Envisaging the global dynamics of a "hyperbolic round table" through tessellation

Engaging with Hyperreality through Demonique and Angelique? (Part #10)

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A symposium of the wise, to celebrate the sesquicentennial of Boston University, selected a tessellation as the metaphor that best captured the spirit of the times -- as noted by Lance Morrow (Metaphors of The World, Unite!, Time, 16 Oct. 1989). This is consistent with the explosion of interest in networking over recent decades, and the articulation of discourse that this has offered over the internet.

Archetypal round table: With respect to encounters of the wise, the archetypal metaphor continually favoured is probably the round table. This implies connectivity beyond the conventions of discourse in relation to a square table (as discussed in the main paper) The constraints and possibilities of discourse configured in this way are discussed separately (Implication of the 12 Knights in any Strategic Round Table, 2014).

In the light of the case for exploration of hyperreality, the question arises as to how a "round table" might be fruitfully reframed beyond the constraints and particularities of Euclidean geometry. How might the "roundness" be understood in terms of a sphere, for example? This would presumably facilitate discourse with respect to issues of a "global" nature (Future Generation through Global Conversation, 1997).

Hyperbolic round table: It could be imagined that some such spherical configuration is already implied in the global connectivity offered by the internet (Spherical Configuration of Interlocking Roundtables: internet enhancement of global self-organization through patterns of dialogue, 1998). . However the question remains as to the pattern of communication between participants when any such table is finite but unbounded -- as might be the case when configured around a sphere, impractical as this may be in reality. One approach is through exploring how mathematicians have sought to represent non-Euclidean connectivity on a flat plane, within a circle. This could then be recognized as effectively a hyperbolic round table. Might this enable higher orders of connectivity to be envisaged?

The pattern of discourse, as a whole, in the case of a conventional round table could indeed be represented by some form of tessellation or tiling, as can be variously envisaged (Interweaving Thematic Threads and Learning Pathways, 2010). In an ideal case the pattern might then be composed of regular polygons (triangles, squares, pentagons, etc) reflecting lines of agreement enclosing areas of consensus. In the regular tessellations of a plane, six equilateral triangles meet at each vertex, four squares meet at each vertex, and three hexagons meet at each vertex. Wikipedia offers an array of images contrasting Euclidean, spherical and hyperbolic tilings (Uniform tilings in hyperbolic plane).

In the images below, the nature of communications suggested by the Poincaré pattern is associated using a superimposed radial gradient indicative of the contrast in clarity and comprehension between the centre and the periphery in that depiction of hyperbolic space. The centre is then reflective of the higher connectivity of the insights of the "angelic" condition.

Exploratory association of radial gradient with use of Poincaré disk pattern

In the following images, the colouring of the radial gradient is reversed to suggest the implication of "demonic" ignorance with respect to the subtler forms of "angelic" connectivity. The centre is then reflective of the distorted connectivity of the "demonic" condition associated with "fallen angels".

Exploratory association of radial gradient with use of Poincaré disk pattern

Communicability around a radially zoned "Dyson sphere" of comprehension: The radial gradients recalls the layers of the atmosphere: troposphere, stratosphere, mesosphere, thermosphere, exosphere. Of fundamental significance to radio communication is the ionosphere, made up of the last three layers. It is ionized by solar radiation and forms the inner edge of the magnetosphere. Its importance to global communications lies in the ability of radio waves to bounce off the ionosophere to more distant locations around the globe. Known as skywave or skip, this refers to the propagation of radio waves reflected or refracted back toward Earth from the ionosphere. The phenomenon is illustrated by many images.

The argument can then be developed by imagining the communication challenge to be "around a globe" rather than "at a table". In this sense the radial gradient is to be understood in terms of nested spheres rather than nested circles. This recalls the hypothetical Dyson sphere imagined as a shell completely encompassing a star (and capturing most or all of its power output).

With respect to the comprehension challenge, the above schematics are then indicative of a reversed condition. The stakeholders seated around the inner surface of the outermost sphere then effectively propagate waves of communication which are bounced off the outer layers of inner spheres. These layers of the "cognitive ionosphere" are spheres "within" rather than "without".

Clearly the smaller the diameter of that layer -- namely the closer to the centre -- the greater the ability to enable communication with more distant stakeholders positioned around the inner surface of the outer shell. Thus the smaller the diameter, the lesser the degree of global incomprehensibility, and therefore the greater the communicability. Just as in radio communication, there is then recognition of a skip zone, or zone of silence, namely a region where communications cannot be received. The larger the diameter, however, the more reduced the communicability enabled.

Indication of communicability "around" a radially zoned "Dyson sphere" of comprehension (communicating stakeholders indicated on outer circle)

To the extent that any inner sphere serves only as a reflecting mirror to communication, it also implies the incomprehensibility of more fundamental aspects of globality. The nesting of the spheres is also usefully indicative of increasing or decreasing orders of self-reflexivity (as discussed above with respect to cybernetic orders of feedback).

As a consequence of its remarkable recent development, the radio communication metaphor offers a powerful language for the exploration of the communicability of global comprehension amongst stakeholders. Noteworthy is the contrast between shortwave communications (as reflected off the ionosphere) and longwave communication (understood as "hugging" the surface as a groundwave). This could enable discussion of the contrasting modes of communication of comprehensibility using the extensive language of radio propagration and the associated radio frequencies and wavelengths.

