated with different numbers of elements. Clearly a match can be attempted purely on the basis of the number (Representation, Comprehension and Communication of Sets: the role of number, 1978).

An unfortunate feature of many such extant checklists is that, because of their simple structure, no attempt is made to consider the relationships between elements of the set -- relationships which may be of considerable importance to the integrity and viability of the set when applied. Agenda 21, as formulated by the UN Earth Summit (Rio de Janeiro, 1992) is an example of an asystemic set of articles in that the relationships between the disparate parts are not considered. A mapping onto a polyhedron may highlight useful questions about relationships implied by the polyhedral pattern. These can lead to useful reconfiguration of the set.

Of particular interest is the case of sets of functions that are considered well-defined and complete. The Myers-Briggs Type Indicator is one example, with its 16 types. These could be mapped onto the vertices of an octagonal prism or the edges of a square antiprism. Clearly the 12 astrological types, as with any other 12-fold set, may be mapped onto a wider variety of convex polyhedra of greater symmetry (vertices: truncated tetrahedron, cuboctahedron, icosahedron; edges: cube, octahedron; faces: dodecahedron,) as well as non-convex forms (edges: tetrahemihexahedron; vertices: cubohemioctahedron, octahemioctahedron, great dodecahedron; faces: octahemioctahedron, great dodecahedron) and less symmetrical forms. Even those with more complicated names are readily comprehensible visually.

Given the importance of duals in relation to the geometry of many polyhedra, the significance of any corresponding mapping is of great potential interest. A dual of a polyhedron is one in which the vertices of the first correspond to the faces of the second. This implies an interesting relationship in alternating, through the dual, between:

Potentially more intriguing is the manner in which the form of a polyhedron may be usefully "decoded", recognizing the specialized interests that have explored this -- such as anthroposophy with respect to "projective geometry" and others with various approaches to "sacred geometry".

Mapping systems to highlight viability: Governance of any kind is called upon to deal with systems of increasing complexity. Such systems may be represented on hierarchical charts or systems diagrams of many kinds -- in two dimensions. Recognizing the arguments of Buckminster Fuller that polyhedra are to be understood as systems, the possible corollary that systems may be represented by polyhedra merits exploration. In that sense complex systems could be, in principle, fruitfully mapped onto complex polyhedra such as to highlight vital complementarity and necessary communication patterns (notably feedback loops).

Seeding organization emergence: crystal / saturation / catalysis ***

Transformation of organization: Identification of pathways, and transitional forms, through which an organization might evolve from one polyhedral form to another such form, possibly more complex. Stella already offers a number of possibilities for such transformation.

Significant issues of cognitive perspective: The cognitive mapping onto a surface that can be formed into a sphere raises interesting issues:

Encompassing disagreement:

Communicating meeting outcomes: The process whereby an integrative synthesis is derived from the insights expressed at a meeting -- the global "sense of the meeting" in Quaker terms -- could be understood in terms of the ability to produce a mapping of them onto a suitable polyhedron. This could then be a visual complement to a press release -- even an index to its elements (on a web page).

Curiously this echoes the intuition associated with the use of gold "nuggets" as a significant meeting product, or even the discovery of "diamonds" in the meeting process. To the extent that the pattern of such insight is reflected in a concluding declaration, its elements could also be usefully mapped onto a polyhedron (Patterning Archetypal Templates of Emergent Order: implications of diamond faceting for enlightening dialogue, 2002; Structure of Declarations: challenging traditional patterns, 1993) possibly even to be associated with song (A Singable Earth Charter, EU Constitution or Global Ethic? 2006). In this context, in relation to discussion of tensegrity structures below, Ronald J. Barnett and Gregory W. Cherry have applied for a patent on Tensegrity Musical Structures.

Communication of more complex forms of organization: Just as a spiral staircase does not lend itself to comprehensible verbal description without any illustration (if only gestures), so there may be many forms of coherent organization that could well depend on the kind of cognitive prosthetic provided by a polyhedron of whatever degree of complexity. As with the spiral staircase, this may enable transition from one level to another. The polyhedron then functions as a mnemonic of a superior degree of order to bullet pointed charts, other checklists and complex organizations charts that do not enhance memorability. Without such support, higher degrees of ordered complexity become essentially unsustainable.

Configuration of the parts, with which people may variously and separately identify, helps to determine whether, as a confguration, a larger and deeper sense of identity and significance emerges. The question is then whether the emergent organization corresponds to a mode of organization with which people are already familiar (experientially) but for whose patterns no adequate description has as yet been found -- as with an "unformed" sense of community or team.

Insightful discussion of the associated communication and comprehension challenges is provided by mathematician Ron Atkin (Multidimensional Man; can man live in 3-dimensional space? 1981; Combinatorial Connectivities in Social Systems; an application of simplicial complex structures to the study of large organizations, 1977) as summarized elsewhere (Social organization determined by incommunicability of insights, 1995).

Such considerations point to the possibility of using interrelated polyhedra, of different degrees of complexity, to map psycho-social issues (over) simply, comprehensibly, and more challengingly -- such as to elicit a greater degree of imaginative engagement. Exploration of the transformations between these degrees of complexity enable learning pathways to be highlighted. They also point to patterns of insight and order that are more likely to be forgotten -- and which are effectively meta-stable and unsustainable, namely which lack adequate mnemonic reinforcement.

Test applications

It is appropriate to consider a wide variety of test cases, which may help to render the use of Stella credible as a pattern language tool in domains in which this might not otherwise be the case:

Self-reflexivity might even be further enhanced by mapping Christopher Alexander's set of patterns onto a suitably complex polyhedron.

Of related interest, in practice, is the possibility of using Stella to enable people to construct and share polyhedral mappings in support of their worldview, much as photographs and video clips are now exchanged worldwide:

Success with a number of these test cases would offer powerful symbols, in polyhedral form, of the integrative identity of topics that are otherwise considered to be regrettably quite fragmented in the understanding of many. Such a possibility was notably seen as relevant on the occasion of the establishment of the Mediterranean Union (Mediterranean Union: a symbolic challenge).

