Requisite helical cognitive engagement within a global brain?

Envisaging a Comprehensible Global Brain as a Playful Organ (Part #7)

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Spiral periodicity of musical scales: The questionable effort to reconcile the circular configuration of 64 hexagrams with a circular pattern of 8 octaves (as in the animation above) highlighted the possibility that a more appropriate design could be achieved through a spiral form. The "overlapping" could then be a feature of helical recurrence at a distinctive pitch level -- a pitch helix, namely a visual representation variously explored in music theory (Diana Deutsch, The Paradox of Pitch Circularity, Acoustics Today 6, 2010, 3; Diana Deutsch, et al, Pitch circularity from tones comprising full harmonic series, The Journal of the Acoustical Society of America, 124, 2008, 1; Roy D. Patterson, Spiral Detection of Periodicity and the Spiral Form of Musical Scales, Psychology of Music, 14, 1986, 1).

In music theory, the spiral array model is an extended type of pitch space. This is a mathematical model involving concentric helices (namely an "array of spirals"). It is understood as representing human perceptions of pitches, chords and keys in the same geometric space as first proposed by Elaine Chew (Towards a Mathematical Model of Tonality, MIT, 2000; Mathematical and Computational Modeling of Tonality: theory and applications. International Series in Operations Research & Management Science. Springer, 2014)

Spiral mapping: The choice of a 2D spiral mapping metaphor for 64 hexagrams (below centre) featured in an earlier argument (Adaptive Hypercycle of Sustainable Psychosocial Self-organization: designing a mapping of a Chinese metaphorical pattern language, 2010).

The emphasis in this argument is on the ability of the human mind to engender and recognize patterns, as discussed in the light of the recent study by Jeremy Lent (The Patterning Instinct: a cultural history of man's search for meaning, 2017), as critically reviewed separately (Patterning Intuition with the Fifth Discipline, 2019). It is in this sense that quite disparate instances of patterning substance may be compared as arising from that proclivity, most notably as argued by George Lakoff and Rafael E. Núñez (Where Mathematics Comes From: how the embodied mind brings mathematics into being, 2000).

A fundamental exercise in patterning, in contrast with the tabular convention is the helical periodic table of chemical elements (G. Schaltenbrand, Darstellung des periodischen Systems der Elemente durch eine rÃäumliche Spirale, Z. anorg. allgem. Chem., 112, 1920). An extract is presented below right. It is important to note that there have been numerous attempts to organize in a comprehensible manner the chemical elements (Internet Database of Periodic Tables, Chemogenesis), with the struggle continuing (D. H. Rouvray and R. Bruce King (Eds). The Mathematics of the Periodic Table, 2005).

Comparable recognition of helical organization
Helical organization of pitch space	Spiral organization of hexagrams	Helical periodic table
Elaine Chew [CC BY-SA 4.0], via Wikimedia Commons		G. Schaltenbrand, 1920

Given the role of the torus variously highlighted above, possibilities of comprehensible representation of helical organization have been discussed separately in relation to its coiling (Visualization in 3D of Dynamics of Toroidal Helical Coils -- in quest of optimum designs for a Concordian Mandala, 2016). Also of relevance is the helical mapping of hexagrams onto a globe by József Drasny (The Image of the Cosmos in the I Ching: the Yi-globe, 2007) as discussed separately (Enhancing coherence through spherical triangulation, 2011).

Spiral dynamics:  In experiential terms, the spiral form can be explored through the framework of spiral dynamics (Don Edward Beck and Christopher Cowan, Spiral Dynamics: mastering values, leadership and change, 1996; Don Edward Beck, et al, Spiral Dynamics in Action: humanity's master code, 2018). This is a structured evolutionary model of adaptive intelligence which has spawned much discussion and (sometimes tangential) integration of concepts by other theorists, such as the AQAL integral theory of Ken Wilber.

Unfortunately most of the images clarifying the spiral stages (tiers) of development are subject to copyright. These are distinguished by colour: Beige: individual survival; Purple: tribal; Red: power and dominance; Blue: follows moral precepts of the group; Orange: autonomy and achievement; Green: egalitarianism and community; Yellow: integration and personal responsibility; Turquoise: collective individualism (Scott Jeffrey, How to Use Spiral Dynamics for Psychological and Leadership Development).

