Toroidal conceptualization

Imagining Toroidal Life as a Sustainable Alternative (Part #7)

Conversation theory: Of particular relevance to this argument has been the manner in which conversation theory has been developed (Gordon Pask and Gerard de Zeeuw, Interactions of Actors: theory and some applications, 1992), with reference to the work of Beer (1994), notably as described by Nick Green (Axioms from Interaction of Actors Theory, Kybernetes, 33, 2004) with respect to triplets of concept resonance:

There are two parts to an interacting participant or Actor. First the P-individual which is a dynamic, productive and incidentally reproductive, adaptive, evolving and learning collection or entailment mesh of concepts. Dawkins' memes (Dawkins, 1976) are a restricted form of P- individual. Second the M-individual which is a mechanical or biological medium e.g. a computer, a brain or a star, which supports the P-Individual and the strains its concepts produce....

The closed toroidal processes which comprise the concepts of P-individuals exist as stable triples in which any pair is analogous to the other and distinguished by the third. Any two concepts may generate the third because of their resonant similarities and differences. The resonance produced by an incident field produces an output radiative field. Knot theory was a matter of some concern to Pask. He coined the term "tapestry" making his entailment mesh structure of the concepts of a participant coherent with knot theory. Whilst loops are always permitted, indeed are the nature of concepts, the intersections of C[onversation] T[heory] become crossings in I[nteraction of] Actors Theory] where crossings up and down with loops define knots. The crossing up or down rule of the knots or links was not decided but a recursive and nested Borromean form seemed most likely. This seems coherent with the superstring theory interpretation of force. (pp. 1433-1435)

Orthogonal form of the Borromean link-isometric view. A putative model of Continuity, the equilibrium of void and not-void, around a void. Borromean rings, as a link of three unknotted loops (such that any two of the loops are unlinked) can therefore be understood as the closure of a braid, as discussed in some detail by Louis H. Kauffman (Knot Logic and Topological Quantum Computing with Majorana Fermions, arXiv, 2013; The Borromean Rings: a tripartite topological relationship, Bridges, 2006).

Representations of Borromean rings
Pask's Stable Concept Triple as Borromean ring	Borromean ring in 3D	International Mathematical Union logo	Scissors-Rock-Paper game

Reproduced from Nick Green (Axioms from Interactions of Actors Theory, 2004)	Reproduced from Kauffman (2006)	Reproduced from Wikipedia

Kauffman notes that as an ordered knot-set, the Borromean rings constitute a "scissors-paper-stone" pattern. Each component of the rings lies over one other component, in a cyclic pattern (left below). (with Red surrounds blue; Blue surrounds green; Green surrounds red)

Borromean rings as an ordered knot-set: Far more controversial is the comprehension of the Christian Trinity to which mystics allude, as discussed and illustrated separately (Vlad Alexeev, Borromean Rings, Impossible World; Symbols of the Holy Trinity, Holy Trinity Amblecote; Borromean Rings, ThisIsChurch.com). The question is whether its integrative function of 3-in-1 and 1-in-3 is best to be presented in terms of a Venn diagram or a Borromean condition.

This distinction is most clearly made and illustrated in an extensive analysis of how Dante Alighieri describes the three rings (tre giri) of the Holy Trinity in Paradiso 33 of the Divine Comedy (Arielle Saiber and Aba Mbirika, The Three Giri of Paradiso XXXIII, Dante Studies, 131, 2013, pp. 237-272). The remarkable interdisciplinary exploration combines insights from speculative theology, geometry and knot theory. It is of particular relevance to the argument here, especially in relation to knots (as discussed further below).

The "scissors-paper-stone" metaphor has been extended to 5-ring and 7-ring Borromean configurations (Marc Chamberland and Eugene A Herma, Rock-Paper-Scissors meets Borromean Rings, Grinnell College, 2014).

Representations of Borromean rings (by contrast with misleading representations)
(3,3)-torus link (circles do not need to bend to form the 3-link, non-Borromean )	3-fold Borromean rings (circles must bend to be able to be woven together)	5-fold Borromean rings	Discordian Mandala (misleading 5-fold interlinkage)

Reproduced from The Three Giri of Paradiso XXXIII (2013)		Reproduced from Chamberland and Herma (2013)	Reproduced from Wikipedia

With respect to any consideration of "9 planetary boundaries", to what extent is it fruitful to explore the possibility that the dilemmas they imply together may derive from the manner in which they are interlocked? Is the quest for the elusive condition of sustainable governance to be compared with that of achieving an interrelationship between them usefully understood in the light of Borromean rings?

Such a framing evoked exploration of the possibility of a 5-fold dynamic configuration of 9-sided "toroids" -- provoked by the pattern of the Discordian Mandala as being emblematic of current incommensurabilities (Concordian Mandala as a Symbolic Nexus: insights from dynamics of a pentagonal configuration of nonagons in 3D, 2016; Visualization in 3D of Dynamics of Toroidal Helical Coils -- in quest of optimum designs for a Concordian Mandala, 2016). The following is one result.

Indicative experimental animations
3 Möbius strips	Configuration 5 9-sided helical toroids	Triangular bonding within triple helix (embedded in octahedral great circles)

Video (mp4); Virtual reality (x3d; wrl)	Reproduced from Visualization in 3D of Dynamics of Toroidal Helical Coils (2016)	Reproduced from Embedding the Triple Helix in a Spherical Octahedron (2017)

Given the current focus on the Triple Helix model of innovation (as noted above), the experimental representations were extended to triple, quadruple and quintuple helices, of which the first is shown above (Framing Cyclic Revolutionary Emergence of Opposing Symbols of Identity: biomimetic embedding of N-tuple helices in spherical polyhedra, 2017).

System holders: Distinct from specific know-how and insights, one of the challenges of such meaning malas is to hold systems of insight as patterns, rather than solely in terms of their individual elements. How can whole systems be effectively held by such devices? How much insight can be rendered explicit in the design and how much must remain implicit? This is the distinction between the reminder dimension and the meditative catalyst dimension. As system holders, the level of understanding is shifted to a more integrative perspective -- the cultural rosary then functions somewhat like an enjoyable conceptual prosthetic! (Designing Cultural Rosaries and Meaning Malas to Sustain Associations within the Pattern that Connects, 2000).

The animation on the right below offers a reminder of the fundamental cyclic archetype of the Ouroboros -- of a snake (or dragon) eating its tail. The animation is a schematic exercise in representation of that process, as discussed separately (Cognitive Osmosis in a Knowledge-based Civilization: interface challenge of inside-outside, insight-outsight, information-outformation, 2017). The central animation illustrates the dynamics of a vortex sustaining a toroid -- as most readily recognized in a smoke ring.

Circular configuration of hexagrams	Animation of vortex ring	Schematic Ouroboros animation

Animated variant at Dynamic Exploration of Value Configurations (2008)	By Lucas V. Barbosa - Own work, Public Domain, Link	Reproduced from: Experimental animations in 3D of the ouroboros pattern (2017)

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Toroidal conceptualization