Figure 4: Application of 12 UN Specialized Agencies to faces of a dodecahedron in Stella
progressive unfolding from 3D to 2D
Application of 12 UN Specialized Agencies to faces of a dodecahedron Application of 12 UN Specialized Agencies to faces of a dodecahedron Application of 12 UN Specialized Agencies to faces of a dodecahedron

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Figure 5: Complexification of the above UN Agency example
(from dodecahedron to icosidodecahedron) using Stella
with subsequent morphing to yet more complex forms (by various methods),
suggesting increasing challenges (and possibilities) of communication and integration amongst the set of agencies so mapped;
a facetting diagram for the icosidodecahedron is also included (as an indication of possible communication pathways)
Morphing of 12 UN Specialized Agencies on faces of a dodecahedron to icosidodecahedron Morphing of 12 UN Specialized Agencies on faces of a dodecahedron to icosidodecahedron Morphing of 12 UN Specialized Agencies on faces of a dodecahedron to icosidodecahedron
Morphing of 12 UN Specialized Agencies on faces of a dodecahedron to icosidodecahedron Morphing of 12 UN Specialized Agencies on faces of a dodecahedron to icosidodecahedron

Mapping possibilities with regular polyhedra

Figure 6: Mapping possibilities of Platonic and Archimedean polyhedra
exemplified by G8 and G20 country groups (or other groups, including the nascent L20)
(see examples of G20 mappings onto icosahedron, icosidodecahedron and rhombicosidodecahedron)
Polyhedra.
(images in illustrated list prepared using Stella Polyhedra Navigator; those in links below can be rotated)		 Polyhedral characteristics
(see comprehensive illustrated table or list)		 Country
Groups
Faces Vertices Edges Total G8 G20
Triangle Square Penta. Hexa. Octa. Deca.
tetrahedron 4 - - - - - 4 6 14 - -
octahedron 8 - - - - - 6 12 26 x -
cube (hexahedron) - 6 - - - - 8 12 26 - -
icosahedron 20 - - - - - 12 30 62 - x
dodecahedron - - 12 - - -- 20 30 62 - -
cuboctahedron 8 6 - - - - 12 24 50 x -
icosidodecahedron 20 - 12 - - - 30 60 122 - x
truncated tetrahedron 4 - - 4 - - 8 18 34 - -
truncated octahedron - 6 - 8 - - 24 36 74 x -
truncated cube 8 - - - 6 - 24 36 74 x -
truncated icosahedron - - 12 20 - - 60 90 182 - x
truncated dodecahedron 20 - - - - 12 60 90 182 - x
small rhombicuboctahedron 8 18 - - - - 24 48 98 x -
small rhombicosidodecahedron 20 30 12 - - - 60 120 142 - x
great rhombicuboctahedron - 12 - 8 6 - 48 72 146 x -
great rhombicosidodecahedron - 30 - 20 - 12 120 180 162 - x
snub cube 32 6 - - - - 24 60 212 - -
snub dodecahedron 80 - 12 - - - 60 150 202 - -

Of particular interest is the manner in which use of mappings (whether for G8, G20 or L20) further down the table (namely more complex) ensures that the direct connectivity required of a given country is reduced. Thus in the G20 mapping onto an icosahedron, all triangular edges border other triangular edges and all verticies connect with other triangular vertices -- there is no "space", whatever flexibility that signifies. In the icosidodecahedron, however, only the vertices of the triangles are connected. In the rhombicosidodecahedron, the triangles do not connect with one another directly. Such complexification introduces "openness" and points to the challenge of "sparseness" rather than the need for "tight bonding" to ensure viability and integrity. Such opennes, as noted below, can only be achieved in practice through embodying tensegrity principles into both communication pathways (and dissociative resistances) between the countries, thereby constituting the emergent whole and its coherence.

Figure 7: Mapping possibilities of duals of Platonic and Archimedean polyhedra
DUALS of Polyhedra
(images in illustrated list; those in links below can be rotated)		 Polyhedral characteristics of duals
(for Archimedean duals, see list of Catalan solids)		 Polyhedra
(as in Figure 6)
Faces (irregular) Vertices Edges Total
Triangle Square Penta. Hexa. Octa. Deca.
tetrahedron 4 - - - - - 4 6 14 tetrahedron
cube - 6 - - - - 8 12 26 octahedron
octahedron (hexahedron) 8 - - - - - 6 12 26 cube (hexahedron)
dodecahedron - - 12 - - -- 20 30 62 icosahedron
icosahedron 20 - - - - - 12 30 62 dodecahedron
rhombic dodecahedron - (12) - - - - 14 24 50 cuboctahedron
rhombic triacontahedron - (30) - - - - 32 60 122 icosidodecahedron
triakis tetrahedron 12 - - - - - 8 18 38 truncated tetrahedron
tetrakis hexahedron 24 - - - - - 14 36 74 truncated octahedron
triakis octahedron 24 - - - - - 14 36 74 truncated cube
pentakis dodecahedron 60 - - - - - 32 90 182 truncated icosahedron
triakis icosahedron 60 - - - - - 32 90 182 truncated dodecahedron
deltoidal (trapezoidal) icositetrahedron - (24) - - - - 26 48 98 small rhombicuboctahedron
deltoidal hexecontahedron - (60) - - - - 62 120 242 small rhombicosidodecahedron
disdyakis dodecahedron 48 - - - - - 26 72 146 great rhombicuboctahedron
disdyakis triacontahedron 120 - - - - - 62 180 362 great rhombicosidodecahedron
pentagonal icositetrahedron - - (24) - - - 38 60 122 snub cube
pentagonal hexecontahedron - (60) - - - 92 150 202 snub dodecahedron  

PART B: Future extension to tensegrity representation

Psycho-social operationalization of polyhedra through tensegrity representation

A major difficulty with abstract forms such as polyehdra is how they engage with psycho-social organization -- irrespective of their inherent aesthetic properties and appreciation of them as "sacred geometry". And, irrespective of recognition of their role in the architecture of biological cells (and radiolaria), this difficulty is evident in efforts to use such forms in the architecture of buildings -- especially those polyhedra of greater complexity.

It is in this respect that learnings from their application through tensegrity -- tensional integrity -- to unusual constructions (such as geodesic domes) are of significance (see Documents relating to Networking, Tensegrity, Virtual Organization). Tensegrity is the means through which polyhedral forms are "operationalized" in building and therefore, with some probability, in psycho-social organization. This raises the question of whether an as yet undiscovered "geodesic" form of psycho-social organization might notably be of great relevance to global governance.

