Potential psychosocial implications of more complex polyhedra

Comparable Modalities of Aesthetics, Logic and Dialogue (Part #19)

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The point was made above that the 4-bit encoding used to distinguishing the 16 logical connectives could be recognized as being only 50% of a richer 5-bit pattern, and only 25% of an even richer 6-bit encoding typical of the original Chinese hexagram which reinforced the influential thinking of Gottfried Leibnitz regarding binary coding. In that light it is of interest to note how any such richer pattern might be coherently represented on a polyhedron. This is indicated in the two animations below using the relatively unique attributes of the 64-edged drilled truncated cube, as discussed separately (Proof of concept: use of drilled truncated cube as a mapping framework for 64 elements, 2015).

Especially noteworthy is the manner in which the form of the 4-dimensional central schematic below is echoed in the 3-dimensional form through the articulation of an internal structure -- effectively hollowing out a cube by "drilling" it as the name indicates. This can be understood as a means of giving 3-dimensional reality to the two "logical" connectives which are so questionably omitted from the set of 16 logical connectives. The drilled truncated cube is therefore capable of "holding" 4x16 distinctions, whereas the recourse to the cuboctahedron and its dual could only "hold" 14, mapped externally onto the faces or vertices respectively. The internal reflection of the outer form in the more complex structure can also be understood as honouring a degree of self-reflexivity otherwise absent from the conventional set of 14 logical connectives -- but implied by the meta-implications of the omitted two (as discussed above).

The inner reflection of the outer form merits consideration in the light of continuong debate regarding subjectivty-objectivity and insideness-outsideness (World Introversion through Paracycling: global potential for living sustainably "outside-inside", 2013; Cognitive Osmosis in a Knowledge-based Civilization: interface challenge of inside-outside, insight-outsight, information-outformation, 2017).

Comparability of fundamental forms -- contrasted with 64-edged drilled truncated cube?
Mapping of 64 hexagram names onto edges of drilled truncated cube	4-statement Venn diagram of a 4-dimensional cube as described by Tony Phillips	Mapping of 64 genetic codons onto edges of drilled truncated cube
	Edges going off in the 4th dimension are shown in green

implications for the coherence of a sonnet? The pattern of the 14-lined sonnet with 10 syllables per line, offers a further challenge to comprehension of the associated meaning and its appreciation -- and the question of "why 14". In exploring further it is useful to recall that the Chinese Yi Jing, with which the hexagrams are associated, is effectively a complex poem and can be recognized in those terms through its dependence on metaphor for its explanation. The corresponding mapping of genetic codons (above right) could readily be explored as lines in the "poem of life".

So framed the array of 140 syllables of a sonnet could possibly be explored as "memetic codons" (M. Pitkänenon, Could one find a geometric realization for genetic and memetic codes? 30 March 2013). It is on these that the poet -- Shakespeare -- could be understood to have drawn and with which he struggled in the act of creation by which the sonnet was engendered as the attractor it proved to be. However crude the experiment, there is then a case for considering how the syllables might be configured through any exercise in mapping them onto a polyhedron. One that is again relatively unique for that purpose is the drilled truncated dodecahedron with 140 vertices, as variously presented below.

Meaningful cognitive coherence implied by a sonnet represented on a drilled truncated dodecahedron (experimental mappings of the 14x10 syllables of Shakespeare's Sonnet 18)
Morphing by truncation to dual with syllables on 140 vertices	Wire frame presentation with syllables on 140 vertices	Animation of (un)folding
Animation prepared using Stella Polyhedron Navigator

An alternative polyhedron of interest for such an exploration is the 1-frequency snub dodecahedral geodesic sphere with 140 faces, as variously presented below.

Clearly the arbitrary mappings above and below invite creative interaction to juxtapose the "memetic codons" on a polyhedron in a manner which relates more meaningfully to the lines of the sonnet. The challenge bears some resemblance to that of solving a Rubik Cube, Here however the attractive outcome is especially cognitive and is primarily recognized through sound, namely an interweaving defined by iambic pentameter (with five pairs of iambs).

