**The-O Ring and The Bull Ring as Spectacular Archetypes (Part #3)**

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Rather than attempting to explore further the tabular articulations, it appears more fruitful to consider how any articulation is mapped to facilitate comprehension, especially that of a higher order consistent with the complexity as it may be experienced and integrated. The use of geometry in this process has been argued separately (*Geometry of Thinking for Sustainable Global Governance*, 2009; *Metaphorical Geometry in Quest of Globality -- in response to global governance challenges*, 2009). It is consistent with the theoretical and symbolic understandings of the variants of "theo". The approach makes extensive use of mapping onto polyhedra as a way of embodying and implying integrity.

**Polyhedral mapping: tetrahedron**: A useful point of departure is to use the simplest polyhedron, the tetrahedron, as offering a surface for the four cognitive modalities on which attention has focused. The animation on the left below is one result. Of significance is the manner in which the polyhedron is contained and engendered by great circles -- a minimum of three of which are required. More are depicted in the image on the right. It is through the triangulation enabled by the minimum number of circles that the polyhedron "emerges" or is rendered sustainable. This has been extensively argued by Buckminster Fuller (*Synergetics: Explorations in the Geometry of Thinking*, 1975), as separately discussed (*Geometry of Thinking for Sustainable Global Governance: cognitive implication of synergetics*, 2009; *Triangulation of Incommensurable Concepts for Global Configuration*, 2011). The cognitive implications have been discussed in relation to the challenge of otherness (*Reframing the Dynamics of Engaging with Otherness*, 2011).

The set of circles is suggestive of the non-linear multidimensionality of an "O-ring" through which the four "theo" variants are related, and by which they are integrated. Somewhat ironically they offer a sense of 3-in-1 and 1-in3 -- with the traditional challenge of its comprehension. Relevant to the cognitive challenge is that the great circles interlock precisely because they are of different orientation. Together they imply what is associated with a sphere and globality -- the degree of integration to which the various forms of "theo" aspire, explicitly or implicitly.

Animations of tetrahedron with great circles(prepared with Stella Polyhedron Navigator) | |

Tetrahedron framed by 3 intersecting great circles(click for animation) | Tetrahedron framed by multiple intersecting great circles (click for animation) |

Another way of considering this pattern is through images such as the following. The interlocked Borromean rings on the left are suggestive of the cognitive entanglement fruitfully associated with any understanding of the great circle interlocking. It is appropriate to note that a representations in two dimensions of that pattern of 3 rings is a classical image of the Christian Trinity. Associating the 4 "theos" with the vertices of the tetrahedron above, each expanded to the point of touching, offers a sense in which they may "pack" together within a spherical context (image on right, below).

Interlocking Borromean rings(by Ron Bennett, image from Wikipedia) | Animation of 4 Theo modalities packed as spheres(prepared with Stella Polyhedron Navigator) |

Any such mapping readily suggests a static perspective which may well be completely inadequate. The circles through which the tetrahedron is constructed usefully imply a dynamic process for the emergence of the configuration -- and for sustaining it over time. This could be related to the arguments of George Lakoff and Rafael Nuñez (*Where Mathematics Comes From: how the embodied mind brings mathematics into being*, 2001) and Marie-Louise von Franz (*Number and Time: reflections leading toward a unification of depth psychology and physics*, 1974) -- recently figuring in a compilation by Alex Bellos (*Alex Through the Looking Glass: how life reflects numbers and numbers reflect life*, 2014). The sense of "getting somewhere" through any of the four modalities might be better framed by cyclic processes consistent with arguments for eternal return.

**Cognitive nature of the interlocking threefold engendering the tetrahedral mapping of "theos"**?: Failure to recognize "The-O Ring" could be considered as directly responsible for systemic neglect. In a separate discussion, inspired by the notorious "poem" on *The Unknown* of former US Secretary of Defense Donald Rumsfeld, sought to clarify the nature of this neglect (*Unknown Undoing: challenge of incomprehensibility of systemic neglect* (2008). Experimental use was made of classical Chinese coding patterns to clarify the relationship between a threefold set (knowing, feeling, acting), as presented in an annex at that time (*Strategic Patterns in terms of Knowing, Feeling and Action*, 2008).