As an additional feature therefore, the capacity in Stella to generate tensegrity representations of polyhedra would be an advantage -- where this is possible and appropriate . A principal reason is that there is a range of arguments suggesting that the viable structure of the more complex polyhedra -- if constructed -- is dependent on the kinds of distribution of forces associated with the tensegrity variant. This is notably true at the level of the biological cell (Donald E. Ingber, Tensegrity and Systems Biology, Journal of Cell Science, 2003).

As defined by R Buckminster Fuller (Synergetics: explorations in the geometry of thinking. 1975):

A tensegrity system is established when a set of discontinuous compressive components interacts with a set of continuous tensile components to define a stable volume in space.

As models, the tensile elements (tendons) can be cables or strings, functioning like a network. The compressive elements can be beams or rods (struts) and function as spacers to prevent the network from collapsing. Despite the generality attributed to tensegrity principles (and explicit in the subtitle of Fuller's study), it should be stressed that most of the literature recognizes only their application to material structures. Of interest to any exploration of tensegrity are the following:

Figure 8: Indications for Construction of Tensegrities from Polyhedra
on the assumption that having polyhedral coordinates in great precision in Stella, possible tensegrity form-fitting could be readily tested by appropriate algorithms

Tensegrity design: The most practical, accessible and focused summary of the challenge (for architectural purposes) is provided by:

Robert Burkhardt:

Valentín Gómez Jáuregui: Tensegrity Structures and their Application to Architecture, 2004

VRML Tensegrity Models (2008), as elaborated by Robert Burkhardt, offer imaginative indications of possible "operationalizations" of polyhedra in the creation of new kinds of psycho-social organization. The following examples (from an extensive list), based on polyhedra, raise the questions: how might organization like that be experienced, how might it be recognized, and how might its emergence be facilitated:

Access to such virtual reality models requires a browser plugin (eg free browser Cortona plugin for PC, Mac OS X, and Pocket PC platforms)

Software support for tensegrity construction:

  • Robert Burkhardt has developed his own software to fabricate his style of double-layer tensegrity dome and sphere from geodesic breakdowns. This a front end which produces input for his tensegrity computation software. The software requires some sort of back end to use the product to finish off the tensegrity computations before moving on to VRML. He believes that the existing software could be adapted to output in a form suitable to another package (such as ANSYS).
  • Diego Budavari and Valentín Gómez Jáuregui are in process of developing a computer programme: Tensegrity Simulator (Simulador de Tensegridad)
  • Jason Evelthon Charalambides (Computer Method for the Generation of the Geometry of Tensegrity Structures, University of Texas, 2004) provides a comprehensive survey of the literature and states the focus of the computer program developed as follows:
    • Tensegrity is a technology that can be applied to structures and its use can influence the construction time efficiency and construction project management in general. However, a significant drawback for a systematic application of tensegrity structures in the building construction industry is the particularly complex geometry that engineers and architects have to generate, in a two or three dimensional virtual or physical environment. The objective of this dissertation was the development of a computer based utility that will facilitate the design professional to devise and construct a specific morphological variation of tensegrity structure systems. The development of this utility was based on a methodology that identified and included parameters that can be associated to the schematic design and design development phases of a design project. The main contribution of the developed computer program is the efficiency with which virtual models of a tensegrity structure can be generated, facilitating the designer in decision making during the design process. Emphasis was given on the development of an interactive graphical simulation/visualization environment for the computer program. This feature assists in the generation and modification of the numerical input, with parameters defined by the user, and allows unobtrusive regenerations of alternate solutions within the computer virtual environment.
    • The author notes that David Georges Emmerich (Structures Tendues et Autotendantes, 1988) developed a systematic method of deriving tensegrity forms from a range of Platonic and Archimedian polyhedra (citing A. Hanaor, Beyond the Cube: the architecture of space and polyhedra, In: F. Gabriel (Ed.), Tensegrity: Theory and Application, 1997)

Relevant tensegrity mathematics: Hugh Kenner (Geodesic Maths and How to Use it, 1976), notably a chapter on choosing a polyhedron.

Sets of Rules (Anthony Pugh, Introduction to Tensegrity, 1976)

  • Circuit pattern associated with polyhedra:
    1. The tendons define the edges of a polyhedron, which need be neither regular nor semi-regular. Four tendons and two struts meet at each junction to form a vertex of the figure, so at least four edges must meet at each vertex of a polyhedron used as a basis for a circuit-pattern figure..
    2. When the struts are joined, they form circuits of struts, hence the name circuit pattern for this strut-tendon relationship. The circuits of struts follow the lines of circuits of tendons, which can be traced on models of appropriate polyhedra.
    3. The circuits of struts interweave with each other, passing under one circuit, then over the next, under the next, and so on.
    4. A network of tendons surrounds a series of circuits of struts, the tendons pulling inwards like the skin of a balloon, the struts pushing outwards like the air in the balloon.
    5. There is a junction of struts and tendons outside the midpoint of each strut, so there will be the same number of junctions as there are struts.
  • Circuit pattern, based on geodesic polyhedra (alternate method), derived from an existing polyhedron (called its principal polyhedron)
    1. Triangulate any faces that are not already triangles
    2. The edges of each triangles are divided into equal numbers of equal parts and then lines drawn between the points established to define a triangular grid on each face
    3. With a sphere drawn through the vertices of the principal polyhedron, lines from the centre through all line instersections to that sphere then constitute the vertices of the geodesic polyhedron -- these are then joined to reproduce the pattern of lines on the surface of principal polyhedron
    4. Circuit pattern tensegrity systems can be based on any such geodesic polyhedron, provided that the faces of the principal polyhedron were subdivided to a frequency which is a multiple of two.
  • Patterns can be based on other polyhedra by following the symmetries of the figure to provide a network of tendons, then to add in the struts.
Unambigious set of rules: It may well be that such a set is "buried" in the many attempts to develop algorithms to derive tensegrities from polyhedra. However such a set does not appear to have been clearly articulated in a manner that lends itself to immediate use.


One of the advantages of such structures is as a representation of a configuration of polarizing factors in social organization -- forces which may be essential to its viability, as discussed elsewhere (From Networking to Tensegrity Organization, 1984; Groupware Configurations of Challenge and Harmony: an alternative approach to alternative organization, 1979; Implementing Principles by Balancing Configurations of Functions: a tensegrity organization approach, 1979).

Just as tensegrity principles are fundamental to the architecture of the biological cell, they may also be fundamental to the coherence of psycho-social organization and community -- to community "architecture". Such tensegrity structures may be especially significant in determining the patterns of (electronic) communication essential within cyberspace to the viability of an organization, a community or some other collective initiative.