Symbolic traces: laurel wreath and Caduceus: It is intriguing to note the traditional and continuing importance attached to the laurel wreath as a symbol of victory and completion -- as well as of death. Curiously the many 2D images of the laurel wreath are intimately associated with the logos of globally oriented institutions, notably the United Nations and its agencies -- and as such their reproduction is severely constrained by intellectual copyright and paywalls. Of further interest is the seeming absence of variants in 3D -- meriting interpretation in the light of enabling comprehension of globality by such bodies. The animation of the Caduceus is an adaptation of a set of screen shots of a 3D model.

It was for this reason that a 3D variant was produced in order to enable distinctive manipulation of the two opposing fronds. Somewhat ironically, the leaves were adapted from an exercise in modelling the Tao (yin-yang) symbol in 3D, notably with the aid of Sergey Bederov, Senior Developer at Cortona3D, as described separately (Stages in imagining a 3D Tao symbol in virtual reality, 2019. The droplet-like shapes composig the symbol have been distorted in the 3D images below.

Helical insights from 3D representation of Laurel wreath and Caduceus
Laurel wreath: static symbol of binary completion or sacred flame	Laurel wreath spiral upright rotation	Laurel wreath spiral inverted rotation	Rotation of Caduceus in 3D

Dragon dance, bullfighting and storyboarding: Of particular interest is the sense in which the conventional static laurel-wreath can be understood as implying a dynamic only evident in 3D as shown in the central animations below. Both are reminiscent of the double dragon dance in quest of a pearl -- so central to Chinese culture and widely depicted in its iconography. The double spiral symbolism is also evident in the Caduceus of Western culture, now a symbol of both healing and commerce. Traditional depictions of the quest for the pearl are reminiscent of the strangely elusive role associated with "global" and even more specifically with the function of a "global brain". Framed as shown by a helical pattern, the challenge to its comprehension is all the greater if avoidance of premature closure is to be recognized as fundamental (Engaging with Elusive Connectivity and Coherence: global comprehension as a mistaken quest for closure, 2018).

With respect to the dance between the two branches of the laurel wreath, this was explored visually in a separate storyboarding exercise regarding the dance between matador and bull. The form of the wreath may be related metaphorically to the tradition of engagement with bull horns, whether in worship in sport or in bullfighting (Transformation of Global Governance through Bullfighting: visual symbols and geometric metaphors, 2009; Game-playing, bull-leaping and laurel wreaths, 2014). Such horns are a useful reminder of the dangerous horns of the dilemmas so characteristic of governance.

Whether the drama is focused on the quest for a pearl or mastery of the bull, the global brain narrative, with its trend toward singularity-collapse, can be understood as fundamentally a dramatic pretence (as suggested above). Is the high drama comparable with the mutual entanglement of the classic tales (Entangled Tales of Memetic Disaster: mutual implication of the Emperor and the Little Boy, 2009; "Big Brother" Crying "Wolf"? But them "wolves" are a-changin' -- them's becomin' "werewolves"! 2013)? Is comprehension of the "pearl" then conflated with comprehension of "singularity" -- confused by misleading warnings and vainglorious posturing, as highlighted by those tales?

Elusive incommensurability of opposites: The conventional laurel wreath is particularly valuable in implying both the possibility of circular completion whilst indicating a form of junction at the base and a mysterious gap at the top -- where the two branches do not connect. The presentation in 3D of the branches moving in relation to each other such as to suggest a spiral form is indication that any connection between opposites is a fundamental challenge to comprehension -- a paradox exemplified by their mirroring of each other.

The challenge is clarified otherwise in the image of the caduceus on the right. Again the nature of the connectivity in cognitive terms is elusive, if not a matter of fundamental risk, as implied by the snake-like forms. The gap merits consideration as a form of cognitive abyss.

One approach to such connectivity is through "correspondences", beyond the simpler assumptions of correlative thinking (A. C. Graham, Yin-Yang and the Nature of Correlative Thinking, The Institute of East Asian Philosophies, 1986). Perhaps appropriately, the validity of correspondences as a bridge is itself a matter of controversy (Theories of Correspondences -- and potential equivalences between them in correlative thinking, 2007).

