Embedding the triple helix in a spherical octahedron

Framing Cyclic Revolutionary Emergence of Opposing Symbols of Identity (Part #4)

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This model is a development of that in the earlier paper, most notably through use of constraining great circles and the octahedron they imply -- as suggested by their use in framing the Pentagramma Mirificum (discussed there in relation to a Quintuple Helix model).

Screen shots of triple helix embedded in octahedral great circles
Octahedral great circles highlighted	Octahedron highlighted	Triple helix highlighted
Interactive 3D versions: x3d; wrl/vrml. Video: mp4

Experimental design constraints applied to all models: The basic approach taken is to develop the 3D models explored in the previous documents in a more systematic manner. To this end emphasis was placed on the following design criteria:

• Interlocking great circles are seen as fundamental to framing the core dynamics of a model -- especially its cognitive and social appeal.
• Circles represented geometrically are potentially to be understood in cyclic terms (whether as systemic feedback loops or otherwise). Such cycles acquire further significance in terms of undertanding of eternal return and enantiodromia through which any starting point s rediscovered and known otherwise for the first time. Mobius, cognitive twist, Coriollis illusion ***
• Understood as the great circles of a common sphere, combinations of circles (variously oriented to one another) are then understood as engendering and framing the spherically symmetrical polyhedra -- notably those most commonly recognized and most readily associated with understandings of coherent organization (tetrahedron, octahedron, cube, etc). Polyhedra can then be understood as a "product of encirclement".
• Whilst great circles can be understood as associated with complementary polyhedral vertices, or axes of symmetry through faces, the focus here is (initially) on those associated with complementary edges of a polyhedron
• The set of such interrelated polyhedra then frames a set of models of which the triple, quadruple and quintuple helices are examples
• This spiral form of the helix, and the dynamic it implies in living systems, is seen as a valuable complement to the static structure of a polyhedron. There is the possibility that the appreciation and attractive power of symmetry is intimately related to a form of helical eye movement by which symmetry is progressively recognized.
• According to the properties of the encircled polyhedron, these frame the embedding of corresponding N-tuple helices which mesh with the great circles by which it is engendered
• The helix, meshing with great circles, is understood as spiralling down through a "cognitive wormhole" at the centre of the model -- to emerge in inverted form as a helix spiralling outward to mesh finally with the great circles -- effectively ensuring systemic closure
• As a design exercise, construction of the models as 3D animations using virtual reality technology, implies a preoccupation with aesthetic coherence seen as fundamental to the interest and communicability of the model.
• In contrast to models articulated in 2D, and depicted on the printed page of academic journals, importance is attached to interaction with the models as enabled by virtual reality technology

The models presented below could be readily improved from a design perspective -- in addition to the interactive facilities which could be added. They have already benefitted to some degree from the guidance of Sergey Bederov of Cortona3D.

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