One of the challenges in deriving tensegrity structures for architectural purposes is what is termed "form-finding" (cf Milenko Masic, et al, Algebraic tensegrity form-finding, 2005). There is a case for understanding the challenge of matching or fitting psycho-social organization to polyhedra (discussed above) as an analogous process of "form-finding". In both cases a polyhedron provides a template for the solution, but the challenge is to find it. Hugh Kenner (Geodesic Maths and How to Use it, 1976) offers guidance on "choosing a polyhedron" (ch. 11) which usefully frames the challenge.

The challenge is to "massage" chaotic network representations into a "global" configuration as primitively envisaged as a challenge to mathematics (Preliminary Clarification of Some Problems of Processing Networks of Entities -- in order meaningfully to map psycho-social relationships, 1973). This may now be more readily feasible with the aid of "polyhedral" network analysis (cf P. Doreian, Polyhedral Dynamics and Conflict Mobilization in Social Networks, 1981).

Process of use

In its current form, Stella necessarily calls for a degree of familiarity. With respect to its relevance to psycho-social organization, one challenge is to make evident which features it is more fruitful to use for what purposes, in what sequence and by whom.

In reflecting on the potential relationship between polyhedral structures and the dynamics of governance, it is interesting to reflect on the success of the online design environment, SodaConstructor -- a Java-based simulator -- open to popular participation. It allows participants to design a wide variety of mechanical systems empowered by parameter-controlled muscle-like dynamics and constrained by "gravity". It is interesting that the Stella application is associated with a "3D minesweeper game", played on polyhedral surfaces -- recalling the popularity of Tetris and Exotic Minesweeper (which uses cells of different shapes: hexagonal, octagonal, square). As with SodaConstructor, users of Stella may post interesting models into a collective repertoire. Several mind mapping software packages make use of hexagonal modelling which therefore help to reinforce the argument for exploring 3D variants.

A proposal has been made to adapt the SodaConstructor approach to the representation or design of complex institutional structures variously empowered by sectoral budget lines (Animating the Representation of Europe: visualizing the coherence of international institutions using dynamic animal-like structures, 2004). Clearly the future evolution of psycho-social organization, based on tensegrity principles, could encourage similar online exploration.

Multi-dimensional heuristic work space

A focus to the confluence envisaged here may be given in terms of the kind of "work space" envisaged in a classical paper by Douglas Engelbart (Augmenting Human Intellect: a conceptual framework, 1962), notably:

Schroeder appears to offer the most significant synthesis, in relation to the above argument, especially given his extensive reference to tensegrity (as the "defining figure" in the workplace he proposes) and his association of it with the pattern language of Christopher Alexander. His study focused on:

... three innovations that suggest an alternative approach to structuring information systems: a multidimensional heuristic workspace, a resonance metaphor for information, and a question-centered approach to structuring information relations. Motivated by the need for space to establish a question-centered learning environment, a heuristic workspace has been designed. Both the question-centered approach to information system design and the workspace have been conceived with the resonance metaphor in mind....

This revised view of the metaphors of space was accompanied by a critical evaluation of the prevailing metaphors for information processes, the conduit and pathway metaphors, which led to the emergence of an alternative, resonance metaphor. Whereas the dominant metaphors emphasized information as object and the movement of objects and people through networks and other limitless information spaces, the resonance metaphor suggests the existence of multiple centers in dynamic proximity relationships...

The federation of multiple autonomous problem-solving spaces, toward goals such as establishing communities of questioners, has become an objective of this work. Future work will aim at accomplishing this federation, most likely by means of the IS0 Topic Maps standard or similar semantic networking strategies.

Schroeder made extensive use of the Axon Idea Processor: a visualization tool for thinkers, as developed by Chan Bok, in elaborating his proposed workspace and illustrating its operation. It allowed total network views and management of relations in the form of clusters as well as explicit links. It also allowed the workspace to be organized in terms of concentric polyhedra -- a feature suggested earlier. Uunlike Stella, Axon does not have a virtual reality mode. Despite that, Schroeder argues that the workspace so enabled is an autonomous, stand-alone environment that affords the possibility of being federated with similar spaces through use of emerging facilities such the IS0 Topic Maps standard.

In his primary focus on the tensegrity in a virtual environment (but not in the virtual reality characteristic of Stella), Schroeder appropriately notes:

It should be noted here that the tensegrity figure in this workspace is only an image, and does not have the capacity to behave in the digital environment as a physical tensegrity would. The figure included here was created based on observation of an existing physical model. The great advantage of the physical tensegrity structures is that the relationships among the forces that determine the structure and the physical stability that results creates an immediate understanding of the whole system. Realizing these characteristics in the digital environment, rather than just representing them, is suggested as an objective of future work.

Schroeder also provides a very helpful detailed summary (pp 199-207) of the advantages of the tensegrity-based workspace, notably in terms of :

Clearly such explorations need to be related to the multi-actor situation of organization emergence and self-organization. Of particular interest is the author's approach to:

Implicitly, combining the above points, is the sense in which the emptiness around which globality is centred brings a special and necessary form of integrative order to "global" networks that are otherwise unconstrained (cf Varieties of nothingness and emptiness, 2008). Arguably it is this centrally referenced emptiness that creates a space, that is potentially isomorphic with that of the proposed workplace of the mind. The approach creates a "context" within which "global" discourse takes place effectively (Future Generation through Global Conversation: in quest of collective well-being through conversation in the present moment, 1997).

"Cognitive circuits"

In a polyhedral pattern language, each polyhedron may effectively be understood as a representation of a "cognitive circuit diagram". This might be considered implicit, even explicit, in Fuller's "geometry of thinking" and intentions, although his insights are undermined and "denatured" by the inability to take account of the cognitive issues of embodiment so ably explored by Lakoff -- and close to Pask's conversation/interaction theories.

Operationalized and enabled as a tensegrity, the featurs and resonant dynamics of the circuit start to become apparent in their relevance to psycho-social organization. There are indeed analogues to the characteristic features of electrical circuits (resistors, capacitors, etc) through which "energy" is engendered, stored and redistributed (cf Electrical Systems as a Guiding Metaphor for Stages of Group Dialogue, 2001). But, in the sense explored by Francisco Varela (The Embodied Mind: cognitive science and human experience, MIT Press 1991, with Evan Thompson and Eleanor Rosch) and enactivism, such a circuit diagram has to be embodied to be enabled.

