The Secret Pathway between Configurations of Otherness?

Mnemonic geometry clues on how to kiss-touch and make-up

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Discourse between disparate domains: There is ever more indication that civilization is faced with a major difficulty in facilitating the engagement between disparate domains. This has long been evident between religions whose insights regarding transcendental unity and harmony do not translate into the relationships between their respective adherents. The pattern is very similar in the case of academic disciplines, between the sciences and the humanities, and between contrasting philosophies. Inter-faith, inter-disciplinary, and inter-ideological dialogue have become exercises in tokenism and virtue signalling with little effective results in practice.

The difficulty is otherwise evident in the relationship between people of different "types", most notably that between people of different personality type -- despite the efforts in team building. Understood otherwise there is an analogous difficulty in the relation between logic and emotion, or of physicality and intuition -- however framed (Interrelating Multiple Ways of Looking at a Crisis, 2021; Ways of Thinking, Perception and Analysis, 2001; Ways of looking at ways of looking -- in a period of invasive surveillance, 2014). Typically this is handled by favouring one or the other in a particular context -- even to the exclusion of the others. The challenge has been partly framed and addressed by the various methods of Edward de Bono (Six Thinking Hats, 1985; Six Frames For Thinking About Information 2008). Their uptake has not responded to the development of the problem -- especially to the conflicts it engenders.

Mathematics and geometry? In this situation it is therefore curious to explore whether mathematics and geometry have insights to offer -- acclaimed as they are as offering the most sophisticated understanding of relationships.

A point of departure is the depiction by science of the relation between two contrasting modalities, of which there are many images.The most obvious is a magnet with fields of force between two poles. This is echoed to a degree by images of right and left brain as related through the hypothalamus. With respect to civilization, this frames a question regarding the hypothalamus and the corpus callosum (Corpus Callosum of the Global Brain?: locating the integrative function within the world wide web, 2014)

Whilst there are many images for a 2-fold relationship, somewhat less evident are the depictions of 3-fold relations. Although in this case imagery of knots and the like -- most notably Celtic knots, the trefoil and the trefoil knot -- frames such understanding. This is especially imbued with a degree of mystery in religious references to any trinity of deities. However although these may well be depicted in two dimensions, it frames the challenge of the form which any 4-fold relationship might take. In this case reference may be made to four-leaf clover and to more complex Celtic knots and related patterns -- even to to the cinquefoil and beyond. Such petal and leaf arrangements clarify the distinction between a degree of unity in which all are linked at a common central point -- an illusion of unity to which many vainly aspire in psychosocial organization -- and other forms of coherent interrelationship (Keith Critchlow, The Hidden Geometry of Flowers: living rhythms, form and number, 2012). .

"Kiss-touch": The approach taken here is to explore with AI the pattern of relationships between 4 modalities of any kind -- as a point of departure highlighted as of special significance (Comprehension of Singularity through 4-fold Complementarity, 2024; AI-enabled Mapping and Animation of Learning Pathways, 2024). This may be readily depicted by placing a fourth orange on top of a configuration of 3 oranges touching one another in a triangular configuration -- a pattern which is widely comprehensible in stacking fruit or cannon balls. Their specific points of contact are commonly referenced in design terminology as a "kiss-touch . It evoked an early discussion by Isaac Newton with David Gregory in 1694 regarding the "kissing number", namely the maximal number of equal size non-overlapping spheres in three dimensions that can touch another sphere of the same size (Oleg R. Musin, The Kissing Problem in Three Dimensions, 13 June 2005).

The specific phrase “kiss-touch” is less common as a lexical unit, but it is transparently descriptive -- especially in scientific or technical settings -- of the moment or locus where two objects meet gently, as in sphere packing, billiards, or gears. It conveys both the delicacy and the exactness of such a contact, borrowing directly from these historical senses of physical and metaphorical closeness. Thus, as used in geometry or packing theory, "kiss-touch" is a natural extension of the established metaphorical and physical meanings of "kiss" as a light, precise contact. As such it featured in the descriptions of tensegrity structures and geodesic domes by Buckminster Fuller, who explored ideas around points of contact, interaction, and energy transfer. This relates closely to the geometric notion captured by "kissing" spheres in mathematics and physics (Synergetics: Explorations in the Geometry of Thinking, 1975)

As the specific point of transition from one sphere or domain to another -- from one "language" to another -- the question is then how best to understand the pattern of transitional pathways between four such domains -- or more -- when such domains are in close touch with one another in communication terms (Pathways in Governance between Logic, Emotion, Spirituality and Action, 2024) . Aspects of the question naturally invite geometrical and other commentary from mathematics, but the concern here is the relevance of such insights to psychosocial interaction. In this light the question relates to the quest for a Rosetta Stone to enable such transition, as discussed previously (Integrative implications of the Rosetta Stone, Philosopher's Stone and Diamond, 2025).

The challenge could be caricatured as determining the "kissing number" in configurations of collective domains -- enabling them to "kiss and make up". The challenge is exemplified in the case of the pattern of 8 major religions (Stephen Prothero, God Is Not One: the eight rival religions that run the world and why their differences matter, Journal for the Scientific Study of Religion, 50, 2011, 1). It is evident in any effort to reconcile the 8 forms of intelligence of multiple intelligence theory, or the contrasting personality types in team building

Of course others claim greater or lesser numbers than eight, and in the case of the disciplines, the Web of Science database includes 241 subjects of study, whereas Wikipedia lists 1475 fields, as noted by Ugo Bardi (Science and the Dragon: Redistributing the Treasure of Knowledge,  Organisms: Journal of Biological Sciences, 5, 2022, 2). The Mathematics Subject Classification offers 64 top-level topics, but with seemingly little concern for how mathematics might be more appropriately organized for their comprehension (Is the House of Mathematics in Order? 2000; Configuring the 64 disciplines of mathematics as a 64-edged drilled truncated cube, 2021).

Transitional manoeuvers: Ironically the question is somewhat analogous to the gravity assist maneuver of spacecraft, and by extension to the hypothetical Interplanetary Transport Network (ITN) -- the collection of gravitationally determined pathways through the Solar System that require very little energy for an object to follow. Given the challenging psychosocial relations within patterns of otherness, the need for an analogous ITN is discussed separately in terms of the Possibility of an "Inter-other Transition Network" (2012). This suggests the value of using mathematics to explore the relations between silos by which communication is currently dysfunctionally fragmented (Mathematical Modelling of Silo Thinking in Interdisciplinary Contexts

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