**Eppur si muove: Biomimetic embedding of N-tuple helices in spherical polyhedra**

Framing Cyclic Revolutionary Emergence of Opposing Symbols of Identity

Symbolic stars vs Strategic pillars; Polyhedra vs Helices; Logic vs Comprehension?

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

Embedding the triple helix in a spherical octahedron

Embedding the quadruple helix in a spherical cube

Embedding the quintuple helix in a spherical dodecahedron and a Pentagramma Mirificum

Embedding six-fold, eight-fold and ten-fold helices in appropriately encircled polyhedra

Embedding twelve-fold, eleven-fold, nine-fold and seven-fold helices in appropriately encircled polyhedra

Neglected recognition of logical patterns -- especially of opposition

Dynamic relationship between polyhedra engendered by circles -- variously implying forms of unity

Symbol rotation as dynamic essential to engaging with value-inversion

References

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The contrast to the geocentric model of the solar system was framed by the Italian mathematician, physicist and philosopher Galileo Galilei (1564-1642). His much-cited phrase, " *And yet it moves*" (* E pur si muove* or

The argument here assumes that much more is based on circles than is conventionally held to be the case. They can be explored as having a fundamental function in engendering order. On the one hand circles are readily understood to frame and identify social circles, their bounded nature, and an implied dynamic -- acquiring one manifestation in the arenas of many games and the hole-like structure of goals in many team sports. This extends to a degree of understanding of rings and cycles -- exemplified by circlets of beads. On the other hand much is understood about the geometry of circles, most notably for this argument through the manner in which as great circles (with a common centre) they variously frame the spherically symmetrical polyhedra (5 Platonic and 13 Archimedean) -- as approximations to a sphere. In this sense those polyhedra might then be better understood in dynamic terms as patterns of cycles, if not as standing wave forms. The contrasting preoccupations of such social circles suggests that they can be usefully recognized as variously "oriented" to one another in a geometric sense.

The interlocking of great circles frames polygons on a polyhedral surface -- typically triangles, squares, pentagons, hexagons, etc. These are matched by corresponding polygons on the opposite side of the polyhedron -- but typically inverted. Between such opposing polygons, it is potentially significant to explore the manner in which spirals link them through the polyhedral centre. The spirals may be triple, quadruple, quintuple, etc -- effectively "engendered" by the sides of the polygon (whether three, four, five, etc), Those sides are themselves defined by the great circles variously oriented to one another. A triangular polygon thus "engenders" a triple helix, a pentagon a quintuple helix, as previously described and illustrated (*Psychosocial Learnings from the Spiral Form of Hurricanes: implications of the triple helix and the 3-fold triskelion as "cognitive cyclones"?* 2017).

Of further relevance, polyhedra have acquired new significance as mapping devices in oppositional logic for ordering patterns of connectivity in Boolean logic and truth tables. Of concern is the extent to which current academic preoccupations with polyhedra are fundamentally impersonal and "unsociable" -- and proud to be so. They are dissociated from psychosocial experience to the point of being essentially meaningless and irrelevant to a civilization in crisis. New forms of cognitive "bridging" are therefore required to the cultural symbols with which people have identified over centuries. The point can be argued with respect to the remarkable work of Buckminster Fuller, so specifically upheld as the "geometry of thinking" (*Geometry of Thinking for Sustainable Global Governance: cognitive implication of synergetics*, 2009).

Given the inspiration via biomimicry of the two-stranded DNA helix, the question then arises as to the contrasting patterns of bonding between the strands of a triple helix, a quadruple helix, or quintuple helix, for example. There is the curious possibility that this pattern of bonding may be very suggestive of common cultural symbols with which different groups variously identify -- notably in star form (whether triangular, square, pentagonal, hexagonal, etc.). Those bonding symbols might then also be understood as "emerging" from the helical patterns, or "sinking" into them through a "cognitive wormhole" at the centre (and inverted through it). The academic modality is seemingly unable to engage effectively with the psychosocial function and holding power of symmetry, readily comparable in its beauty with the hypnotic power of a whirlpool. How indeed do symbols become psychoactive?

In moving and transforming the symbols may rotate in ways suggestive of new kinds of insight, especially in 3D visualization (*Envisaging NATO Otherwise -- in 3D and 4D? Potentially hidden faces of global strategy highlighted through polyhedra*, 2017). The movement in relation to the spiral "strings" might also be usefully associated with patterns of sound of mnemonic value, as explored through sonification.

An earlier experimental exploration of this kind -- of far simpler structure -- is presented separately (*Dynamic Exploration of Value Configurations: interrelating traditional cultural symbols through animation*, 2013; see video). An application of current relevance is separately described (*Middle East Peace Potential through Dynamics in Spherical Geometry: engendering connectivity from incommensurable 5-fold and 6-fold conceptual frameworks*, 2012).

Many symmetrical symbols appear to serve as inadequately acknowledged "holding patterns" for significance. With flags bearing symbols of unity accorded overriding significance on every official occasion, those symbols would seem to merit "upgrading" as devices for heralding governance of change in an information-based society.** Is the assumption of the silent static nature of logos and cultural symbols essentially misleading?** Is there indeed a valuable possibility to * Eppur si muove*?

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