Transforming vehicles of identity between global and toroidal forms

Visualization in 3D of Dynamics of Toroidal Helical Coils (Part #12)

Globality: On a spherical world, intensely preoccupied with "globalization", it might be asked how a torus could be understood to be of any relevance. It is indeed already a challenge to comprehend the full significance of globality -- despite the irony of the widespread focus on the balls of the variety of ball games. Supposedly these provide a degree of complex experiential insight into globality. Much of geometry can also be understood as a quest for more precisely articulated understanding of it (Metaphorical Geometry in Quest of Globality -- in response to global governance challenges, 2009).

That exploration gave focus to the question of what geometry serves as a vehicle of personal or collective identity, whether it be points, lines, planar surfaces or the like, as explored in more detail (Engaging with Globality -- through cognitive lines, circlets, crowns or holes, 2009). Clearly there is a sense in which people identify with a "point" when making one -- or with a "line" when pursuing a line of argument. The identification with a surface is evident in the case of land ownership (with regard to which there is so much conflict), or with the cubic volumes of a dwelling or place of work. More generally is it somehow with a "sphere" that the wholeness of personal identity is associated -- and hence a degree of resonance with globality?

In a period of crisis, scoring a "point" is clearly of relatively limited value, as exemplified by blame-games (Collective Mea Culpa? You Must be Joking! Them is to blame, Not us! 2015). Framing strategy as pursuit of a particular "line" might well be as questionable as iconic imagery of a witch riding a broomstick or that from Dr Strangelove of Aircraft commander Major T. J. Kong riding the bomb down. Control of "territory" and space, could be questioned as a cognitive analogue to the military approach to full-spectrum dominance (Bruce Gagnon. The Pentagon's Strategy for World Domination: full spectrum dominance, from Asia to Africa, Global Research, 2014)/

The issue can be presented more provocatively through questioning the possibility of associating any "plan" -- especially when conventionally framed by a spreadsheet -- with an understanding of globality (Spherical Accounting: using geometry to embody developmental integrity, 2004; Adhering to God's Plan in a Global Society: serious problems framed by the Pope from a transfinite perspective, 2014). Given the mysterious nature of holes, speculative provocation can be extended further (Is the World View of a Holy Father Necessarily Full of Holes? Mysterious theological black holes engendering global crises, 2014).

Toroidal vehicles: One indication of the relevance of a torus is that the planetary globe travels through (and defines) a toroidal tunnel around a sun vital to life on that globe. The torus thus becomes of significance from a temporal perspective -- an orbit ensuring the systemically healthy dynamic of the seasons. That the solar system is moving as a whole, with the planetary orbit then tracing a helix, is a different matter.

The interplay between that orbit and the central sun has been a theme of reflection, symbolism and metaphor since the beginning of civilization. A striking example is provided by the symbol of the Ouroboros and its relation to the sense of eternal return. Of potential relevance however is any shift from preoccupation with globality and its distorting representation on planar projections, to a perspective which encompasses the toroidal orbit and the mysterious role of the sun.

In this period of increasing recognition of crises, a question is within what vehicle (or life raft) can crises be fruitfully navigated?. A powerful toroidal image has been elaborated in the form of a doughnut by Oxfam (Kate Raworth, A Safe and Just Space for Humanity: can we live within the doughnut?, Oxfam, 2012). This serves as a cognitive device to hold an understanding of planetary resource boundaries -- another geometrical notion. That argument can however be transformed into one applicable to cognitive and psychosocial boundaries (Exploring the Hidden Mysteries of Oxfam's Doughnut: recognizing the systemic negligence of an Earth Summit, 2012; Recognizing the Psychosocial Boundaries of Remedial Action: constraints on ensuring a safe operating space for humanity, 2009).

Requisite toroidal complexity: A doughnut could however be recognized to be excessively simplistic in a context calling for requisite complexity in cybernetic terms. Hence the argument above in terms of a complex of toroidal helices in dynamic relation to one another. Can meaning be feasibly associated with such complexity and mapped fruitfully onto it to provide a vehicle for identity? Clearly visualization offers a means of rendering comprehensible complexity which is otherwise articulated primarily in mathematical equations for the few.

