Dynamic bonding patterns in n-tuple helices engendering n-fold rotating symbols

Framing Cyclic Revolutionary Emergence of Opposing Symbols of Identity (Part #3)

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The question here is how a symbol emerging from an n-tuple helical pattern might variously interrelate the separate helical pathways as it rotates (and as it might descend into the central cognitive wormhole). The examples below are purely suggestive with each vertex to be understood as in contact with a helical strand. Each line in the symbol may alternative between various conditions -- as alternate types of bond. The perspective on the symbols presented is looking "down" or "up" the axis of the n-tuple helix, as indicated in the screen shots above.

Understood generally, various approaches to such bonding patterns might be recognized:

• Geometry of stars: As noted below, in a number of cases a valuable approach has been initiated under a pseudonym within the GeoGebra environment. That 2016 study offers an interactive presentation of 5-, 6-, 8-, 9-, 10- and 12-pointed stars.
• Magic stars: There is a long-standing interest in magic stars, most notably in recreational mathematics in relation to magic squares and magic polygons (Mutsumi Suzuki, Magic Stars; Cleve Moler, Magic Stars, 2014). The most comprehensive approach is that of Harvey Heinz (Magic Stars, 2009).
• Systematics: One articulation, with a significant cognitive dimension, is offered by John Bennett through the study of multi-term systems, termed systematics (General Systematics, Systematics, 1, 1963, 1), culminating in a magnum opus (The Dramatic Universe: search for a unified vision of reality, 1956).
• Sacred geometry: A valuable illustrated summary of the set of polygons from the perspective of sacred geometry is provided by Tomo Perisa (Relations of descriptive and inscribed circles of basic polygons, Sveta Geometrija), including the 11-sided and 13-sided polygon: hendecagon and tridecagon. The perspective is that which tends to inform the elaboration and attraction of symbols, whether sacred to religion, to an ideology or to heraldic identity -- variously imbuing them with significance.
• Software applications: The name of the Google Ngram Viewer, , and possibly its ultimate intent, suggests that it might develop into a device which could detect sets of relevance to this concern with bonding patterns.. As currently understood an n-gram is a contiguous sequence of n items from a given sequence of text or speech. Currently the viewer charts frequencies of any set of comma-delimited search strings using a yearly count of n-grams found in sources printed from 1500 to th present.
• Freemasonry: The symbolism of this influential world view includes an unusual form, termed The Camp, composed of five co-centric forms: a nonagon, a heptagon, a pentagon and triangle, enclosing a circle. An animation of the counter-rotation of such a nested set of polygons is presented in connection with a discussion of Ninefold configuration in practice and its comprehension constraints (2016).
• Traditional symbols: A systematic approach to the traditional symbolic significance of a range of such symmetrical patterns is provided by the Hermetic Order of the Golden Dawn (Polygrams and Polygons). Despite the historical controversy associated with that order, it is appropriate to note the importance attached to it by the poet W. B Yeats, prior to his resignaion -- especially given his seminal poem The Second Coming (1920), whose inspiration for the "gyre" therein is clarified by Neil Mann (Geometry).

The Second Coming by W. B. Yeats (first verse)

Turning and turning in the widening gyre The falcon cannot hear the falconer; Things fall apart; the centre cannot hold; Mere anarchy is loosed upon the world, The blood-dimmed tide is loosed, and everywhere The ceremony of innocence is drowned; The best lack all conviction, while the worst Are full of passionate intensity.

Triangular bonding? The following animation is suggestive of the pattern of bonding in the case of a triple helix, notably with respect to its initial application to academia (universities), business and government. Of related interest is its relevance to the three pillars of sustainability -- economic, social, environmental -- as most recently recognized by the 2005 World Summit on Social Development. This contrasts with the three pillars of the European Union, defined in instititutional terms. The Three Pillars of the United Nations are human rights, peace and security, and development.