Challenge of representation and comprehension: The Shao Yung circle of hexagrams can be presented in 3D (below centre). The rotation highlights a potential problem of representation using current X3D norms in that the hexagrams may be "reversed" when particular techniques are used. Thus one perspective on the circle, during the rotation, corresponds correctly to the original; the other is incorrect. This is inadvertently indicative of a potential problem of comprehension. The misrepresentation can be corrected with other techniques. The issue is of relevance in that the left hand branch in the original is characterized by 4 sets of hexagram "houses" with an unbroken line at the bottom; the 32 hexagrams in the 4 "houses" in the right hand branch have a broken line at the bottom. Marking the start of each "house" by rendering the hexagram in red helps to clarify the issue.

The opposing "branches" of the circle can be angled with respect to each other in 3D, as was done with those of the laurel wreath above. Two variants are presented below which serve to indicate other potential challenges to representation and comprehension -- despite the technical problem of reversal in the central image having been rectified. These arise from how the opposing branches are angled with respect to each other. The variant on the left corresponds more closely to the original; that on the right has one branch reversed.

Use of a central sphere in the angled variants serves to recall the pursuit of the pearl in the traditional dragon dance. The issues of representation serve to highlight the challenges of mutual comprehension between opposites -- most notably between right and left.

Rotation in 3D of angled hexagram branches illustrating challenge of "reading"
Angled hexagram "branches"	Circle of hexagrams in 3D	Angled hexagram "branches"

The possibility of confusion in the comprehension between opposites is further illustrated in 3D by the presentation in rings of the rows of the Shao Yung central square arrangement (below left). The order needs to be variously "twisted" and "inverted" to achieve correspondence between the upper and lower trigrams in the stack of rings. This is a reflection of the mirror imaging embodied in the sequa re arrangement.

Reconciliation of 3D organization of hexagrams in rings (by row) with Shao Yung square (by row)
Shao Yung square arrangement	Rings by row (no twist)	Rings by row (with inversion of upper set)	Rings by row (with inversion and twist)

Challenge of 7-fold versus 8-fold organization: octaves versus hexagrams: As noted above with respect to the grouping of 7 major scale keys into an "octave", the 8-fold organization is achieved because one of the keys is comprehended as common to a second octave. This "confusion" is resolved by recognizing that the pattern is helical as shown in the central image below -- in contrast with that on the left. The question then arises as to whether the challenging "twist" in the organization of the matrix of hexagrams (above left) is clarified by organization in a helical pattern.

One experiment with such an organization is presented below right in which the upper portion of the matrix is mapped onto the red helix and the lower onto the green. The convention adopted is to position the hexagram with unbroken lines at the lowest position of the green helix, with the hexagram of broken lines in the highest position of the red helix -- offering a degree of correspondence with the matrix. In this variant it is either the lower or the upper trigram which is common between the distinctive helix turns -- whether vertically or horizontally. Other variants can be explored.

Comparison of 3D helical organization of musical pitch with a helical pattern of hexagrams: the challenge of 7 versus 8
Ring configuration of pitch succession	Helical organization of musical pitch	Double helical organization of hexagrams
The animations have been constructed using X3D Edit, allowing parameters to be variously modified using a text editor. The models here (and above) have many technical defects which could be readily circumvented with greater skill.

Chirality (handedness); The example of double helical organization of hexagrams (above right) is arguably an instance of a pattern exhibiting a relatiely high degree of coherent complexity coducive to comprehension ("CCCC"?) -- and as such potentially indicative of the organization of the global brain. As a resonance hybrid, this might however be more appropriately associated with alternation between alternative helical patterns.

Examination of the variant shown highlights the issue of chirality -- namely the direction of rotation of the helix, with the two variants being mirror images of each other. This is a primary characteristic of DNA (Emily Singer, New Twist Found in the Story of Life's Start, Quanta, 26 November 2014; Youri Timsit, DNA Self-Assembly: From Chirality to Evolution, International Journal of Molecular Sciences, 14, 2013, 4 Vsevolod A. Tverdislov. Chirality as an Instrument of Stratification of Hierarchical Systems in Animate and Inanimate Nature, 2012).