As noted above, one focus to the argument can be expressed in terms of the requisite toroidal design of a nuclear fusion reactor -- from which it is claimed for public relation purposes that that the "energy of the sun" can be readily and efficiently obtained. Does this suggest the case for exploring the design of an analogous "reactor" of relevance to psychosocial energy -- whether individually or collectively -- as separately argued (Enactivating a Cognitive Fusion Reactor: Imaginal Transformation of Energy Resourcing (ITER-8), 2006)? Given the reference to the Ouroboros, it is delightfully ironic that a particular preoccupation in the case of the nuclear reactor is framed as containing the "snake-like" dynamics of the circulating plasma.

Topology: Reference to the investment in such complex technology based on the torus suggests the merit of further exploration of the interplay between torus and sphere -- succinctly clarified by the following animations.

Animations of toroidal complexification
Torus-to-Sphere transformation	Trefoil knot	Clifford torus
Animations reproduced from Wikipedia

Made by User:Kieff		Made by Jason Hise

Mathematics, especially topology, has a wide array of insights of potential relevance in this regard -- further to the pointers highlighted above. What might these suggest with respect to optional vehicles for individual or collective identity?

Especially intriguing is the sense of a cognitive challenge somewhat analogous to achieving orbit of the globe -- both conceptually (through an understanding of globality and spherical geometry) and through the technology required. As argued, "technomimicry" may well offer a pathway for creative innovation.

Appropriateness of a hypersphere as a vehicle for identity? As noted above, every reason is offered to constrain any sense of identity to the simplest geometric forms in 2D or 3D. Collective may embody these in heraldic signs, symbols on flags, or statues. In addition to the insights of mystics, this constraint has been variously challenged by mathematicians (Ian Stewart, Flatterland, 2001; Dionys Burger, Sphereland: a fantasy about curved spaces and an expanding universe, 1965). Especially relevant is the question if Ron Atkin (Multidimensional Man: can man live in three dimensions? 1982).

The conceptual adventures of mathematicians and astrophysicists are an invitation to associate personal and collective identity with ever more complex topology. The hypersphere (especially the 3-sphere) is one example of the space within which a Concordian Mandala might be more appropriately located to enable any such sense of identity. The possibility has notably been highlighted by Mark A. Peterson, pointing out that language in Dante's Divine Comedy suggesting that he visualized his universe in the same way (Dante and the 3-Sphere, American Journal of Physics, 47, 1979; S. Lipscomb, Art Meets Mathematics in the Fourth Dimension, Springer, 2014, chapter 2; and extensive discussion thereof, Dante and the 3-Sphere, Science and Philosophy Chat Forums). Perhaps appropriately, the 3-sphere is also known as a glome -- a term employed for the fictitious kingdom of the novel of C. S. Lewis (Till We Have Faces, 1956).

Despite some articulation of perceptions for those living in such geometry, the principal inadequacy of the insights of mathematicians -- is that they naturally tend to avoid any exploration of the embodiment of identity therein or by such spaces. This is the challenge articulated by George Lakoff and Mark Johnson (Philosophy in the Flesh: the embodied mind and its challenge to western thought, 1999) and in the subsequent argument (George Lakoff and Rafael Nuñez, Where Mathematics Comes From: how the embodied mind brings mathematics into being, 2001).

It is in this sense that any study of the organizing function of the brain is especially valuable, as with that of Arturo Tozzi and James F. Peter (Brain Activity on a Hypersphere). The authors note that:

Current advances in neurosciences deal with the functional architecture of the central nervous system, paving the way for "holistic" theories that improve our understanding of brain activity. From the far-flung branch of topology, a strong concept comes into play in the understanding of brain signals, namely continuous mapping of the signals onto a "hypersphere": a 4D space equipped with a donut -- like shape undetectable by observers living in a 3D world. Here we show that the brain connectome may be regarded as a functional hypersphere.... We anticipate that this introduction to the brain hypersphere is a starting point for further evaluation of a nervous fourth spatial dimension, where mental operations take place both in physiological and pathological conditions. The suggestion here is that the brain is embedded in a hypersphere, which helps solve long-standing mysteries concerning our psychological activities such as mind-wandering and memory retrieval or the ability to connect past, present and future events.