Meaningful cognitive coherence implied by a sonnet represented on a 1-frequency snub dodecahedral geodesic sphere (experimental mappings of the 14x10 syllables of Shakespeare's Sonnet 18)
Mapping onto 140 faces (vertices 72)	Mapping onto 140 vertices of dual (72 faces)	Unfolded polyhedral net
Animation prepared using Stella Polyhedron Navigator

Relevance of quantum field theory to cognitive organization? Quantum mechanics now represents an exciting frontier framed by the cautionary comment of Richard Feyman (an iconic exemplar) with words to the effect that: if anyone claims to understand quantum physics, they do not understand quantum physics. An early effort to derive implications for international relations is that offered by Alexander Wendt (Quantum Mind and Social Science: unifying physical and social ontology, 2015; video; interview), as discussed separately (On being "walking wave functions" in terms of quantum consciousness? 2017).

With such precautions, it is appropriate to note how an understanding of a 14-fold pattern has been articulated in the obscure literature of the field -- notably characterized by unfamiliar references to Feynman diagrams, amplituhedron theory, associahedron, permutohedron, and cyclohedron (Pascal Lambrechts, et al, Associahedron, Cyclohedron and Permutohedron as compactifications of configuration spaces, Bulletin Belge. Math. Soc. Simon Stevin, 17, 2010)). In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behaviour and interaction of subatomic particles (List of Feynman diagrams). They have been recognized as the most succinct representation of present knowledge about the physics of quantum scattering of fundamental particles and are variously described and depicted (Tim Evans, 'Diagramology': Types of Feynman Diagram, 2nd January 2018).

For Nima Arkani-Hamed, et al (Cosmological Polytopes and the Wavefunction of the Universe, September 2017):

The contribution of each Feynman diagram to the wavefunction of the universe is associated with a certain universal rational integrand, which we identify as the canonical form of a "cosmological polytope", which have an independent, intrinsic definition, making no reference to physics. The singularity structure of the wavefunction for this model of scalars is common to all theories, and is geometrized by the cosmological polytope.

As noted by Robin Ticciati (Quantum Field Theory for Mathematicians, Cambridge University Press, 1999), all Feynman diagrams constitute a high-dimensional polytope. At second order there are 14 connected Feynman diagrams which represent scattering processes (p. 108). As confirmed by I. T. Todorov (Analytic Properties of Feynman Diagrams in Quantum Field Theory, Elsevier, 2014), a class of primitive diagrams for scalar meson-nucleon scattering consists of 14 Feynman diagrams, as shown below

14 Feynman diagrams for scalar meson-nucleon scattering
2D array	Mapping onto cuboctahedron
Reproduced from I. T. Todorov (Analytic Properties of Feynman Diagrams in Quantum Field Theory, Elsevier, 2014, p. 52-53)	Reprojection of array on left

It is perhaps therefore somewhat unsurprising to discover diagrams corresponding to those used to represent the set of 14 logical connectives (presented above).

Representations of an associahedron
Hasse diagram of the Tamari lattice T4	Faces of the associahedron of order 4 labelled with ovals ("vacuum diagrams")	Associahedron as an order-4 truncated triangular bipyramid (animation)
Tilman Piesk, Public domain, via Wikimedia Commons		TED-43, CC BY 3.0, via Wikimedia Commons

As noted by Bo Feng and Yaobo Zhang (Note on the Labelled tree graphs, arxiv.org, 4 September 2020; submission to Journal of High Energy Physics), all Feynman diagrams having non-compatible poles, can be removed leaving diagrams all having a singular pole, as exemplified in the case of the remaining 14 Feynman diagrams up to a sign (as shown below right). The authors reproduce images from an earlier paper (Xiangrui Gaoa, Song Hea, and Yong Zhang, Labelled tree graphs, Feynman diagrams and disk integrals, arxiv.org, 14 October 2017; Journal of High Energy Physics, 2017, 11). This clarifies the relation of Cayley functions to Feynman diagrams and reference to "poles".