It is such cognitive circuits that sustain the patterns of behaviour on which governance is dependent -- and which it endeavours so desperately to seek to change. They constitute the sustaining tracery of communication pathways and feedback loops.

PART C: Meta-patterning considerations from other cultural perspectives

The challenge of any pattern language lies in what makes the pattern meaningful. For psycho-social organization, what is "goodness of fit" in a cognitive, organizational and strategic context where comprehension is critical to communication, credibility, coherence and viability?

A core issue is any limitation on the capacity of the human mind to "grasp" sets of elements of any degree of complexity and how any such constraints may be circumvented in the organization of principles, categories, organizations or strategies as discussed elsewhere (Representation, Comprehension and Communication of Sets: the role of number, 1978). Polyhedra represent an interesting way of bypassing the constraints of hierarchical and networking metaphors -- combining their advantages and compensating for their respective weaknesses.

Patterns as enabling emergence of a "quality without a name"

With respect to the challenges of governance, much has necessarily been made of issues relating to the tangibles (food, health, etc) on which economists so readily focus, and to the intangibles manifest over time (freedom of information, freedom of religion, etc) which are also a focus of politics. And, in the attention given to the pattern language of Christopher Alexander (discussed above), the focus has been on the process of design that the methodology enables in whatever domain. Little attention is given to his declared, and extensively described, purpose of using such patterns to enable the emergence of what he describes as the "quality without a name" (Timeless Way of Building, 1979).

In A Pattern Language (1977), in that spirit, Alexander has done much to clarify what would here be termed the elven pathways fundamental to providing a subtle sense of a desirable "place to be" or a "sense of place" -- of feeling "at home".

There is a central quality which is the root criterion of life and spirit in a man, a town, a building, or a wilderness. This quality is objective, but it cannot be named... every place is given its character by certain patterns of events that happen there. These patterns are always interlocked with certain geometric patterns in the space... To reach the quality without a name we must then build a living pattern language as a gate. (pp ix-xi)

Industrialized society has however come to recognize aspects of its importance under the term "quality time" or in the increasing difficulty for top corporations to retain valuable executives. But the point was made long ago by the realization that "man cannot live by bread alone". This perspective is further discussed elsewhere (Walking Elven Pathways: enactivating the pattern that connects, 2006).

The merit of Alexander's achievement has been to create a qualitative bridge between the technical considerations of architecture, so evident in discussion of tensegrity, and the qualitative purpose of the spaces so created. This might be said to go beyond the "heuristic" workspace of the mind which Schroeder's work seeks to enable. Put succinctly, what is the quality characteristic of the space associated with any particular pattern and the manner in which it is embodied and lived?

Curiously, at best, the architectual aesthetic is very sensitive to the qualities of space created. It is in this sense that the use of strategic confifgurations of pillars can be fruitfully revisited. Associated with those pillars, for example, are the spaces between those pillars and the tension associated with the relation of those spaces to the pillars that define them. It might be argued that it is the purpose of the strategic pillars to create desirable "places to be". Were the spaces to be filled, extending the bounding pillars into a wall, "sides" are then created in ways that were noted as problematic in the earlier study.

Quality of space as a central challenge of governance

Seemingly quite dissociated from the explicit challenges of governance, there is an extensive literature on the spirit of place and psycho-social attachment to the land:

It is quite clear that it is precisely such attachment which provides a prime focus for major conflicts, notably the unresolved cycles of violence in the Middle East focused on the symbolism of Jerusalem. Implicit, but unmentioned in that context is the epistemological significance of more poetic understandings of space, as exemplified by:

The question to be asked, as an extension of Schroeder's concern with a workplace of the mind, is the nature of the quality associated with particular enabling polyhedral configurations. The disciplines of architecture and design claim sensitivity to the appropriateness of spaces for particular functions. This is reflected in the design of parliaments, meeting rooms, think tanks and retreat centres.

Beyond the concrete however, what are the patterns that enable configurations of thought of relevance to the challenges of governance? How does a pattern language provide appropriate cognitive scaffolding for the emergence of viable strategy? How is engagement with such patterns enabled?

"Spirit of team": vibration and resonance

The sense of quality associated with space is also sensed to be associated with psycho-social organization. Typically this quality is named in jargon such as "vibrations". People refer to the degree of "resonance" they experience (or do not) with particular initiatives and groups of people. The most frequent recognition of such "spirit" is in the experience of "team spirit" in sports, and its analogue in the military.

Various efforts have been made to elaborate such understanding:

Insights from Chinese culture

This exploration was inspired by the use of "pillar" as the overriding European strategic metaphor, as discussed in the earlier paper (Towards Polyhedral Global Governance: complexifying oversimplistic strategic metaphors, 2008). Polyhedra have been presented there as configurations of such pillars in any more coherent approach to governance. With respect to global governance, involving a variety of cultures, it is therefore useful to seek complementary insights from the quite distinct Chinese culture -- especially if there is any tendency to assume that Eurocentric strategic thinking is to be considered appropriate worldwide.

Geometric elaborations of polyhedra, developed by the logic of geometric transformation from Platonic polyhedra, can be understood as patterns implicit in the Platonic forms -- as implied by the potential for such transformation. Corresponding elaborations and implications may be found in the Chinese logical elaboration of two classic coding systems, each a pattern language par excellence:

There is a visual irony to the fact that the European "pillars", and the "unfilled spaces" between them, can be fruitfully compared to the complete (yang) and broken (yin) lines constituting any hexagram. It is of course a fact that the I Ching was long used as tool of governance in classical times -- and was required study for entry into the Chinese civil service.

These patterns are then qualitatively associated, through their respective metaphorical representations, in

The set of patterns calls for exploration as a pattern language (Conditions of Objective, Subjective and Embodied Cognition: mnemonic systems for memetic coding of complexity, 2007; 9-fold Higher Order Patterning of Tao Te Ching Insights: possibilities in the mathematics of magic squares, cubes and hypercubes, 2007).

This of course raises the question of the qualitative implications that may potentially be associated with the patterns of the other Platonic forms, as developments of the less complex ones. Both the icosahedron and the dodecahedron suggest mappings of:

The integrative (global) nature of such sets is a real challenge to comprehension in terms of how the parts play off against each other. It is for this reason that the relationships are typically embodied in metaphor and stories, an approach explicitly favoured by the authors of The Wisdom of Teams (1993).

