Triangulation, connectivity and "stitching": enabling coherent global system dynamics

Middle East Peace Potential through Dynamics in Spherical Geometry (Part #10)

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The role of triangulation has been reviewed separately with respect to triadic logic, triadic dialectics, triadic strategic applications, triadic conceptualization and
triadic education and learning (Triangulation of Incommensurable Concepts for Global Configuration, 2011). This follows from earlier exploration of Spherical Configuration of Categories to Reflect Systemic Patterns of Environmental Checks and Balances (1994).

That discussion notably referred to the focus of R. Buckminster Fuller who argued extensively for the fundamental importance of triangulation as the basis for the stability of structures, notably with respect to his application of spherical triangulation to geodesic domes (Synergetics: explorations in the geometry of thinking, 1975). He demonstrates the need for omnitriangulation as a fundamental requirement of system integrity:

Not until we have three noncommonly polarized, great-circle bands providing omnitriangulation as in a spherical octahedron, do we have the great circles acting structurally to self-interstabilize their respective spherical positionings

It is possible therefore that the integrity of psychosocial systems, and the connectivity of the "patterns which connect" of Gregory Bateson, involve an "omnitriangulated" emotional engagement.

Fuller's insights have been applied separately in the explorations of the structural requirements for the possible polyhedral  organization of governance (Towards Polyhedral Global Governance: complexifying oversimplistic strategic metaphors, 2008; Configuring Global Governance Groups: experimental visualization of possible integrative relationships, 2008; Geometry of Thinking for Sustainable Global Governance: cognitive Implication of Synergetics, 2009).

The argument has also been developed in relation to a degree of intuitive understanding of geometry upon which strategic discourse relies through metaphor (Experience of Cognitive Implication in Fundamental Geometry: unexamined metaphoric framing of strategic discourse, 2012). This noted the case made from the perspective of cognitive psychology by George Lakoff and Rafael Nuñez (Where Mathematics Comes From: how the embodied mind brings mathematics into being, 2001).

Replication of distinct polyhedral patterns (from above) to facilitate illustration of connectivity (below)
5-fold Star	6-fold Star

The following images show how the lines defining the stars connect together in the truncated icosahedral pattern -- also indicating where necessary connectivity is not (quite) possible in two-dimensions. Note that the function of the external polygons as "vehicles" or templates is now of secondary significance -- although those forms are now engendered within each star form, and coloured accordingly..

Illustration of the two patterns of connectivity between neighbouring star configurations
Connectivity of 6-fold Stars around a 5-fold Star	Connectivity of 5-fold Stars around a 6-fold Star

The "lines" in the image above also call for reflection on how they might function in systemic terms, especially with respect to the emergent dynamics of any such system:

• according to a biological metaphor, they would constitute segments in vital metabolic pathways (shown above) -- only completed when connected in three dimensions.
• according to an electrical circuit metaphor, they would constitute elements of a wiring diagram or circuit board -- again only enabling functionality when the connections are established in three dimensions (as in the case of supercomputer memory organization).
• according to a brain metaphor, the links could be understood as neural synapses, namely a structure that permits a neuron to pass an electrical or chemical signal to another cell (susceptible of simulation as in the ambitious Blue Brain Project)
• according to the (dialogue) communication metaphor, these elements can be understood as discourse threads -- only enabling integrative discourse when appropriately connected in three dimensions.
• according to a legal metaphor in which the linking elements could be understood in terms of vital relationships between elements of a  (global) agreement, notably as a guarantee of appropriate checks and balances, and reporting obligations.
• according to a learning metaphor in which the links were indicative of learning pathways through which knowledge is built progressively, notably ensuring integrative linkage between the elements of knowledge acquired.
• according to an architectural metaphor in which structural elements may have distinct functions in distributing tension or compression (or both), as is especially obvious in the case of the geodesic domes whose polyhedral structure and dynamics may well be compared to that of a football

Other suggestive metaphors, readily recognized as having greater cognitive relevance, include:

• a concept map, namely a diagram showing relationships among concepts, as a graphical tool for organizing and representing knowledge.
• an argument map, namely a visual representation of the structure of an argument in informal logic -- typically including the components of an argument such as a main contention, premises, co-premises, objections, rebuttals and lemmas
• a cognitive map (known also as mental maps, mind maps, cognitive models, or mental models) indicative of a pattern of psychological transformations by which information can be acquired regarding the metaphorical spatial environment.
• a conceptual graph, namely a formalism for knowledge representation.
• a topic map, namely as a standard for the representation and interchange of knowledge

Potentially more intriguing are the implications suggested by:

• reflecting and refracting optical systems whose functioning depends on the mutual angle of orientation of their elements. As a metaphor associated with "vision", these merit careful attention with regard to the capacity to present and focus an image -- as understood with respect to strategy development and eliciting consensus. The cognitive implications of mirroring of notably attracted extensive study.
• aerial arrays dependent for their functioning on the orientation of their elements with respect to one another
• associated with the optical metaphor is the facetting required to enhance the brilliance of precious stones -- with which human values are so intimately related (Patterning Archetypal Templates of Emergent Order: implications of diamond faceting for enlightening dialogue, 2003)

The pentagon and the hexagon, as shown above, are intimately related in geometrical terms to a form of triangulation from which the 5-fold and 6-fold stars emerge. The question which might fruitfully be asked in relation to the earlier arguments concerning the cognitive significance of polyhedra and triangulation, is whether there are systemic implications suggested by the metaphors mentioned immediately above.

The patterns suggest the significance of connectivity being enabled or disenabled in dynamic systemic terms -- readily understood in the operation of a model train set in which traffic is allowed or disallowed. If such connectivity were to be understood as sequentially phased, as is characteristic of electrical systems, is there then a sense of effects analogous to those of the rotation operations of motors and dynamos? How might cognitive "cycles" be identified as characteristic of the global organization of the pentagon-hexagon patterns in a truncated icosahedron (cf. Cognitive Cycles Vital to Sustainable Self-Governance, 2009)?

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