Animation of 3-fold bonding pattern

Any such configuration of three strategic pillars offers a variety of interpretations of the dynamic between the functions they represent -- notably in the contrast between a "broken pillar" and a "solid pillar", where the former may suggest relative weakness, or possibly an implicit, passive or fallow condition. A solid pillar might reflect a disproportionate emphasis. The animation offers eight configurations consistent with the Chinese BaGua pattern of trigams composed of solid and broken lines. The triangular configuration (or a sixfold elaboration) may be speculatively recognized as a kind of strange attractor, as suggested above (El-Attractor -- Timeless Complex Dynamic Health, Wealth, Stealth / Youth, Couth, Truth, 2007).

Quadrangular bonding? Various possibilities merit consideration in relation to the pattern of bonding in the case of a quadruple helix. Especially intriguing is the challenge of any relationship to the Christian cross and to the extremely charged symbol of the swastika in its different orientations. The animations on the right were developed in a separate argument which notably referred to the pattern of Knight's moves in chess across a matrix of 3x3 cells (Swastika as Dynamic Pattern Underlying Psychosocial Power Processes: immplicate order of Knight's move game-playing sustaining creativity, exploitation and impunity, 2012).

Animations of 4-fold bonding pattern
Quadrangular	Left-facing Swastika	Right-facing Swastika

Inspired by the possibility of representing the basic pattern of 8 distinctions of the I Ching trigrams in the triangle above, the animation on the left is a tentative exploration of an approach to configuring the tetragrams of the associated Chinese classic, the Taixuanjing (Canon of Supreme Mystery), as discussed separately (9-fold Magic Square Pattern of Tao Te Ching Insights -- experimentally associated with the 81 insights of the T'ai Hsüan Ching, 2006). Although unsuccessful it focuses the question of how to develop complex patterns of notation, as discussed further below. Of further potential relevance is a consideration of 4-dimensionality (Four-dimensional requisite for a time-bound global civilization? 2015).

As indicated above, strategic concerns may be articulated in terms of four pillars. For example the United Nations identifies four foundational pillars, four pillars of a global counter-terrorism, as well as four pillars of transitional justice (truth, justice, reparation, guarantees of non-recurrence), with UNESCO identitying four pillars of learning, and the World Bank identifying the four pillars of the knowledge economy, with four pillars of human rights distinguished by UNDHR (personal rights, relationships with a social group, civil liberties and political rights, and economic rights). The World Economic Forum identifies four pillars of economic corruption. Proposals have been made to extend to four the three pillars of sustainability (discussed above), namely with the inclusion of culture. OECD distinguishes four central pillars of inclusive growth.

Pentangular bonding? In arguing for recognition of a 5-fold dynamic, it is somewhat ironic to note the correspondence between fundamental traditional understandings of East and West, as exemplified by the pentagrams of Hygeia and Wu Xing discussed separately (Five-fold cognitive dynamics of relevance to governance? 2015). The following images were reproduced from an earlier discussion (Cycles of enstoning forming mnemonic pentagrams: Hygiea and Wu Xing, 2012). The animation is suggestive of the pattern of bonding in the case of any quintuple helix. The image on the right derives from System Dynamics, Hypercycles and Psychosocial Self-organization: exploration of Chinese correlative understanding (2010).

Pentangular bonding patterns
Hygeia (Hugieia) Pentagram of Pythagoreans	Animation of 5-fold bonding pattern	Chinese 5-phase Wu Xing cycle	Hypercycle representation
Reproduced from Hygiea entry in Wikipedia (G. J. Allman Greek Geometry From Thales to Euclid, 1889, p.26) with labels added		Adapted from Wu Xing entry in Wikipedia Interaction arrows: black=generating; white= overcoming	Reproduced from entry in Principia Cybernetica

Of relevance to bonding explorations, a more rigorous interactive approach to the 5 pointed star is offered under a pseudonym in the GeoGebra environment. A detailed mathematical description is provided by Harvey Heinz (Order-5 Magic Stars, 2003).