There have been many exercises in interpreting the relevance of the I Ching and the Tao Te Ching to issues relating to organizational strategy and management. Perhaps most relevant to this argument is the classical text of Cheng Yi (1033-1107) as translated by Thomas Cleary (The Tao of Organization: the I Ching for group dynamics, Shambhala, 1995), who comments:

The Tao of Organizations analyzes relationships and power configurations within groups. Taking into account both the subjective and objective dimensions of these structures, it is extraordinarily subtle and complex. The relationship between interpersonal and intrapersonal forces... is the central focus of the explanation.

Cleary adds an appendix on Cheng Yi's instigation of "inner design" (or noumenalism) intended to bring out hidden dimensions in classical Chinese works -- obscured because of the use of rigidly fixed interpretation schemes, inappropriate in any context of transformation. He comments at length on how such principles were consciously applied by the legendary Japanese industrialist Matsushita Konosuke (1894-1989).

More generally the challenge of comprehending the interrelationship between seeming disparate facets of a pattern may be seen as a challenge of "correlative thinking" of which the Chinese understanding of yin in relation to yang offers an example (A. C Graham, Yin-Yang and the nature of Correlative Thinking, Institute of East Asian Philosophies, 1986). The issue for governance is how credible is any degree of correspondence and to what degree of confidence does it give rise in strategy development and implementation (cf Theories of Correspondences and potential equivalences between them in correlative thinking, 2007).

Pattern dynamics: change, alternation, resonance

As noted above, the further work of Christopher Alexander and his team has been on The Nature of Order (2003-2004) in which the the mostly static patterns from A Pattern Language have been amended by more dynamic sequences, which describe how to work towards patterns.

As a set of patterns in its own right, the I Ching is also known through its translation as the Book of Changes -- namely the transformation between conditions symbolized by particular patterns. This focus on change is also evident in its companion classic, the T'ai Hsüan Ching.

It is interesting that at the molecular level, the benzene molecule fundamental to the structure of organic matter, is explicitly recognized to be based on a pattern of resonance in the bonding arrangements between its six constituent carbon atoms. As such it is known as a resonance hybrid -- characterized by the fact that its enduring viability is based on the alternation between bodning configuration. Such molecular resonance is recognized when no single conventional model, using only even number of electrons shared exclusively by two atoms, can actually represent the observed molecule. It might be expected that some analogue would be operative at the psycho-social level (cf Configuration of modes as a resonance hybrid, 1986).

Curiously whilst healthy democracy and governance is understood to be based on alternation of power between political parties or coalitions, there is little attention to the need to explore this principle in relation to healthy governance and the formulation of strategy. In certain forms of farming, the value of crop rotation is recognized as essential to systatainable exploitation of fields. It might be inferred that this offers learnings for alternation between strategies (Sustainable Cycles of Policies: crop rotation as a metaphor, 1988). More generally it might be inferred that sustainable governance is dependent on alternation between patterns (Development through Alternation, 1983).

In the light of such arguments, using the pattern coding language of the I Ching, the systemic relationships between various sets of patterns have been explored as an experiment (Transformation Metaphors -- derived experimentally from the Chinese Book of Changes (I Ching) for sustainable dialogue, vision, conferencing, policy, network, community and lifestyle, 1997).

Possibilities of "variable geometry" in psycho-social organization

Within the above context current use of the term "variable geometry" fails to distinguish between different configurations, understood simply as alternatives, and the dynamics of shifting between configurations in response to different conditions. The latter is the essence of structural nimbleness (cf Wanted - A New Social Entity: role of the potential association, 1971).

An excellent metaphor is the switch consciously made, according to circumstances, between different tactical configurations in rugby and other such team sports. These configurations may be named or numbered to be called out by the captain during a game. They are termed formations in association football, as distinct from the formations in American football. They refer to the position players line up in before engaging during the course of play. There are both offensive and defensive formations with many in both categories.They also feature in computer simulations (see Team formations).

For purposes of governance, "variable geometry" has been defined in a European context as follows:

'Variable-geometry' Europe is the term used to describe the idea of a method of differentiated integration which acknowledges that there are irreconcilable differences within the integration structure and therefore allows for a permanent separation between a group of Member States and a number of less developed integration units. (Europa Glossary) [more]

As noted by Warren Mason and Susan Penksa (The Variable Geometry of Security Cooperation: a policy framework for European integration. 2004), the development in Europe of a vast array of new structures of cooperation -- both public and private -- has left analysts reaching for conceptual tools with which to frame these phenomena. Charles Grant develops the argument further (Can Variable Geometry Save European Enlargement, 2005). Its relevance operational relevance has been highlighted by Patrick Joachim Dunphy (Variable Geometry Europe -- patching together what works in the fight against hard-core cartels: carrots, sticks, custody and leniency, 2007). Mike Goldsmith (Variable Geometry, Multilevel Governance: European integration and subnational government in the new millennium, 2003) reviews the extent to which concepts such as variable geometry and multilevel governance remain useful in aiding understanding of the processes of change through which EU territorial politics are passing.

Variable geometry has been notably recognized in relation to trade (Paolo Guerrieri and Stephen S. Cohen, The Variable Geometry of Asian Trade, 1994. The approach has also been envisaged for the UN system (Alternation between Variable Geometries: a brokership style for the United Nations as a guarantee of its requisite variety, 1985) and within the UN system. As reviewed by C. Patel (Single Undertaking: a straitjacket or variable geometry? 2003) and discussed by Andrew Cornford (Variable Geometry for fhe WTO: concept and precedents, 2004) who notes that the idea of was raised during the Uruguay Round when the constitution of the new multilateral organization (eventually leading to the WTO) was under consideration. Indeed, as noted by the Swiss National Science Foundation,

World trade rules are not marked by a uniform 'geometry'. Exceptions are made to general principles for defined groups of nations, for specific principles and specific issues. Regional trade arrangements (RTAs) and Special and Differential Treatment (SDT) are the two most important exceptions. [more]

The relevance has been explored in relation to the "multilateral" trading system, notably at a meeting of World Trade Institute (The Single Undertaking After Cancun: diversity and variable geometry in the WTO System, 2004).