One understanding of the distinctive "inner spheres" is offered by the colour-coded contrasts of spiral dynamics, notably as summarized from an integral theory perspective by Ken Wilber (Ken Wilber Summary of Spiral Dynamics model).

Beige: Archaic-Instinctual Purple: Magical-Animistic Red: Power Gods Blue: Mythic Order

Orange: Scientific Achievement Green: The Sensitive Self Yellow: Integrative Turquoise: Holistic

A visual presentation of these as nested levels or stages might then take the form of the image on the right (below) -- although the software used did not permit the nested spheres to be coloured distinctively according to such a convention.

Images suggestive of the challenge of deriving and communicating insight (generated using Stella Polyhedron Navigator)
Communicability constrained by nested spheres of comprehensibility	Array of "panelists" in quest of focus and inspiration?

Consistent with earlier discussion of the truncated icoshedron, this has been used as the approximation to the outermost sphere on which the stakeholders are held to be positioned. The advantage of the use of a spherically symmetrical polyhedron is that it is also suggestive of the territory with which each stakeholder is identified (from a flat Earth perspective) -- namely the set of polygons constituting the faces of the polyhedron.

Partial unfolding of that polyhedral net (in the image on the right) suggestively recalls the array of elements in a parabolic antenna ("dish antenna") characteristic of a satellite dish and of some radio telescopes. Clearly the processes of dialogue may well be configured to focus insight associated with thre stakeholders represented there -- as "panelists" (Meeting Focus: a Description -- Towards Transformative Conferencing and Dialogue, 1984). The unfolding configuration also recalls the arguments for the Dymaxion global map of Buckminster Fuller, projected onto an icosahedron.

The partial unfolding around the nested spheres also recalls the form of a petalled flower as a traditional metaphor for inspiration, especially in the light of the study by Keith Critchlow (The Hidden Geometry of Flowers: living rhythms form and number, 2011)

Integrating stakeholder perspectives: Both sets of the earlier images are helpful in contrasting quasi-hierarchical order (images on right) with the potentially more appropriate hyperbolic order (images on left). The latter is a particular challenge to conventional comprehension but is representative of that challenge to an interesting degree. The regularity of the hierarchical form (on the left) is that to which many initiatives aspire, whether as the unity of science or as the heavenly order of religion. It reflects the synthesis which is the aspiration of dialogue at a round table of the wise -- or deprecated as the demonic order ruled by Satan (as implied by the lower right-hand image).

A hyperbolic understanding of the global integration of a "round table" (images on the left) is especially suggestive of the nature of connectivity of communication around the table. In the radial zones closest to the periphery, the primary mode of communication could then be understood as being in binary 2-fold terms. Zones towards the centre are then progressibely more indicative of the increasing subtlety of 3-fold, 4-fold, and n-fold communication. This understanding offers the possibility of further articulation (in a subsequent discussion).

Especially useful is the sense in which the proximity of points around the circumference of the table is indicative of the greater facility of dialogue -- in binary terms, between those points adjoining each other. The facility of communication between points more distant is thus to be understood as more constrained -- that passing in larger arcs, across the table, being more especially so. The manner in which all arcs meet the circumberence at right angles is helpfully indicative of how absolutely "right" is held to be the associated perspective.

There is then the question of the global connectivity of those "seated" around the circumference of the table (as stakeholders of contrasting qualities) -- potentially framed in terms of the 72 angels or demons, or perhaps 144 (given both angelic and demonic possibilities). Is the perspective from each seat to be identified with one such quakitative insight or pattern -- or perceived to be so identified? More intriguing is the degree of integration implied by the non-binary connectivity of points within the table and by the areas thereby circumscribed. Understood as dynamc containers of complex insights, these would be indicative of subtler and more challenging insight towards the centre of the table -- implying ever greater degrees of self-reflexivity.

Paradoxical comprehension: Visualizing the challenge to comprehension can be taken further by interrelating the contrasting understandings -- specualitively -- within a pattern of interlocking Möbius strips as indicated below. This recalls the challenge suggested above by Borromean rings.

Experimental integration of 4 contrasting perspectives within 4 Möbius strips

Use was previously made of interlocking Möbius strips (in a schematic Denkmodel), to frame issues of some relevance to the above argument, as discussed separately (Psychosocial Energy from Polarization within a Cyclic Pattern of Enantiodromia, 2007)

Experimental visualization of three-stage transformation between "global" and "local"
.	Stage 0/1	Stage 2	Stage 3/0
Explication
Ambiguity Contradiction
Implication

uch schematics are suggestive of the requisite cognitive complexity for engagement with "global" integration. They recall the many efforts to represent schematically the adaptive cycle proposed by the Resilience Alliance, and promoted as necessary for strategic resilience by Thomas Homer-Dixon (The Upside of Down: Catastrophe, Creativity, and the Renewal of Civilization, 2006), as discussed separately (Adaptive Hypercycle of Sustainable Psychosocial Self-organization, 2010) .

Representation of adaptive cycle in complex systems (with many variants available via Google images)
(adapted from a variant of Thomas Homer-Dixon, The Upside of Down, 2006)	(as reproduced from Panarchy, as described by The Sustainable Scale Project)

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