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Toroidal embodiment, knottedness and being a torus?


Imagining Toroidal Life as a Sustainable Alternative (Part #8)

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In the light of the challenge of experiencing oneself as well-rounded, as argued above, is experiencing oneself as a torus just as feasible (if not more so) as experiencing oneself as global or spheroid in some way? The arguments of an earlier section suggest that there is an intuitive appreciation of circlets and ring formation. These could be said to have as much credibility as a global form -- if not more. It could be useful to explore the history of the discovery, appreciation and use of the ball in comparison with the necklace or bracelet -- especially given the importance attached to that of the wheel.

Multidimensionality: This argument has further implications in the light of the assumptions too readily made regarding personal human experience of "globality" and "wholth" (Wholth as Sustaining Dynamic of Health and Wealth: cognitive dynamics sustaining the meta-pattern that connects, 2013).

To the extent that one assumes oneself to be "rounded", is this as a 2-sphere -- superficially, as with a bubble? Or is there an understanding of depth for which a 3-sphere would be more appropriate? More challenging -- if not inspiring -- is human identity better understood in terms of an N-sphere, with N being commensurate with the insights of physics? Such topological considerations could clarify the manner in which spherical experience of coherence could be transformed into toroidal experience -- with corresponding topological distinctions between 2-torus and N-torus.

Are human aspirations to "freedom" far more usefully recognized as being associated with such multidimensionality, as implied by arguments from various perspectives:

For some, speculation extends to the nature of "multidimensional humanity" and to "transdimensional humanity" (Alice Bryant and Linda Seebach, Multidimensional Potential of Human Beings).

Such language merits comparison with the results of recent neuroscience research which indicates the remarkable possibility of cognitive processes taking up even up to 11-dimensional form in the light of emergent neuronal connectivity in the human brain. As summarized:

Using mathematics in a novel way in neuroscience, the Blue Brain Project shows 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)

With respect to "being a torus" in any way, to what extent are the following animations suggestive of dynamics which could be associated with such understanding?

Turning a punctured torus inside-out A stereographic projection of a Clifford torus performing a simple rotation through the xz plane. 4D flat torus projected into 3-dimensions and rotated on a fixed axis. Ring torus becomes a horn torus, then a spindle torus, and finally degenerates into a sphere.
Turning a punctured torus inside-out Projection of a Clifford torus performing a simple rotation Animation of 4D flat torus projected into 3-dimensions Animation of torus-sphere transformation
By w:en:User:Surot - English Wikipedia, Public Domain, Link Created by Jason Hise with Maya and Macromedia Fireworks. Claudio Rocchini [CC BY-SA 3.0], via Wikimedia Commons User:Kieff [Public domain], via Wikimedia Commons

Topological semiotics? Any relevance of the torus to this argument is significantly framed both by topology and semiotics. This perspective was introduced by Algirdas Julien Greimas (Toward a Topological Semiotics), however, it is less evident from the following how the role of time is to be understood as complementing the role of space:

Space as form is therefore a construction that chooses only certain properties of "real" objecrs and only one or other possible levels of its own pertinence, to signify. It is obvious that all construction is an impoverishment and that the emergence of space makes most of the richness of expanse disappear. What it loses in concrete and lived fullness is compensated for, however, by multiple increases in signification: by becoming signifying space, it simply becomes another "object"...

Hence, contrary to what happens in the production of nonscientific discourse, where, for example the temporalization and the spatialization of the models are procedures of normal enunciation, semiotic models are considered to be achronological, realizable at all times and in all places, but independent of their realization. (Extract from The Social Sciences: a semiotic view, 1976-1990)

Greimas is especially known for his introduction of the Greimas semiotic square, discussed separately with respect to its possible three-dimensional implications (Square of opposition and Crossed quadrilaterals: the cognitive challenge? 2019).