This gave rise to the following table relating that threefold set to a fourfold set. The table has been adapted to associate the four "theos" with that fourfold pattern.

Association of generic conditions with classical Chinese binary codes (amended from version in Strategic Patterns in terms of Knowing, Feeling and Action (2008) which functions as a key to the various classical patterns of 64 hexagrams of the I Ching, presented there: Fu Xi, Jing Fang, King Wen and Mawangdui -- with links to specific commentary) | ||||

binary coding in hexagram | "theo" | Knowing(upper pair in hexagram) | Feeling(middle pair in hexagram) | Acting/Action(lower pair in hexagram) |

"theorem" | known knowns (KK) | feltknowingly (FF) | doneknowingly (DD) | |

"theory" | knownunknowns (KU) | | doneunknowingly (DU) | |

"theology" | unknownknowns (UK) | unfeltknowingly (UF) | undoneknowingly (UD) | |

"theosophy" | unknownunknowns (*K) | unfeltunknowingly (*F) | undoneunknowingly(*D) |

The cells of the table have been shaded here to suggest degrees of overlap in significance relating to the distinct "theos".

In the light of the manner in which 3-fold and 4-fold category systems can be considered as interrelated, that exercise noted the complementary categorization of conditions of the *Tao Te Ching*, notably as experimentally related to another classic of that period, the *T'ai Hsüan Ching* (*9-fold Magic Square Pattern of Tao Te Ching Insights: experimentally associated with the 81 insights of the T'ai Hsüan Ching, *2006). This gave rise there to an alternative pattern of strategies ordered in the light of the *Tao Te Ching* and *T'ai Hsüan Ching* with links to their commentaries. Rather than the binary coding system of the *I Ching*, giving rise to 64 hexagrams, it uses a **ternary coding system**s that gives rise to 81 tetragrams. This may be explored in the light of the keys offered experimentally by the following table.

The cells of the table have been shaded here to suggest degrees of overlap in significance relating to the distinct "theos".

**Polyhedral mapping: Szilassi: **Of interest, when the 4 "theos" are mapped onto polyhedral faces, is the fact that all 4 faces of the tetrahedron are in contact with each other -- a feature unique amongst polyhedra, with the exception of the far more complex Szilassi polyhedron (one of the Stewart toroids).. With seven faces, the latter suggests the possibility of using it to map either:

- Version A: the 4-fold pattern of faces attributed to the tetrahedron, with the significance associated with the minimal set of 3 great circles, thereby distinguishing:
- on the inner 3 faces, the significance attributed to the great circles -- whatever that might be considered to be
- on the outer 4 faces, the significance associated with the 4-fold "theo" modalities

- Version B: the 7-fold articulation of theology (explored in the first part of this paper as a template applicable to the other modalities of "theo"), distinguishing:

These alternative patterns are presented in the images below.

Animations of the 7-faced Szilassi polyhedron (prepared with Stella Polyhedron Navigator) | |

Version A Cognitive modalities of "theo" (theory, theorem, theology, theosophy) | Version B Theology-related articulation as a template (for theory, theorem, theology, theosophy) |

"External" faces | "External" faces |

"Internal" faces | "Internal" faces |

The argument here is that the Szilassi polyhedron is suggestive of the requisite complexity through which a set of "theos" can be related -- the hexahedral face to which each is attributed sharing an edge with each of the 6 other faces in the configuration. As an unusual cognitive jigsaw puzzle, appropriately resistant to simplistic assumptions about how science and religion might be related, further insight may be obtained from the following animation. This shows how the polyhedron can be decomposed into the parts which are conventionally recognized -- and the challenge of comprehending how they might be drawn together into an integrative configuration.

Animation of folding and unfolding of Szilassi polyhedron (prepared with Stella Polyhedron Navigator) |

[See also the interactive Parameterized Szilassi Polyhedron presented as a Wolfram Demonstration Project] |

With respect to further development of this argument, it is appropriate to note that the Szilassi polyhedron has the topology of a torus -- with a central hole framed by the "inner" faces. It is therefore suggestive of a toroidal form to whatever can be associated cognitively with the O-ring, as indicated below.

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