Necessity of encompassing a hole -- with a dodecameral mind?

Time for Provocative Mnemonic Aids to Systemic Connectivity? (Part #11)

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Significance of a "hole"? Of potential significance is the manner in which some form of hole may be depicted at the centre of the various round tables -- a hole which continues to feature in many summit seating configurations. This relates to consideration of a higher value being associated with the centre of a dodecahedron, of the drilled truncated cube, and of the dodecagonal configuration.

The mysterious nature of a hole is explored by Roberto Casati and Achille C. Varzi (Holes and Other Superficialities. 1994) and can be provocatively extended to current strategic challenges (Is the World View of a Holy Father Necessarily Full of Holes? Mysterious theological black holes engendering global crises, 2014).

The nature of a hole can also be explored in terms of "something missing" and "necessary incompleteness", as discussed with respect to the work of biological anthropologist Terrence Deacon (Incomplete Nature: how mind emerged from matter, 2012; The importance of what is missing, New Scientist, 26 November 2011). The argument can be explored with respect to the Szilassi polyhedron and its unfolding (Reframing nothing as a vital focus for sustainability, 2014).

The matter is of relevance to the question of any 13th presence at a table (such as at the Last Supper), but is usefully exemplified by the role of the truncated tetrahedron at the centre of the 12-fold configuration of Archimedean polyhedra. The tetrahedron may be similarly understood as at the centre of a 4-fold configuration of Platonic polyhedra. The map above indicating distinctive relationships pathways between spherically symmetrical polyhedra usefully highlights their distinctive positions -- and the manner in which the terahedron and the truncated tetrahedron are "transcendent".

Given the historical point of departure to this argument with the mysterious Roman dodecahedron (depicted above), there is an aesthetic irony to the possibility that the implication of configuration of any cognitive mapping device around a hole merits greater attention -- as an integrative necessity for 12-fold variety. The argument is potentially of greater relevance in the case of the significance attached to the Chinese puzzle balls -- especially in the depth of focus required to carve them. There is of the curious correspondence in English between "hole" and "whole", potentially to be understood in terms of "wholth" (Wholth as Sustaining Dynamic of Health and Wealth: cognitive dynamics sustaining the meta-pattern that connects, 2013).

Drilled truncated dodecahedron: On that basis, of potential interest is the manner in which each of the 12 positions can be understood as centred on a hole through the centre of a drilled truncated dodecahedron -- which appropriately reflects the structure of the Roman dodecahedron with its 12-holes of unknown functionality.

Animation of drilled truncated dodecahedron Faces=260 (5 types); Edges=420 (8 types); Vertices=140 (3 types)
All faces non-transparent	Outer faces transparent
Animations generated using Stella Polyhedron Navigator

Again it should be stressed that this is an argument for any pattern of 12 distinctions, with the Zodiac serving primarily as a familiar example. Deprecated or not by some, it should also be emphasized that few if any 12-fold patterns currently articualated endeavour to be more than a checklist; they do not make any effort to articulate systemic relations among the 12.

Dynamics around a hole? Given the argument above with respect to achieving cognitive liberation from the static structure of the cube -- and "entrapment" therein -- it might then be asked whether the complex structure depicted above is simply "a better mousetrap". As a challenge of suggestive (if not provocative) design, several approaches to rendering such structures dynamic can be envisaged.

• the application through which the drilled truncated dodecahedron was generated (Stella Polyhedron Navigator) offers a variety of facilities for transforming such a polyhedron through morphing, and for embedding that morphing in a cycle between various extremes
  
  
• rather than using a polyhedral framework, emphasis could instead be placed on the dynamics of movements of 12 (or 24) distinctive entities. This approach has been explored using a three-dimensional variant of the pattern of the Basque lauburu -- in the light of the geometry of its construction, which is notably distinct from polyhedra in that it is based on curves rather than straight edges. This enables a switch in metaphor from visualization to the disparate "voices" so characteristic of discourse (Improvisation in Multivocal Poetic Discourse: Basque lauburu and bertsolaritza as catalysts of global significance, 2016). Experimental animations are presented separately (24-fold Pattern Implied by Dynamics of the Lauburu in 3D Visualization of the interplay of sets of voices in discourse, 2016).
  