Psychoanalytic appreciation of torus: Arguably the greatest attention to any sense of "being a torus" has been the focus of Jacques Lacan as a psychiatrist and psychoanalyst. His understanding in that respect is most accessibly articulated by Owen Hewitson (From the Bridges of KÃönigsberg: why topology matters in psychoanalysis, LacanOnline.com, 9 January 2015; Why Topology Matters in Psychoanalysis, LacanOnline.com, 1 March 2015). Points made there of relevance to this argument include:

  • Throughout the 1970s he busied himself with rings of string, weaving knots and lattices in an attempt to construct a topological model that represented the human soul. Referring to one topological form which especially fascinated him during these years â-- the Borromean knot â-- his biographer Elisabeth Roudinesco wrote that these were the years that Lacan lived on "planet Borromeo".
  • Topology offers a way of thinking about any particular space whatsoever. The human reality of our everyday lives can be considered as a topology, composed of points, arranged into neighbourhoods, within sets.
  • He was trying to represent the unconscious -- topographically, structurally, dynamically -- solely with recourse to models based on the purely two-dimensional and three-dimensional spaces of Euclidean geometry.
  • The hallmark of topology is that it deals with figures which retain their properties regardless of their deformations. You can bend and stretch and manipulate topological space as much as you like but it still retains its essential topologically significant characteristics.
  • For Lacan: "A torusâ-... is the structure of neurosis, in as much as desire can, from the indefinitely innumerable repetition of demand, be looped in two terms. It is on this condition at least that the contrabanding of the subject is decided â-- in this saying that is called interpretation." (L'Etourdit, 1972)
  • Human suffering itself exhibits the characteristics of a toric loop: "Man goes round in circles because the structure, the structure of man, is toric"... adding yet another extension to his proposition: "The world is toric" (Seminar XXIV, 14 December 1976).
  • For Lacan: "Topology is not 'designed to guide us' in structure. It is this structure." (L'Etourdit, 1972). Topology is about a delimitation of space. This could be signifying space, physical space, relational space â-- its topological properties persist regardless of the nature of the components of that space, or the deformations that space undergoes. We are instead talking about a space that can be conceived regardless of the elements that compose it.
  • This is perhaps what Lacan's legacy means in psychoanalysis today. At the end of his life, and at the end of his adventures in topology, we are left the "knots programme"e; as the new therapeutic paradigm.

More recently a very careful exploration of the signifiance of the torus, partially in the light of the above, has been made by the psychologist and philosopher Steven Rosen (2004, 2006, 2008, 20014), as discussed below in the concluding section.

Toroidal organization of music perception? Music is much valued for the coherence it offers in the lives of people. Of interest is therefore the degree to which this relates to engagement with the challenges of the times, as discussed separately (Implication of Toroidal Transformation of the Crown of Thorns: design challenge to enable integrative comprehension of global dynamics, 2011). Of particular relevance to this argument is the discussion there of Musical facilitation of integrative comprehension (2011), noting the rich consideration given to the cognitive role of music by Douglas Hofstadter (GÃödel, Escher, Bach: an Eternal Golden Braid, 1979).

The earlier discussion noted the manner in which the torus may be used as a representation of harmonic space. The results of psychoacoustic experiments of the inter-key relations of all major and minor keys can be represented geometrically on a torus (Carol Krumhansl and Edward Kessler, Tracing the dynamic changes in perceived tonal organization in a spatial representation of musical keys. Psychological Review, 89, 1989, 4, 334-368; Benjamin Blankertz, et al, Constant Q Profiles and Toroidal Models of Inter-Key Relations -- ToMIR, 1999). The keys are located on the surface of a torus, in which the circle of fifths and the parallel and relative relations between major and minor keys are represented.

As noted by Roger Shepard at that time::

Increasingly adequate accounts of musical pitch are provided by increasingly generalized, geometrically regular helical structures: a simple helix, a double helix, and a double helix wound around a torus in four dimensions or around a higher order helical cylinder in five dimensions. A two-dimensional "melodic map" of these double-helical structures provides for optimally compact representations of musical scales and melodies. A two-dimensional "harmonic map", obtained by an affine transformation of the melodic map, provides for optimally compact representations of chords and harmonic relations; moreover, it is isomorphic to the toroidal structure that Krumhansl and Kessler (1982) show to represent the psychological relations among musical keys (Geometrical Approximations to the Structure of Musical Pitch Psychological Review, 89, 1982, 4)

More relevant to the toroidal clarification sought here is the remarkable subsequent work of Dmitri Tymoczko (The Geometry of Musical Chords. Science, 313, 2006, pp. 72-74; A Geometry of Music, 2011). This focuses on the Tonnetz (tone-network) of musical tuning and harmony, namely a conceptual lattice diagram representing tonal space (Dmitri Tymoczko, The Generalized Tonnetz, Journal of Music Theory, 56, 2012, 1). In the latter he questions whether the (equal-tempered) Tonnetz is in fact truly toroidal, as argued in terms of Neo Riemannian music theory, among others. For Tymoczko:

Previous theorists have unanimously answered this question affirmatively, to the point where one would almost court ridicule to suggest otherwise. But our discussion has given us reason to be more circumspect... The assumption that the Tonnetz is toroidal is an example of music theorists implicitly attributing structure to their models over and above that contained in their mathematical formalism.