14-fold representations of a set of Feynman diagrams from quantum physics
Associahedron of (PT({1, 2, 3, 4, 5, 6}))2	Polytope of the Next-to-Star graph with n=6	Representation following removal of non-compatible poles
Reproduced from Bo Feng and Yaobo Zhang (Note on the Labelled tree graphs, arxiv.org, 4 September 2020; submission to Journal of High Energy Physics)

Implications of a 14th Archimedean polyhedron? Given the recognized relationship between Feynman diagrams and polytopes, visualization of the set of 14 of interest here might otherwise have been explored in terms of the widely referenced set of Archimedean semi-regular polyhedra -- although conventionally held to be 13 in number. However Branko Grünbaum has pointed out a frequent error in which authors define Archimedean solids using a local definition -- thereby omitting a 14th polyhedron, namely the elongated square gyrobicupola, or pseudo-rhombicuboctahedron (An Enduring Error, Elemente der Mathematik, 64, 2009, 3). This has 18 square faces and 8 triangular faces. Further clarification is offered by separately (14th Archimedean Solid: the Archimedean solid that isn't, RobertLovesPi.net, 10 June 2014; Evelyn Lamb, A Few of My Favorite Spaces: The Pseudo-Rhombicuboctahedron -- the tortured psyche of a misunderstood solid, Scientific American, 27 March 2018; T. R. S. Walsh,  Characterizing the Vertex Neighbourhoods of Semi-Regular Polyhedra, Geometriae Dedicata, 1, 1972).

Depending on how the precise definition is "tweaked", the number of "Archimedean solids", can be considered as:

• 13 (the conventional list)
• 14 (including the pseudo-rhombicuboctahedron / elongated square gyrobicupola, but not the mirror images)
• 15 (including mirror images of two of the solids)
• 16 (including mirror images and the pseudo-rhombicuboctahedron)

Indeed if only 13 polyhedra are to be listed in the set, the definition must use global symmetries of the polyhedron rather than local neighborhoods. The 14th meets a weaker definition of an Archimedean solid in which the faces surrounding each vertex are of the same types (i.e. each vertex looks the same from close up), so only a local isometry is required. Conventionally considered to be one of the 92 Johnson solids, it is not usually considered to be an Archimedean solid, even though its faces consist of regular polygons that meet in the same pattern at each of its vertices -- because unlike the 13 Archimedean solids, it lacks a set of global symmetries that take every vertex to every other vertex.

The 14th is locally vertex-regular â-- the arrangement of the four faces incident on any vertex is the same for all vertices; this is unique among the Johnson solids. However, the manner in which it is "twisted" gives it a distinct "equator" and two distinct "poles", which in turn divide its vertices into 8 "polar" vertices (4 per pole) and 16 "equatorial" vertices. Not being vertex-transitive, this is an additional reason for omitting it from the 13 Archimedean solids.

It could be argued that the controversial distinction between globl and local perspectives is at the heart of many psychosocial issues, as argued in one particular case (Local Reality of Overcrowding -- Global Unreality of Overpopulation: comprehensible reframing of engagement with global issues via metaphors of proximity, 2019). The subtlety of the distinctions between 12, 13 qnd 14 could even be said to have been enacted in the dynamics of the drama associated with the Last Supper -- of such fundamental symbolic importance to Christianity.

In the animation on the left, "12" Archimedean polyhedra are associated with the 12 vertices of a cuboctahedron, rotating around an omitted 13th at the centre (the truncated tetrahedron). The pattern is reproduced from Psychosocial Implication in Polyhedral Animations in 3D (2015) in relation to the closest packing criteria for the 12.. By contrast, in the central animation, "14" Archimedean polyhedra are associated with the 14 vertices of a rhombic dodecahedron -- the dual of the 14-faced cuboctahedron. The added "14th" is distinguished by a green colour.

In the animation on the right, the 14 Feynman diagrams associated with scalar meson-nucleon scattering (reproduced above from I. T. Todorov, 2014), are experimentally arrayed on the 14 vertices of a rhombic deodecahedron -- in order to explore correspondences with the central array. No effort has been made here in the attribution of the diagrams to that end.

Animation of contrasting polyhedral arrays (using distinctive design metaphors)
Cuboctahedral array of 12 Archimedean polyhedra (attached vertex to face)	Rhombic dodecahedral array of 14 "Archimedean polyhedra" (vertex to centre)	Rhombic dodecahedral array of 14 Feynman diagrams
Animations prepared with the aid of Stella Polyhedron Navigator

No attempt has been made to position the individual polyhedrsa in relation to one another in the configuration. Such possibilities would presumably follow from the work in the Chinese papers cited above.