With the origin of life held to be associate with preference for one chiral form, the question might be asked whether the emergent organization of the global brain is to be similarly understood or whether it takes the form of a resonance hybrid. The insights of biology into the fundamental role of DNA supercoilding are potentially relevant to any development of the psychosocial argument with respect to the enoding offered by hexagrams (Ruggero Cortini, Chiral theory of DNA supercoiling, Imperial College, 2013; Paolo Bettotti, et al, Structure and Properties of DNA Molecules Over The Full Range of Biologically Relevant Supercoiling States, Scientific Reports, 8, 2018, 6163).

In response to the current tragedy of polarization in society and their challenges to governance, such indications suggest that what is required is not so much an approach to "conflict resolution" -- potentially oversimplistic under the circumstances -- but rather a focus on chirality and its "resolution". These may only be comprehensible through topological forms of higher dimensionality, as suggested by fundamental physics. In that respect, issues relating to compactification of "extra dimensions" are then a primary challenge to their comprehension.

Double helix, triple helix and juggling patterns: Through the biological role of DNA, the fundamental importance of helical organization has evoked considerable attention. Relatively little attention has been given to its psychosocial implications, if only in metaphorical terms (Climbing Elven Stairways: DNA as a macroscopic metaphor of polarized psychodynamics, 2007; Walking Elven Pathways: enactivating the pattern that connects, 2006). Reference to the triple-stranded variant of DNA is less widely made.

Attention has however recently been focused on the Triple Helix Model of Innovation which refers to a set of interactions between academia, industry and governments, to foster economic and social development. This focus may imply that it is a specific instance of forms of psychosocial organization which merit attention -- gi en the importance various attached to triadic organization and symbolism, as discussed separately (Contrasting the implications of "triple helix" -- cognitive and otherwise, 2017; Psychosocial Learnings from the Spiral Form of Hurricanes: implications of the triple helix and the 3-fold triskelion as "cognitive cyclones"? 2017).

The Triple Helix model have evoked attention to more complex possibilities. Especially noteworthy is consideration of a Quadruple Helix approach by the a Committee of the Regions of the European Union (Using the Quadruple Helix Approach to Accelerate the Transfer of Research and Innovation Results to Regional Growth, 2016). A Quadruple and quintuple innovation helix (Q2IH) framework has been developed. Of potential relevance is the relationship to more conventional devices for articulating strategic preoccupations, as discussed separately (Symbolic stars vs Strategic pillars; Polyhedra vs Helices; Logic vs Comprehension?, 2017). This frames the question of possible Biomimetic embedding of N-tuple helices in spherical polyhedra (2017).

Curiously, but appropriately (given the challenges of governance), a triple helix is characteristic of the "braiding" of one juggling pattern (as illustrated below) as mathematically understood (Governance as "juggling" -- Juggling as "governance": dynamics of braiding incommensurable insights for sustainable governance, 2018).

Comparison of helical molecular structure with patterns of innovation and juggling
Variant of the double helix	Triple-stranded DNA	Triple Helix	3-ball juggling pattern as a braid
Reproduced from Wikipedia	DubOOIan [CC BY-SA 4.0]	University of Melbourne	Monash University, Melbourne

Supercoiling of higher degree as characteristic of global brain organization? It is intruiging that organic life appears to resolve the challenges of chirality through processes associated with supercoiling of DNA. As noted by Wikipedia, supercoiling is important in a number of biological processes, such as compacting DNA, and by regulating access to the genetic code, DNA supercoiling strongly affects DNA metabolism and possibly gene expression Additionally, certain enzymes are able to change DNA topology to facilitate functions such as DNA replication or transcription (Brad A. Krajina, et al, Large-Scale Conformational Transitions in Supercoiled DNA Revealed by Coarse-Grained Simulation, Biophysical Journal, 111, 2016, 7; Twists and Turns of Life: patterns of DNA supercoiling, ScienceDaily, 4 April 2016).

As noted by the latter:

The classic double helix structure that one associates with DNA is but an extremely limited view of its physical 'shape'. The molecule that holds the codes of life is capable of further winding itself into myriad complex shapes called 'supercoils' that are capable of affecting gene expression patterns.... DNA molecules are wound and rewound into complex structures that condense their immense lengths to a fraction of their actual size in order to fit their long strings of information into microscopic cells. But this 'packed' DNA that fits neatly into a cell also needs to be 'unpacked' periodically for gene expression and replication.