Neo-Riemannian theory typically assumes enharmonic equivalence, which wraps the planar Tonnetz graph into a torus. In the generalization of Tymoczko, many of the geometrical representations associated with neo-Riemannian theory are unified into a more general framework representing the voice-leading Tonnetz as a more complex "nontoroidal, three-dimensional structure whose individual octahedra are the duals of the cubes in "Cube Dance", and in which major, minor, and augmented triads are all on an equal footing". This is far from constituting an increase in comprehensibility in comparison with the relative simplicity of the images below (left and centre). That in the centre reflect reference in the literature to a "chicken wire" pattern of tones of the Tonnetz. One complexification, for example, potentially consistent with the arguments of Tymoczko, is the twisted toroidal knot of a hexagonal "chicken wire" (below right). The question is how the human brain is able to recognize coherence in complexity, whether consciously or intuitively in some way.

Emmanule Amiot. The Torii of Phases. International Conference on Mathematics and Computation in Music, 2012 *** chain mail

Inter-key relations of all major and minor keys
(toroidal representation)
 Neo-Riemannian Tonnetz
(animation of a toroidal view)		 Toroidal knot
(animation)
Geometric representation of inter-key relations in music Toroidal animation Neo-Riemannian Tonnetz Animation of toroidal knot
Derived from psychoacoustic experiments by Krumhansl and Kessler (1989) By Davidwbulger - Own work, Public Domain, Link Adaptation of a model developed
by Adrian Rossiter with Antiprism

Orbifolds, moonshine and wallpaper: Given the role of an orbifold in ordering musics (Dmitri Tymoczko, The Geometry of Musical Chords, Science, 313. 5783, 7 July 2007, pp. 72 - 74), it is intriguing that orbifolds have a recognized role with respect to the connectivity of "moonshine", as in the mathematics of moonshine theory theory (Michael P. Tuite, Monstrous Moonshine from Orbifolds, 1992) -- indeed a key figure in its development has been John Conway, who developed the notation by which orbifolds are characterized, as noted separately (Potential Psychosocial Significance of Monstrous Moonshine: an exceptional form of symmetry as a Rosetta stone for cognitive frameworks, 2007).

With the challenge of such "monstrous" symmetry to comprehension, is it possible that folk intuitions of "taming monsters" through music are indicative of powerful truths -- as discussed separately in relation to a possible periodic table of beliefs (Systematic Visual Representation of Musical Possibilities on an Orbifold, 2007)? Beliefs are then perhaps to be understood as the most generic forms of cognitive property. Curiously the animation on the right above, given the association to musical scales, is suggestive of the kind of archetypal "scaly monster" which is anathems to some religions (another example being even more suggestive in this respect).

Similarly intriguing is the potential role of orbifolds in justfying the improbable coherence of the UN's 17 Sustainable Development Goals. There is a strange possibility that the coherence of such a 17-fold pattern might be related, if only unconsciously, to the so-called wallpaper group (or plane symmetry group). This is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern (orbifold notaion of wallpaper groups). Such patterns occur frequently in architecture and decorative art, especially in textiles and tiles as well as wallpaper. It has been proven that there are only 17 distinct groups of possible patterns. Cynically it could however be argued that the 17 SDGs can be understood as "global wallpaper" for decorative purposes.

Collective relevance? Most curious is the seeming degree of irrelevance of the toroidal thinking of both psychoanalysts and musical theorists to the collective challenges of society and to comprehension of their representation. Despite the preoccupation with the unconscious in the case of some, there is no link (especially in topological terms!) to the space of discourse of such as John Ralston Saul (The Unconscious Civilization, 1995). In the case of psychoanalysis this neglect presumably follows from that of Freud in relation to Carl Jung's concerns with the collective unconscious. It could be argued that the irrelevance of the non-natural sciences to the natural sciences is echoed to a degree by the irrelevance of the collective to the psychological sciences with their focus on the individual.