Psychosocial implications? Such correspondences invite speculative reflection (Potential of Feynman Diagrams for Challenging Psychosocial Relationships? Comprehending the neglect of an unexplored possibility, 2013; Credibility of Psychosocial Analogues of Feynman Diagrams: Cognitive engagement with challenging visualization, 2013). The question framed by the argument above is whether Bach, Shakespeare, and others, have some intuitive sense of a 14-fold array of cognitive modalities -- central to their much appreciated creativity. Missing is recognition that their various expressions of that intuitive pattern, in aesthetic media or otherwise, involve the use of particular (and seemingly incommensurable) "pattern languages" -- in the absence of any adequate facility to express that intuition otherwise.

Of some relevance is the manner in which ignorance of more complex possibilities is factored into such modelling, or not, as might be inferred from the arguments of Nicholas Rescher (The Strife of Systems: an essay on the grounds and implications of philosophical diversity, 1985; Pluralism: Against the Demand for Consensus, 1993; Ignorance: On the Wider Implications of Deficient Knowledge, 2009; Unknowability, 2009).

Aspects of the argument have been developed separately (Potential for Coherence through Engaging Strategic Poetry: memorable cycles of subdivision enabling viable governance, 2021)

Neural implications for cognitive coherence and strategy articulation? The composition of a sonnet in the light of the above presentations then calls for consideration in the light of the investigations of the Blue Brain Project using mathematics in a novel way in neuroscience. This has shown that:

... the brain operates on many dimensions, not just the three dimensions that we are accustomed to. For most people, it is a stretch of the imagination to understand the world in four dimensions but a new study has discovered structures in the brain with up to eleven dimensions -- ground-breaking work that is beginning to reveal the brain's deepest architectural secrets..... these structures arise when a group of neurons forms a clique: each neuron connects to every other neuron in the group in a very specific way that generates a precise geometric object. The more neurons there are in a clique, the higher the dimension of the geometric object...

The appearance of high-dimensional cavities when the brain is processing information means that the neurons in the network react to stimuli in an extremely organized manner. It is as if the brain reacts to a stimulus by building then razing a tower of multi-dimensional blocks, starting with rods (1D), then planks (2D), then cubes (3D), and then more complex geometries with 4D, 5D, etc. The progression of activity through the brain resembles a multi-dimensional sandcastle that materializes out of the sand and then disintegrates. (Blue Brain Team Discovers a Multi-Dimensional Universe in Brain Networks, Frontiers Communications in Neuroscience, 12 June 2017)

Of interest in this respect, as noted above, is the use of artificial intelligence to generate sonnets (Jey Han Lau, Trevor Cohn, Timothy Baldwin, Julian Brooke, and Adam Hammond. Deep-speare: A joint neural model of poetic language, meter and rhyme. Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics (Long Papers), 2018), summarized by the authors as This AI Poet Mastered Rhythm, Rhyme, and Natural Language to write like Shakespeare, IEEE Spectrum, 30 April 2020).

Such insights merit consideration in the light of the elaborate interweaving of many of the above arguments (however seemingly disparate) in the consideration of the mathematical abstractions of space groups. In mathematics, physics and chemistry, a space group is the symmetry group of an object in space, usually in three dimensions. In crystallography, space groups are also called the crystallographic or Fedorov groups, and represent a description of the symmetry of the crystal.

There is indeed a degree of recognition of "cognitive space" by psychology, understood by analogy to location in two, three or higher dimensional space, in order to describe and categorize the thoughts, memories and ideas (Marcelo Soria-Rodríguez, Total Cognitive Space: from AI to organizational changes, Towards Data Science, April 2021; David J Piggins, Cognitive Space, Perception, 4, 1975). The term has been adopted as the name of a major machine intelligence corporation having with a particular focus on space operations. Unfortunately these preoccupations are not informed by the mathematical abstractions of "space groups", nor are those abstractions in any way informed by the cognitive challenges of comprehension.

The presentation of this argument could be usefully challenged due to the absence of any attempt to articulate it in poetic form, as explored by Sam Illingworth (Are scientific abstracts written in poetic verse an effective representation of the underlying research? F1000 Research, 2016; The Poetry of Science). Could the section headings of the argument convey a higher order of connectivity if presented in the form of a sonnet?

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