Use of Concordian Mandala for a 3D ordering of value polarities

Con-quest Aesthetically Reframed via the Concordian Mandala (Part #8)

Relationship to icosahedron: The relationship between 9-fold and 5-fold configurations also lends itself to exploration in terms of the icosahedron, The latter has triangular faces but can be understood as the interlacing of 12 pentagonal patterns (each composed of 5 triangles). The icosahedron is of particular interest as the focus for the management cybernetics of Stafford Beer in articulating the group integration process trade marked as syntegration (Beyond Dispute: the invention of team syntegrity, 1994). One review of this process, using electronic protocols, is provided by Paula Boyle and Peter Lloyd (Web-Weaving, 2007). This follows from earlier experiments organized by Stafford Beer and Gordon Pask, as separately described (Metaconferencing: discovering people / viewpoint networks in conferences, 1980).

Further insight is also to be derived from Stafford Beer's recognition of the association between the enneagram and the icosahedron, which he describes as emerging from collaboration with Joseph Truss -- in a chapter on The Dynamics of Icosahedral Space (Beer, 1994, pp. 196-209):

But it is a matter of great interest that in the whole of the literature... the enneagram occurs as a plane figure. Nowhere had there been the slightest hint that a three-dimensional manifestation existed... No wonder the search took so long, given that the diagram was discovered spread across four vertical planes... The icosahedron is the actual origin of the enneagram... (p. 206)

As Beer notes with respect to detection of the enneagram "hanging" within the icosahedron:

Consider: if it can be detected when the icosahedron stands on one vertex, it must be detectable when the model stands on any vertex. Moreover, if it is present when these two poles are aligned, it must be present when any two poles are aligned. Thus it comes about that points 4 and 5 on the enneagram refer to any side [meaning edge] of the icosahedron -- which therefore enfolds 30 three-dimensional, four planar, enneagrams... The icosahedral model conceived as a spinning sphere could be regarded as "an interpenetration of phi-ness"... an interpenetration of three-dimensional four-planar enneagrams... (p. 206-207).

A representation of the enneagram embedding is discussed separately (Representation of Creative Processes through Dynamics in Three Dimensions: global insight from spherical reframing of mandalas, the zodiac and the enneagram, 2014). This includes the following

9-fold enneagram embedded within an icosahedron
with addition of an indicative central sphere (constructed by manual modification of a virtual reality model of the icosahedron generated by Stella Polyhedron Navigator software)
View of enneagram associated with only one pattern of vertices of icosahedron (view in 3D with virtual reality plugin)	View of enneagram from left image with the icosahedron framework hidden (view in 3D with virtual reality plugin)

Note the colour coding and positioning of the icosahedral vertices -- which offer guidance when rotating the above models in virtual reality in order to render visible the enneagram pattern. Green-Magenta links of the enneagram are the only links embodied within icosahedral edges (on the left), where they are invisible. Three of the 12 vertices, positioned on the vertical axis (of the image on the right), do not form part of the 9-fold enneagram pattern (Red, Cyan and Black). The various possibilities for rotating the models in three dimensions affect the proportions of the enneagram as portrayed and the relative visibility of the Cyan and Black vertices.

The earlier discussion offers additional visual renderings. Beer remarks with respect to mentation, both in individuals and groups, and the role of an information set (an " infoset"):

It is at least possible that the multiple enneagrammatic structure, reverberating as it does, provides a complex of linkages to constitute such a "corporate brain" that would then give rise to its own [infosettic] consciousness. (p. 208)

Icosahedral analogue? Clearly such an argument could well apply to the Concordian Mandala. This suggests the merit of seeking a polyhedral analogue to the icosahedron in 3D -- potentially one of greater complexity. This is consistent with the interest noted above in a 3D variant of the Kalachakra Mandala.

This 3D approach is further justified by the visual "deficiences" of the standard rendering of the Discordian Mandala (most notably the variable lie width). This suggests that the nonagons should indeed be tilted between the (proximate) surrounding pentagonal configuration and the (distant) pentagonal confiuration in the centre. The changing width of the lines of a given nonagon is consisistent with this assumption. The question is then whether the 5 nonagons could be arrayed across such an implied polyhedron within which they are effectively embedded.