A similar argument can be made with regard to the remarkable insights of music theorists in a period when music is proving ever more valuable to the lives and experience of people who have little interest in music theory and geometric representation of tones. If indeed humans order their engagement with music through a toroidal form of some kind, is this indicative of the possibility of using such insight to enable new modes of engagement with the strategic challenges of the times?

The value of being able to engage more fruitfully with strategies through music has been argued separately (A Singable Earth Charter, EU Constitution or Global Ethic? 2006). Why is this not envisaged in relation to the UN's 17 Sustainable Development Goals -- or to "doughnut economics"? How might the emerging insights of music theory contribute to this?

Seemingly also neglected in both cases is the experiential dimension -- beyond the capacity to provide "academic" explanations of the challenge for the individual. Whilst the world can be held to be "toric", the challenge is how the individual is able to experience reality in such terms -- however clear the topological modelling. This recalls the challenge for the individual (described above) of relating Flat Earth experience to the subtle understanding of global reality. It is in this sense that the elusively intuitive associations to the torus (through symbolic artefacts such as rings, circlets, and the like) hold such insights to a valuable degree. How is toroidal embodiment to be comprehended -- namely "being a torus"?

The sophistication of the arguments for various forms of representation of music itself offers a metaphor for the distinctions individuals may be free to make with regard to imagining "global" organization. The professional disagreements, whether between psychoanalysts or music theorists, also lend themselves to representation by the distinctions made by music theorists -- as distinctive tones and chords in a "polyphony" elusively and ironically echoing recognition of the traditional music of the spheres -- encompassing (professional) "discord". How are "discordantly" contrasting perceptions held by a representation, of greater or lesser sophistication, which must necessarily vary in dimensionality to accommodate the perceiver?

Even a planar Tonnetz would not constitute a meaningful framework for engagement with music by many. How might the higher dimensional representations -- whether toroidal or not -- be recognized as matching with that engagement? On the other hand is music unconsciously organized by the brain in such higher dimensional terms -- as suggested by the research noted above? Such questions are relevant to the meaning purportedly offered by "doughnut economics" and the credibility it is hoped to inspire (without any effort to enable this)?

So framed, imagining toroidal living may be an option for an individual. How might such toroidal embodiment be extended to the collective -- as has been done in a questionable manner, and to a strange degree, with respect to "global" and "globalization"? More challenging is how fundamental such toroidal organization might be (D. K. F. Meijer and J. H. Geesink, Is the Fabric of Reality Guided by a Semi-Harmonic, Toroidal Background Field? Neuroquantology, 17, 2019, 4; International Journal of Structural and Computational Biology, 2018).

Knottedness: The significance for Lacan of the Borromean rings with respect to individual pathology can be related to that of the legendary Gordian knot with which Alexander the Great was confronted. The Borromean-like form of the so-called Discordian Mandala could be said to exemplify such a knot, as argued separately (Mapping grossness: Gordian knot of governance as a Discordian mandala? 2016). The legend has been used by other authors:

Commenting on a session of the World Economic Forum, John Jullens argues that: It's as if the global economy is being strangled by a gigantic Gordian knot from which it cannot untangle itself (The Gordian Knot of Global Economic Growth, Strategy-Business, 15 October 2013).

In the absence of depictions of such a knot in cognitive terms, the implication that the dilemmas of global management might be explored topologically as a knot merits consideration in the light of the mathematical interest in the endless knot, the trefoil knot, the cinquefoil knot, and the septfoil knot (as shown below). This would be consistent with the psychological significance associated with knot topology by Jacques Lacan and R. D, Laing -- in respect of individual "self-governance". (R. D. Laing, Knots, 1972; Jean Michel Vappereau, Knot: the theory of the knot outlined by Jacques Lacan, Lacanian Works, July 1996). Especially intriguing with respect to the higher dimensionality of string theory, is the choice of "string" as a metaphor, given the contrasting importance attached to "knot" as a metaphor by psychoanalysis.