On the occasion of a multi-stakeholder dialogue and networking event for addressing the challenge of making ambitious targets of the world community a reality, the Helsinki Process on Globalisation and Democracy (2005), Susan George (Variable Geometry to Design Positive Outcomes, 2005) stated:

In my view, variable geometry is the most useful political concept to emerge from the Helsinki Process. It is the recognition that no one institution, or type of institution, can solve by itself the problems we confront today. We need a cooperative framework in which various actors abandon their turf-wars and work together.

The question is whether comprehension of the possibilities would be enhanced by the software facilities outlined above through which a large repertoire of configurations and relationships can be explored.

Curiously, with respect to the suggestion above regarding sports teams and previously (Engaging with popular games), the pattern of numbers defining the formations is reminiscent to that used to characterize polyhedra:

Figure 9: Comparison of number patterns in football formations and polyhedra
(possibly suggesting that one might be represented by the other)
Examples of nomenclature in offensive or defensive formations in American football strategy Examples of polyhedral symmetry groups
(typically further extended in vertex arrangements)
  • 4-3-4: Four-Three defense
  • 3-4-4: Three-Four defense
  • 5-2-4: Short Yardage defense
  • 4-2-5: Nickel defense
  • 4-1-6: Dime defense
  • 3-1-7: Prevent defense
  • 3-3-2: tetrahedral
  • 4-3-2: octahedral
  • 5-3-2: icosahedral
  • 3-6-6: tetrahedral truncated
  • 4-6-6: octahedral truncated
  • 5-6-6: icosahedral truncated

If there proves to be a useful mapping from football tactics to polyhedra, given the extensive repertoire of the latter, this would then raise the possibility of unforeseen tactics in football -- as envisaged with respect to the emergence of unforeseen governance strategies. It would also offer a means to render credible more complex strategies of governance in response to the problematique -- a form of Rosetta Stone for strategy.

Given the variety of ways in which polyhedra can be represented, notably to foresee possibilities of transformation between them, it may be useful to note the manner in which Stella allows for the display of the cell diagram of any polyhedron -- typically in the form of an inverted hierarchy:

A cell diagram is a graph indicating how stellation cells relate to each other. The layers of cells are shown as layers of nodes, each node representing one cell type. Edges of the graph connect nodes from one layer to nodes of the next layer if the two cell types share a common face, so the cell type at the bottom of the edge supports the cell type at the top of the edge.

Understanding of polyhedra may thus be usefully framed as providing a conceptual interface between hierarchy and network -- typically problematic in psycho-social organization portrayed in 2D -- using forms in 3D with degrees of symmetry that render complexity comprehensible. Such understanding also connects with traditions of "sacred geometry", even through "faceting diagrams".

In this context it is appropriate to note the many comments in the literature on the polyhedral structure of a football, notably in relation to the fullerene molecules (see also Understanding Sustainable Dialogue: the secret within Bucky's Ball? 1996).

Comprehension of strategically appropriate patterns through the fourth dimension

It is repeatedly noted that the strategic challenges of the times now call urgently for extraordinary thinking -- beyond the conventional patterns that continue to exacerbate those challenges. As a prime advocate, Edward de Bono is instigating a World Council for New Thinking. As noted earlier, mathematician Ron Atkin asks whether man can indeed live in three dimensions (Multidimensional Man; can man live in 3-dimensional space? 1981). Another mathematican, Marcus du Sautoy (Finding Moonshine: a mathematician's journey through symmetry, 2007) makes the point:

For centuries composers, writers, artists, choreographers and architects have pludered the mathematical world in search of new structures to stimulate them creatively. (Creative Calculations, The Guardian, 29 April 2008)

In also enabling the exploration of fourth dimensional polyhedral patterns (regular polytopes), Stella might then be said to act as a kind of bridging psychopomp. It traces virtual pathways through the unknown and the uncertain to facilitate Walking Elven Pathways -- enactivating the pattern that connects (2006). The appropriateness and relevance of such exploration of the paradoxical order of the hyperreality of the emerging knowledge society has been highlighted through the subtle aesthetics of such as semiotician Umberto Eco (Travels in Hyperreality, 1973) -- effectively calling for hypercomprehension (Hyperaction through Hypercomprehension and Hyperdrive: necessary complement to proliferation of hypermedia in hypersociety, 2006).

As a mathematician, Marcus du Sautoy has worked with musicians and dancers to explore how new ideas of mathematics can be woven into a piece of theatre pushing all boundaries (The 19th Step, 2008) -- inspired by a work of Jorge Luis Borges, author of the Library of Babel (1941), a knowledge-edifice of hexagonal-shaped rooms in a four-dimensional context.

For governance the challenge might indeed be framed as the Comprehension of Appropriateness (1986) -- for which a polyhedral pattern language is a vital "revolutionary" cognitive support (Metaphoric Revolution: in quest of a manifesto for governance through metaphor, 1988). The essence of this challenge is making the subtlety of appropriate connectivity credible. The mathematician's response to "moonshine" usefully frames one approach to this challenge (Potential Psychosocial Significance of Monstrous Moonshine: an exceptional form of symmetry as a Rosetta stone for cognitive frameworks, 2007).

Homeostatic equilibrium: necessary "human sacrifice" to the "gods"

The dynamic equilibrium characteristic of tensegrity structures makes evident the challenge of the sustainability of psycho-social systems through what can be mapped onto them. In the earlier paper, it was argued that the deities of any pantheon can be usefully associated with the implicit polyhedra for mnemonic purposes (Re-membering the Dodekatheon). The self-stabilizing processes of a tensegrity, through the manner of interaction of its parts under stress, could then be understood as the manner in which the particular "gods" of which it is composed require appropriate "sacrifice" -- to be appropriately "honoured" -- to ensure the continuing integrity of the system as a whole.

How should such "gods" then be understood?

It is the interplay of any set of "deities" that is then fundamental to the integrity and identity of the system mapped by the tensegrity used. Any such sense of integrity is typically much challenged in the current context, whether with respect to natural, psycho-social or subjective systems. Of great potential interest is then the nature and "appropriateness" of the "sacrifice" that must be made within that framework to particular deities "ruling" any of its parts. In tensegrity terms, this is the manner in which particular parts must necessarily be constrained in response to the equilibrating functions of the local (partial) systems within the tensegrity as a global system.