Of particular relevance to the above argument, the authors offer the following images illustrating the structure of a hypersphere (or glome). They note that the shape of the glome is ever changing, depending on the number of circles taken into account (in the left hand image) and their trajectories (see video by Niles Johnson, A visualization of the Hopf fibration). The other images illustrates depiction of a hypersphere as two spheres glued together along their spherical boundary, giving rise to a Clifford torus (see animation above, showing a stereographic projection of a Clifford torus performing a simple rotation through the xz plane). The image on the left is a suggestive illustration of how periodicity might be cognitive embodied, whether in the case of the organization of music or as the elements of the periodic table.

Alternative representations of a hypersphere
Arrangement of circles in 4D	Clifford torus	2 Spheres glued together

Tozzi and Peter also make use of a single image (one phase from Johnson's video, also figuring in the Wikipedia commentary on Hopf fibration), presented in the following animation using changing colour values as a suggestion of the dynamics within which brain organization might be associated. Use of a similar technique can be used with respect to the 6-dimensional Calabi-Yau manifold of significance to superstring theory, with which the extra dimensions of spacetime are conjectured to be associated, as well as mirror symmetry. Given its relevance to the branes of astrophysics, such speculation has been explored in terms of hypothetical correspondence between global brane and global brain (Global Brane Comprehension Enabling a Higher Dimensional Big Tent? 2011).

Animations suggestive of higher-dimensional brain functioning
Hypersphere (Hopf fibration)	Calabi-Yau manifold

Such triggers for the imagination are a reminder of the unexplored wealth of 4D polytopes -- otherwise known as polychora (Comprehending the shapes of time through four-dimensional uniform polychora, 2015).

The typical approach of mathematics is also somewhat misleading in that it appears to emphasize static organization of structures in 4D (effectively snapshots) when the dynamics may be especially significant to embodiment of identity therein. It is in this sense that explorations of the organization of music by the brain is especially valuable (Dmitri Tymoczko, The Geometry of Musical Chords. Science, 313, 5783, 7 July 2006, pp. 72-74; A Geometry of Music: harmony and counterpoint in the extended common practice, 2011). This is discussed separately (Engaging creatively with hyperreality through music, 2016).

It is therefore of considerable interest to note the results of psychoacoustic experiments by C L Krumhansl and E J Kessler (Tracing the dynamic changes in perceived tonal organization in a spatial representation of musical keys, Psychological Review, 1982) of the inter-key relations of all major and minor keys can be represented geometrically on a torus -- as shown by Benjamin Blankertz, Hendrik Purwins and Klaus Obermayer (Constant Q Profiles and Toroidal Models of Inter-Key Relations -- ToMIR, 1999) in the following image.

Geometric representation of the inter-key relations
of all major and minor keys
(derived from psychoacoustic experiments by Krumhansl and Kessler)

Geometric representation of the inter-key relations of all major and minor keys of music

For other relevant publications see
Music Cognition Laboratory

Following the arguments of Douglas Hofstadter (I Am A Strange Loop, 2007; Gödel, Escher, Bach: an Eternal Golden Braid: a metaphorical fugue on minds and machines in the spirit of Lewis Carroll, 1979), the challenge for collective organization may even be understood as the organization of such loops in "hyperreality" as experienced (Sustaining a Community of Strange Loops: comprehension and engagement through aesthetic ring transformation, 2010). Hence the case for articulating patterns of global organization through music (A Singable Earth Charter, EU Constitution or Global Ethic? 2006).

Arguably there is then a case for appreciating the forms of "hypercomprehension", as can be variously argued:

Given that one of the topological methods of "construction" of a 3-sphere variant of a hypersphere is understood in terms of "gluing together" the boundaries of a pair of 3-balls, the cognitive paradox of relevance to strategic comprehension can be highlighted by framing the "glue" in terms of a Möbius strip as depicted below. The image has been discussed separately (Embodying Strategic Self-reference in a World Futures Conference: transcending the wicked problem engendered by projecting negativity elsewhere, 2015).

30 Future Global and Conferencing Challenges for Humanity
(self-referential adaptation of the 15 Global Challenges of the Millennium Project)

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