  
• the animations of the cuboctahedron above with its 12-vertices (with which the individual Archimedean poyhedra can be associated as shown), suggests that these might be represented in an animation in which they emerge from the holes of the drilled truncated dodecahedron, and sink back into its centre. There is an instructive challenge to this possibility in that the configuration of vertices of the cuboctahedron does not match that of the holes. This is interesting in that the cuboctahedron is especially significant in enabling a cycle through the Platonic forms, most notably the icosahedron of 12 vertices -- widely illustrated by "jitterbug" videos from Buckminster Fuller. As the dual of the dodecahedron, the vertices of the icosahedron necessarily match those of the drilled truncated dodecahedron. Associating the 12 Archimedean polyhedra with the vertices then allows them to be used in a dynamic of "emergence and reabsorption" from the latter (as suggested by the animation below).
  

Screen shots of animation of 12 spheres in icosahedral configuration emerging from the drilled truncated dodecahedron
Video mp4; Interactive variants of animation wrl, x3d (can be switched to wireframe rendering) Animations generated using Stella Polyhedron Navigator

Beyond the use of colours to distinguish the pattern of 12, these may be appropriately replaced by the actual Archimedean polyhedra (not to scale), as shown below.

Screen shots of animation of 12 Archimedean polyhedra in icosahedral configuration emerging from the drilled truncated dodecahedron
Video mp4; Interactive variants of animation wrl, x3d (can be switched to wireframe rendering) Animations generated using Stella Polyhedron Navigator

Psychosocial implications of a "dodecameral mind"? Much is made of the bicameral nature of the human mind with its two hemispheres. Much is also made of two-hemisphere division of global society: East-West, North-South, "two cultures", and the like. The integration of such hemispheres is a challenge (Engendering Viable Global Futures through Hemispheric Integration: a radical challenge to individual imagination, 2014; Corpus Callosum of the Global Brain? Locating the integrative function within the world wide web, 2014).

In Greek mythology, the Twelve Olympians deities, were also collectively known as the Dodekatheon (Ian Rutherford, Canonizing the Pantheon: the Dodekatheon in Greek Religion and its Origins, 2010). In Plato's dialogue -- Phaedrus -- a relation between these 12 gods and the 12 signs of the Zodiac is recognized (Carlos Albuquerque, The Twelve Olympians in the Zodiac, 13 May 2013; Ken Gillman, Twelve Gods and Seven Planets, Considerations, 11, 1996, 4).

To the extent that those deities, as with the signs of the Zodiac, can be understood as external projections of a potential internal psychological organization, there is therefore also a case for imagining the operation of a "dodecameral mind" (Internalizing a "dodekatheon" to inform the "dodecameral mind", 2009). This is consistent with the approach explored by Thomas Moore (The Planets Within: the astrological psychology of Marsilio Ficino, 1993), as separately discussed (Composing the Present Moment: celebrating the insights of Marsilio Ficino interpreted by Thomas Moore, 2001). It is also consistent with the argument of Joseph Campbell (The Inner Reaches of Outer Space: metaphor as myth and as religion, 1986).

There is a case for recognizing the contrasting qualitative modalities as characteristic of multiple intelligences, as articulated by Howard Gardner: Frames of Mind: the theory of multiple intelligences (1984). These can be understood as modalities or worldviews of different "orientation", readily distinguished as cognitive or cultural biases (Systems of Categories Distinguishing Cultural Biases, 1993). Gardner has since articulated some 10 "intelligences", fulfilling 8 citeria, with tentative suggestions for "additional intelligences".

The relation to astrological types is the subject of commentary (The intelligence quotient of the zodiac natives, 8 May 2015). The theory is considered controversial, especially in the light of the prevailing theory of general intelligence and the lack of empirical evidence. Unfortunately for those deprecating the lack of evidence for such variety in favour of a singular intelligence, their preference has not engendered more fruitful approaches to the variety of perspectives which give rise to the intractable differences in discourse -- now effectively tearing global society apart. An attitude typical of "heartless heads"?

The Zodiac, as with any set of deities, might be usefully explored as a case study of the controversial interplay of cultural heritage, deprecation of the past, popular familiarity, and potentially simplisitic comprehensions of intelligence (proving "unfit for purpose" in terms of engaging with the challenges of society).

The number-based formalism of patterns of polyhedra suggest a more fruitful approach to mapping the variety of 12-fold strategic perspectives in play -- and to rendering them comprehensible through animations. Rather than deprecating patterns of insights because of their content or antiquity, greater insight could be derived from exploring the patterns in their own right, as argued separately (Representation, Comprehension and Communication of Sets: the Role of Number, 1978). Such exploration is consistent with the cognitive issues raised by George Lakoff and Rafael E. Nunez (Where Mathematics Comes From: how the embodied mind brings mathematics into being, 2000).

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