This frames the question as to whether analogues of some kind may prove vital to comprehension of the problematic challenges of polarization in psychosocial processes (DNA Supercoiling as a Pattern for Understanding Psycho-social Twistedness, 2004). More intriguing is why so little attention is given to that possibility -- given the crisis of polarization in society. In this respect, given current preoccupation with the Triple Helix model of innovation, a suitable provocation is offered by the insight from biomolecular dynamics: The triple helix does not represent the most thermodynamically stable structure that can be adopted by two complementary...strands. (DNA Supercoiling Is Required for Intramolecular Triplex Structures, Science Direct).

Supercoiling of DNA	Continuous 9-winding helical spiral	Double helical spiral	5-coil pattern
User:Notahelix, User:JoKalliauer [CC BY-SA 3.0], via Wikimedia Commons	Reproduced from Towards a 3D visualization of toroidal counter-coiling dynamicss, 2016

It is presumably within this topological context that the mysteries of psychosocial mutation and morphogenesis are to be clarified -- notably in the light of te insights of René Thom (Structural Stability and Morphogenesis: an outline of a general theory of models, 1972). The possibility is an invitation to further specualtion (Reframing the Dynamics of Engaging with Otherness: triadic correspondences between topology, Kama Sutra and I Ching, 2011). Is the Shao Yung circle of hexagrams itself to be understood as a supercoil?

Global brain comprehension from an "axial" perspective? Little reference is made to the axial perspective of helical organization, as notably illustrated in the case of DNA (below left; and see especially Reginald Brooks, GoDNA: The Geometry of DNA. 2001). It could be argued that a significant number of centrosymmetric logo-style devices, by which organizations and strategic initiatives are identified, could be understood as "axial" perspectives on complex helical structures that are less readily presented or comprehended (World Guide to Logotypes, Emblems and Trademarks of International Organizations). Could the UN's 8-fold Millennium Development Goals and the 16 (+1)-fold Sustainable Development Goals be more appropriately addressed through recogition in terms of their 8-fold and 16-fold helical entanglement respectively?

As yet to be explored is whether and how appropriate helical arrangement of hexagrams offers an axial view of the distiguishing trigrams in the final turn of the helix in a manner corresponding to the alternative BaGua arrangements shown below centre.

Comparable representations of axial views implying helical entanglement ?
Axial view of DNA	Logo of Scientific and Medical Network	Alternative BaGua trigram arrangements		Sustainable Development Goals	Spiral keyboard
		Fuxi - Earlier Heaven	King Wen -- Later Heaven
		Reproduced from Wikipedia		United Nations	Spiral keyboard, Wifflegif

Given their valued cognitive role, many mandalas and yantras might well be understood from that perspective -- potentially as "standing waves" -- each possibly to be understood in terms of a "brain scan".

Animations of standing wave as vibrations of a circular membrane
Standing wave in two dimensions.	Higher harmonic standing wave on a disk
Oleg Alexandrov [Public domain], via Wikimedia Commons;

Spiral and helical staircases: Whilst many of the helical patterns indicated above can be considered abstractions, it is remarkable that experience of helical movement in a spiral staircase is widely recognized and appreciated. This common experience is especially relevant to this argument in that it is consistent with the above-mentioned arguments regarding the embodiment of knowledge through movement. It is potentially for this reason that such staircases have long been a feature of buildings of symbolic significance, as discussed by George L. Hersey (The Monumental Impulse: architecture's biological roots, 2001). Wikipedia offers a List of ancient spiral stairs. He notes the degree of interest of Leonardo da Vinci in this pattern and its relationship to the helical structure of many snail shells. Leonardo was also the architect of a much-famed staircase (Da Vinci Designed a Double Helix Staircase at the Château de Chambord, Ancient Origins, 2018).

Both double spiral and double helix staircases are possible, with two independent helical stairs in the same vertical space, allowing one person to ascend and another to descend, without ever meeting if they choose different helices. Fire escapes, though built with landings and straight runs of stairs, are often functionally double helices, with two separate stairs intertwined and occupying the same floor space.

In contrast with the abstractions above, it is also remarkable that a number of websites enable anyone to design a spiral staircase (but not a double helical variant), with an extensive choice of parameters and possibility of visualization of the result:

• Spiral Stair Geometry CalculatorÂ
• 3D Spiral staircase calculator
  
• Calculating the size of a spiral staircase
• Define the dimensions of your spiral staircase

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