As indicated above, concens with strategy and principle may be articulated in terms of five pillars, as with the Five Pillars of Islam or the Five Pillars of US cyber security. The fundamental principles of Wikipedia are summarized in terms of "five pillars". OECD distinguishes five pillars of effective development action. UNFPA distinguishes five pillars of population and development. Of relevance is the widely appreciated distinction by Peter Senge (The Fifth Discipline: the art and practice of the learning organization, 1990). The strategic significance attributed to The Book of Five Rings is noted below.

Hexangular bonding? The following animation, as presented in the previous paper, is suggestive of the pattern of bonding in the case of a six-fold helix.

Animation of 6-fold bonding pattern

Of relevance to bonding explorations, a more rigorous interactive approach to the 6 pointed star is offered under a pseudonym in the GeoGebra environment. A detailed mathematical description is provided by Harvey Heinz (Order-6 Magic Stars, 2003). Understood in terms of 6 pillars, examples include: The Six Pillars of Self-Esteem (1995) articulatd by Nathaniel Branden; The Six Pillars of Character; Six Pillars of Knowledge Economics; Six Pillars of Peace. As argued by David Grebow, missing from the above-mentioned "fifth discipline" articulation is a learning technology required to teach and continuously support the other five disciplines (The Sixth Discipline, Brandon Hall Group, 2013).

Heptangular bonding? Importance is conventionally attached to the effective size of decision-making groups, notably in relation to information processing capacity (George A. Miller, The Magical Number Seven, Plus or Minus Two: some limits on our capacity for processing informationt and size, Psychological Review, 63, 1956). Patterns of bonding in such contexts therefore merit consideration. As noted above, this raises distinctive problems in the case of the great circle approach advocated here and the related absence of symmetrically regular spherical polyhedra. Of interest is the recent elaboration of a "ritual of the heptagram" in the neopagan tradition -- specifically noting forms of relationship within the pattern which can be undestood as distinctive forms pf bonding.

Images of 7-pointed star, implying a 7-fold bonding pattern
Reproduced from Wikipedia		Copyright Shutterstock

A detailed mathematical description is provided by Harvey Heinz (Order-7 Magic Stars, 2003). It is potentially appropriate to note that the wrapping of DNA around the nucleosome core results in 7-fold compaction of DNA (N. Ramaswamy, et al, Structure of D-DNA: 8-fold or 7-fold helix? The EMBO Journal, 1983). With respect to recognition of any set of "seven pillars", many examples exist. Most memorable is perhaps the Seven Pillars of Wisdom (1926), as articulated by T. E. Lawrence, but featuring in Proverbs 9:1 (Henry M. Morris. The Seven Pillars of Wisdom, Institute for Creation Research). Other examples include the The Seven Pillars of Life (Science, March 2002) articulated by Daniel E. Koshland, the Seven Pillar of Ecosystem Management (Landscape and Urban Planning, 1998) articulated by R. Lackey, and the Seven Pillars of Democratic Governance (Synergy Associates, 2009) by Mel Gill.

Octangular bonding? The following animation is suggestive of the pattern of bonding in the case of an eight-fold helix.

8-fold bonding pattern
Animation	Image

Of relevance to bonding explorations, a more rigorous interactive approach to the 8 pointed star is offered under a pseudonym in the GeoGebra environment. A detailed mathematical description is provided by Harvey Heinz (Order-10 Magic Stars, 2003). Expressed in terms of pillars, the Dali Lama and Archbishop Desmond Tutu have articulated Eight Pillars of Joy. Google allegedly operates in terms of Eight Pillars of Innovation. Extensive reference is made to the articulation of Eight Pillars of Prosperity (2011) by John Allen.