Use of the metaphor "human sacrifice" in this context provides a useful systemic context for the "sacrifices" that humans, collectively or individually, are currently being challenged to make in the interests of sustainable governance. It is of course useful to note the association to the mortal sacrifices that individuals were called upon to make in the past -- and which government may continue to expect of them to protect "the motherland" (but not "the planet?). As with the ritual sacrifices of the past, whether of animals or humans, it is unfortunate that their systemic significance has been lost with the rejection of the abhorrent characteristics that presumably were then considered necessary to render such significance memorable. Ironically, however, few regulatory measures, including legislation, are now introduced through modern governance without some form of preliminary mortal sacrifice to justify them.

On a personal level, how sacrifices are variously made may continue to be an issue in response to the different challenges of living in a resource constrained society. It is perhaps from this perspective that the seeming rigidity of systems of values, such as the Confucian, should be revisited in terms of the systemic significance attached to calls to "honour". The same applies to the promotion of "family values" whether by government or religion. These could be fruitfully positioned in relation to the "freedoms" (from such values) that tensegrities may also be seen as encoding. The relevance of such an argument may be seen in the analogous role played by calls for "respect" -- and the dgree of importance attached to it -- within the seemingly most alienated sectors of society.

A tensegrity based on a Platonic or Archimedean polyhedron is especially valuable in highlighting the interlocking cycles underlying its structure -- the "great circles" around it in geometrical terms -- in contrast to those associated with the "cognitive circuits" (discussed above). Elsewhere it has been argued that these fruitfully point to a cyclic understanding of identity in dynamic rather than static terms (Emergence of Cyclical Psycho-social Identity: sustainability as "psyclically" defined, 2007 ). It is in this sense that traditional articulations of sets of values and prescriptions with which many identify through religions can be fruitfully seen as offering indications of systemic relevance (Navigating Alternative Conceptual Realities: clues to the dynamics of enacting new paradigms through movement, 2002). The work of Alexander on dynamics in relation to pattern emergence is clearly relevant.

Challenge of a new language for governance?
"Why do we put so much emphasis on audio-visual means of portraying goal, trend, condition, projection, and alternative? Partly because so many valuable participants in decision-making have dramatizing imaginations... They are not enamoured of numbers or of analytic abstractions. They are at their best in deliberations that encourage contextuality by a varied repertory of means, and where an immediate sense of time, space, and figure is retained".
(Harold Lasswell, The transition towards more sophisticated procedures. In: D. B. Bobrow and J. L. Schwartz (Ed.). Computers and the Policy-making Community, 1968, pp. 307-314)

CONCLUSION: "Globality", viability and comprehensibility

The key question is whether the possibilities of representation on a form -- whose integrity only fully emerges virtually in 3D -- augments the range of possibilities of negotiating necessary constraints in practice in order to achieve such integration. A crude experiential metaphor is whether a game of bridge is more feasible, interesting and sustainable where there is a recognized need for 2 pairs of players such that, if only 3 are immediately available, ensuring the presence of a fourth is desirable. A related illustration is the (arbitrary) statutory restriction on numbers of members of some elite clubs.

The simplest regular polyhedra may be considered the least "global" -- as exemplified by "spherical" -- in that, compared to more complex Platonic or Archimedean polyhedra, the facets are most distant from the circumsphere through the vertices. "Globality" is therefore essentially implicit in these simpler and most readily comprehensible cases. It is however the simplest that are the most easiest to construct in practice and are the most stable.

The ease of representation of the more complex polyhedra in Stella lies in the many advantages of a virtual environment. Ironically it might be argued that articulation of any complex sets of values -- as virtues -- is also much easier in a "virtual environment", unconstrained by the challenge of implementation in practice.

The structural viability of the more complex polyhedra relies on tensional integrity in practice -- namely an artful balance between compression and tension elements ("checks and balances") that is the focus of tensegrity architecture. The "art" of achieving this lies in the requisite variety of the multiplicity of differently oriented facets of such polyhedra -- that together, in psycho-social terms, reflect the sustaining culture of the whole.

It should be stressed however that, in the case of psycho-social organization, "in practice" is concerned with the necessary patterns of communication between the parts of the whole -- the necessary feedback loops. In a society constructed on information flows through cyberspace, viable "virtual organization" is therefore dependent on how these communication pathways get designed into the structure and reflected in e-mail exchanges, for example.

This is the significance of the tensegrity "operationalization" of polyhedra. It provides a map of both the requisite communication patheways and the necessary separators (or insulators) in order for a new degree of order to emerge -- with sufficient robustness to redistribute stresses globally, thus dynamically resisting any tendency of its constitutive network to collapse. It may be fruitful to consider the "insulators" as regulators (as provided by governance through directives and regulatory authorities), with the flexible communication pathways as exemplifying the "freedom" essential to resilience and the aspirations of the governed.

The challenge indeed lies in "bridging the chasm" between the simple and the complex, the "virtual" and the "concrete". This is a generalization of the challenge of "speaking across the chasm of frame conflict" in human communication (fruitfully discussed by Schroeder) -- to which the different orientations of the faces of polyhedra draw attention. It is also recognized as the two-culture chasm between the "sciences" and the "arts" -- exemplified in the realm of governance by the cognitive challenge of heads of state inspired by poetry (Poetry-making and Policy-making: arranging a marriage between Beauty and the Beast, 1993).

More provocatively and self-reflexively, this is also the challenge of interrelating the various threads (architectural, cognitive, and otherwise) of the above argument -- whose proponents might be variously seen as living on different faces or planes of a polyhedron as yet to be identified.

It is these principles that need to be embodied in tensegrity psycho-social organization and strategy -- if viable coherent "global" governance of any "integrity" is to be achieved, whether at the global, regional or personal scale. In this context, the set of polyhedra offer a range of explicit articulations of "global" -- especially through the dynamics of their tensegrity variants.

It is important to recognize that the above focus is on enabling exploration of possibilities and interpretations. The concern is not with closure on any particular set of definitive interpretations or understandings. In the application of such a "pattern language" to governance in uncertain times, it is the capacity to be cognitively sustained in the exploration of possibility and potential that is most to be valued in the quest for "sustainable development" and "quality of life". The art of global governance may turn out to be well illustrated by the capacity to explore and promote more integrative metaphors -- possibly mapped onto the transformational potentials of the set of spherically symmetrical polyhedra.

The possibilities of a polyhedral pattern language therefore lie in its capacity to trigger alternative and complementary ways of thinking about intractable governance issues -- eliciti9ng creativity appropriate to the much sought "paradigm shift".

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