Challenges to imagining toroidal life?
Trefoil knot Cinquefoil knot Septfoil knot Seifert surface
bounded by a set of Borromean rings
Cinquefoil knot Septfoil knot Seifert surface bounded by a set of Borromean rings
By Jim.belk Animation: MichaelFrey (talk) - Own work, Public Domain, Link  By Jim.belk - Own work, Public Domain, Link By Jim.belk - Own work, Public Domain, Link Reproduced from Wikipedia
[more images]

Given the widespread understanding of knots, and their widespread use as metaphors of strategic and existential challenges, their representation is of interest in enabling imaginative reframing -- especially if a toroidal form is understood as twisted into a knot. It is in that sense that a Seifert surface is of potential interest as a representation of the Gordian knot of sustainable development, especially in the light of the "planetary boundaries". The boundary of such a surface is a given knot or link. A range of more complex illustrations is offered by Carlo H. Séquin (Torus Immersions and Transformations. University of California at Berkeley, 2011).

Could the simple representation of "doughnut economics" be more appropriately addressed in strategic terms if its degree of knottedness was recognized -- if only in experiential terms? It is surely appropriate to argue that society is in a "knotted" condition for which insights from Lacan and others would offer indicators (whatever their limitations). More challenging is the possibility that collective organization of "global governance" merits understanding in the light of the Borromean ring condition -- as argued separately (Engaging with Elusive Connectivity and Coherence: global comprehension as a mistaken quest for closure, 2018). The argument was adapted there to the symbolic Flag of Europe (Imagining the Flag of Europe otherwise?; Borromean challenge to comprehension of any trinity?).

Knot vs Torus: The interplay between "knot" and "torus" merits recognition in the light of the earlier argument regarding the distinction between 2-sphere and 3-sphere. Is the knotted thread to be considered as one-dimensional, 2-dimensional, or 3-dimensional -- namely is it a hollow tube, potentially of toroidal form? If that torus is effectively a container or conduit, however knotted, what circulates in experiential terms within that conduit? The question can be variously discussed (Circulation of the Light: essential metaphor of global sustainability? 2010; Enactivating a Cognitive Fusion Reactor: Imaginal Transformation of Energy Resourcing (ITER-8), 2006).

A particular focus to the institutional challenge is given by widespread interest in the Triple Helix model of innovation, namely to the set of interactions between academia, industry and governments, to foster economic and social development. Curiously little attention within that framework is given to the extent to which the three helical threads could be "tubular" rather than notionally thread-like. More curious is that little attention is given to what happens at the two "ends" of the helix -- an important consideration in the case of the double-helix form of DNA, by which the model was originally inspired. Of particular value however is the focus on enabling "innovation" -- with its cognitive and experiential implications, otherwise divorced from the preoccupations of the natural sciences.

An obvious question is how any reconciliation of the "doughnut economics" perspective with the Triple Helix perspective is to be comprehended -- if the latter is itself to be understood as toroidal in some manner. Both are necessarily susceptible to various forms of knottedness, at least as (mistakenly) perceived and/or experienced. Aspects of the question are addressed separately (Requisite curvature: reconciling the Triple Helix, the Triskelion and the Borromean condition, 2018).

Toroidal embodiment: Of some relevance is the sense in which an individual necessarily embodies toroidal forms, if only through the human digestive system, and most notably the gastointestinal tract. In that sense a human being is very much a torus -- or rather a segment of a torus. Other portions of the torus are external to the human body, appropriately indicative of the manner in which the human being is embedded in the environment, giving reality to preoccupation with recycling and dependence on it. Many animals can be seen in such terms, especially the simplest -- as with the earthworm most obviously.

Missing is insight into the psychosocial analogue, given the manner in which "digestion" of information is part of the process of "innovation". A human being, as conventionally conceived, is then a "slice out of a cycle" -- a cyclic which tends to go unrecognized.

It is appropriate to note that references to "toroidal embodiment" are typically totally divorced from psychosocial considerations -- as might be expected. Ironically, most of the many references are to patents. At another extreme, that insight is however echoed in poetic terms (Antonin Tuynman, Technovedanta: a technological meta-knowledge philosophy beyond science and religion). In addition to its real world correlations, the torus has also been used speculatively to illustrate certain concepts of the so-called "subtle energy" worlds, most notably in the esoteric study of sacred geometry (The Torus in Metaphysics, Harmonic Resolution; The Metaphysics of Our Spirituality within the Universal Toroidal Field, OpenHand, 2018).

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