Enneagram bonding? As noted above, the enneagram is a notable feature of a particular school of thought which associates modes of thought and personality characteristics with the vertices of a variant of the 9-pointed star. A detailed mathematical description is provided by Harvey Heinz (Order-8 Magic Stars, 2003). Of relevance to bonding explorations, a more rigorous interactive approach to the 9 pointed star is offered under a pseudonym in the GeoGebra environment. Understood in terms of pillars, examples include: Stephen Sideroff (Nine Pillars of Resilience and Success, 2015), Gunnar Sevelius (Nine Pillars of History, 2010), and the Nine Pillars of Conservatism of the European Young Conservatives.

Especially relevant to the argument here for increasing the mnemonic characteristics of symbols, musical notes are associated with those vertices -- as presented in an animation in the enneagram entry in Wikipedia. That image is adapted in the following simplistic animation which points to the possibility of associating bonding relationships with complex patterns of memorable chordal relationships through sonification.

Animation of 9-fold bonding pattern

Adapted from Wikipedia

In the mathematical disciplines of topology, geometry, and geometric group theory, the Heawood graph is a basic feature in discussion of the subtleties of orbifolds (for "orbit-manifold"). An orbifold is a generalization of a manifold. It is a topological space (an "underlying space") with an orbifold structure. Seemingly incomprehensible to most, orbifolds have been applied to music theory. As discussed separately (Musical implications of orbifolds for comprehension of questioning dynamics, 2014), there is the possibility that the distinctive cognitive feel for logical distinctions and connectivity might be associated with chords -- in the light of the work of Dmitri Tymoczko (The Geometry of Musical Chords, Science, 2006; A Geometry of Music, 2011):

A musical chord can be represented as a point in a geometrical space called an orbifold. Line segments represent mappings from the notes of one chord to those of another. Composers in a wide range of styles have exploited the non-Euclidean geometry of these spaces, typically by using short line segments between structurally similar chords. Such line segments exist only when chords are nearly symmetrical under translation, reflection, or permutation. Paradigmatically consonant and dissonant chords possess different near-symmetries and suggest different musical uses.

Decagonal bonding? A detailed mathematical description is provided by Harvey Heinz (Order-10 Magic Stars, 2003). Of relevance to bonding explorations, a more rigorous interactive approach to the 10 pointed star is offered under a pseudonym in the GeoGebra environment, this is associated with an overlapping variant of which screen shots are presented in the animation below. Especially interesting is the manner in which the pattern is generated using 7 circles and 4 lines.

Animation of 10-fold bonding pattern

Adapted from GeoGebra

There are numerous examples of articulations in terms of "ten pillars": Ten Pillars of Economic Freedom, Ten Pillars of Buddhism, Ten Pillars of Financial Independence, Ten Pillars of Successful Strategic Planning, Ten Pillars of Successful Technology Implementation, The Ten Principles of the UN Global Compact.

Hendecagram bonding? A detailed mathematical description is provided by Harvey Heinz (Order-11 Magic Stars, 2003). Their symbolic use is exceedingly rare, although one detailed description of its various symbolic articulations (as an endekagram) is noted by the Hermetic Order of the Golden Dawn (Polygrams and Polygons).

Dodecagram bonding? Considerable importance is conventionally associated with 12-fold configurations, most notably juries and the round tables of the wise (Checklist of 12-fold Principles, Plans, Symbols and Concepts: web resources, 2011). It could therefore be assumed that the challenge of shifting patterns of bonding would invite intensive study. Of particular relevance is the approach of Arthur Young (The Geometry of Meaning, 1976) as separately discussed (Typology of 12 complementary dialogue modes essential to sustainable dialogue, 1998).

Of relevance to bonding explorations, a more rigorous interactive approach to the 12 pointed star is offered under a pseudonym in the GeoGebra environment. As might be expected, there are many articulations in terms of "twelve pillars" including: the 12 pillars of competitiveness of the World Economic Forum, 12 pillars of wisdom as the ultimate intelligence test (New Scientist, 26 October 2010), 12 pillars of well-being articulated by Rick Hanson, 12 pillars of trust, articulated by Jane Anderson.

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