In the quest for such a polyhedron, initial clues are provided by polyhedra offering a proximate (larger) pentagonal form and a distant (smaller) pentagonal form. Other clues are offered by the number of vertices or sides. Especially intriguing is the possibility that the embedding may only be achieved through two sets of 5 nonagons, namely 90 sides or vertices -- with 10 implied pentagons.

This would then suggest that separate mapping templates could be used for a "discordian" and a "concordian" ordering of value polarities. This would be consistent with one sense of traditional mandalas in which both are essentially illusory -- as conventionally understood.

Polyhedra suggestive of the possibility of embedding 5 regular nonagons (potentially as 2 complementary sets of 5)
Truncated icosahedron (90 edges of 2 types)	Drilled truncated icosahedron (90 vertices of 15 types)

Images generated by Stella Polyhedron Navigator software

Initial efforts to construct the mandala in 3D using virtual reality software were only partially successful, due primarily to the relative incompetence of the author. A degree of greater skill would enable the possibility to be explored more appropriately -- notably in relation to Borromean ring interlocking with regular nonagons. Some preliminary results are indicated by the following images.

Concordian / Discordian Mandala in 3D (stages in preliminary experiment -- "work in progress")

In considering this 3D mandala as an integrative mapping surface in relation to the 2D approach, of particular interest is the extent to which the pentagonal features (as yet not evident in the right hand image above) are emergent effects of perspective and therefore of cognition. With respect to the table above, it is these 9 sets of intangibles (indicated by pentagons) which are primarily associated with the rows of that table, whereas the more tangible systemic linkages are associated with the 5 columns of that table (indicated by nonagons).

The global interlocking of the nonagons is usefully consistent with considerations of quantum entanglement, whether with respect to the subtlety of human values or the complexity of so-called wicked problems. The challenge to comprehension necessarily calls on an aesthetic dimension, even one of mythopoeic nature, as envisaged by J. R. R. Tolkien as author of the epic tale The Lord of the Rings (1954). This has proven to be of exceptional global appeal, as discussed separately (Relevance of Mythopoeic Insights to Global Challenges, 2009).

Comprehension of the configuration of the Concordian Mandala could then be allusively compared to that tale -- described as governed by a complex "metric", with the scenario classically synthesized in poetic form:

Three Rings for the Elven-kings under the sky,
Seven for the Dwarf-lords in their halls of stone,
Nine for Mortal Men doomed to die,
One for the Dark Lord on his dark throne
In the Land of Mordor where the Shadows lie.
One Ring to rule them all, One Ring to find them,
One Ring to bring them all and in the darkness bind them
In the Land of Mordor where the Shadows lie.

As might be expected, there is indeed confusion to the configuration with which the concept of concord may be associated.

An entry in the Unionpedia: the concept map on Borromean rings and Three Rings notes that The Three Rings are fictional artifacts in Tolkien's legendarium (as indicated in the poem above). Although others have recognized this association, most remarkable is the mathematical work extending Borromean and Brunnian rings to new higher order versions, as explicated with numerous images (citing Tolkien) by Nils A. Baas (New States of Matter Suggested by New Topological Structures, International Journal of General Systems, 2012). This notes that in recent years some strange and counterintuitive states of matter have been observed in cold gases and nuclear systems.

The question to be asked is whether "strange and counterintuitive" insights are now urgently required with respect to viable configurations of human values variously conducive to concord and discord. Does Tolkien's intuited set of 20-plus rings lend itself to understanding otherwise, especially given the possible configuration of a "9-ringed" mandala -- for Mortal Men doomed to die?

Given the appeal of his speculative spirit, is discord to be understood as a strange form of "dis-ease" calling for more appropriate framing by metaphor (Memetic and Information Diseases in a Knowledge Society, 2008)? Is his set of rings somehow consonant with a yet-to-be-recognized set of metabolic pathways, mysteriously configured such as to sustain life (Memetic Analogue to the 20 Amino Acids as vital to Psychosocial Life? 2015). How might a Concordian/Discordian mandala be consistent with:

One Ring to rule them all, One Ring to find them,
One Ring to bring them all and in the darkness bind them