Complementary visual patterns: Ouroboros, Mbius strip, Klein bottle

Cognitive Osmosis in a Knowledge-based Civilization (Part #3)

The focus in the following examples is on some form of injestion dynamic. However "inappropriate", it is of course of great interest to recognize examples with respect to ejestion in a later portion of what is necessarily to be understood as a cycle. The patterns presented here are all based on the torus, derived from it, or associated with it in some way. In its more commonly recognized form as a ring or halo, its appreciation as a much-valued symbol should not be forgotten.

Of particular relevance is the topological relation between the torus and the sphere -- given the importance currently associated with globalization and the challenges of the spherical globe as a widely indicated symbol. It is in this respect that the seeming improbability of sphere eversion (without tearing or creasing) merits attention. This possibility was first proven by Stephen Smale (Classification of Immersions of the Two-Sphere, 1958). The process is demonstrated in various videos (Sandra Streams: Turning a Sphere Inside-outt, YouTube, 1994; Justin Bruckman, How to Turn a Sphere Inside Out, YouTube, 2007).

Given the fundamental importance of the symbol, the implications of eversion for any form of global understanding and insight are another matter, as separately argued (World Introversion through Paracycling: global potential for living sustainably "outside-inside", 2013).

Whether the symbolic focus is through the sphere or the torus, of particular interest is the sense they offer of enveloping -- possibly through "graspng" and "grabbing". Related processes include:, englobing, encompassing, enwrapping, enclosing and encircircling. The latter is highlighted in the phrase "circling the wagons". Such processes variously imply a sense of embodiment -- with the possible further implication of entitlement. A particular sense of the process is offered by the processes in the game of go, whose significance with respect to the promise of artificial inelligence have been recently highlighted (AlphaGo Zero: Google DeepMind supercomputer learns 3,000 years of human knowledge in 40 days, The Telegraph, 18 October 2017).

Torus -- its inversion and eversion: Standard tori fall into four categories. namely the relatively familiar ring torus, a horn torus with the rotating circle touching the axis of rotation, or a self-intersecting spindle torus, with the sphere considered as a degenerate case. Inversions of the standard tori are known as cyclides (or Dupin cyclides), of which the ring cyclide is especially relevant to the following visualizations. Others include the horn cyclide and the spindle cyclide.

Animation of torus-sphere transformation
(Reproduced from Wikimedia)

Discussed in:

Comprehension of Requisite Variety for Sustainable Psychosocial Dynamics: transforming a matrix classification onto intertwined tori (2006)
Enabling Wisdom Dynamically within Intertwined Tori: requisite resonance in global knowledge architecture (2012)
Visualization in 3D of Dynamics of Toroidal Helical Coils -- in quest of optimum designs for a Concordian Mandala (2016)
Toroidal configurations as fruitful loopery (In: Changing Patterns using Transformation Pathways, 2015)
Cognitive implication of toroidal forms and dynamics (2011)
Toroidal cognitive container? (2011)

Animation of torus inversion/eversion

Videos showing torus eversion (inspired by the possibility of sphere eversion):

Arnaud Chéritat: Torus eversion: turning the torus inside out (YouTube, 2017)
Greg McShane: Torus eversion (YouTube, 2008)

Images on the left from Carlo H. Sèquin (Torus Immersions and Transformations). See commentary and imagery on smooth torus eversion (Mathematics Stock Exchange, 2013)

Ouroboros (Uroboros): The ouroborus symbol has been of fundamental importance to alchemy and continues to be so, as extensively studied by Carl Jung (Psychology and Alchemy, 1944; Mysterium Coniunctionis; an inquiry into the separation and synthesis of psychic opposites in alchemy, 1970) and by Marie-Louise von Franz (Alchemical Active Imagination, 1997). The ouroboros is central to a current exhibition on Never Ending Stories: the loop in art, film, architecture, music and cultural history (Kunstmuseum Wolfsburg) as summarized and illustrated by Joobin Bekhrad (The Symbol that Spanned Millennia, BBC: Art History, 4 December 2017).

The BBC summary uses as its primary image one of a double dragon-serpent. This features with related images in an extensive account by Iona Miller (Jungian Genealogy: Jung on the Serpent). The coloured image, variously available, has been reversed and depicted in a black-white version below.

Image of double dragon-serpent Ouroboros
Reverse colour	Black-and-white

Original image from Clavis Artis, an 18th century manuscript on alchemy attributed to Zoroaster (Biblioteca dell'Academia Nazionale dei Lincei, Rome)

The symbol continues to be widely valued in jewellry and iconography, with many imaginative variants in torcs, pendants and clasped bracelets. Of considerable significance in this respect is its appeal as a symbol by-passing a plethora of explanations and commentary. However it is the manner in which the intertwining of its elements is explored that merits recognition of an intuitive appreciation of forms of complexity understood through symmetry. Some forms recall the widely recognized symbol of infinity with which the ouroboros is closely associated. That on the right below recalls the association with the dream of August Kekulé enabling the discovery of the structure of the benzene molecule so fundamental to organic life.

Contrasting depictions of the ouroboros -- and its associations with infinity symbol

The relation between two dragons in variants of the symbol can be usefully compared with continuing popular appreciation of the dragon dance of Chinese culture. The dragon is often led by a person holding a spherical object representing a pearl. There are many depictions of intertwined dragons in pursuit of a pearl. Other cultures have engendered comparable symbols as with references to a pair of serpents. The omphalos in the temple of Apollo in Delphi is associated with the myth of the meeting point of two eagles sent by Zeus. Images of the Mayan deity Quetzalcoatl tend to be considered comparable to the ouroboros.

The coiling of two serpents around an axial rod forming the caduceus dates from the Mesoptamian deity Ningishzida and has associations with trade, eloquence, negotiation, alchemy and wisdom. [As a symbol of healthcare, it has been mistakenly appropriated as a symbol of medicine in modern times -- confused with the Rod of Asclepius of Greek mythology, the internationally recognized symbol of medicine, but having only a single serpent.]

The ouroboros has been used to frame considerations of cosmology, notably in the form of a frequently reproduced image by Nancy Ellen Abrams and Joel R. Primack (Cosmic Uroboros, 2006; The View From the Center of the Universe, 2007). For a civilization in crisis globally this suggests a final cognitive frontier -- in several senses (Being the Universe : a Metaphoric Frontier, 1999; Metaphors To Die By: correspondences between a collapsing civilization, culture or group, and a dying person, 2013). This would follow from the arguments of Henryk Skolimowski (The Participatory Mind: a new theory of knowledge and of the universe, 1994).

Related issues are discussed in:

With respect to intuitive recognition of the challenges of some form of "cognitive osmosis", it could be argued that this is evident to a degree in current interest in a Triple Helix concept in sustaining innovation -- as promoted by the Triple Helix Association. This is partially inspired by the double helix of DNA -- with which both the single-serpent and double-serpent varieties of DNA can be readily compared (Jeremy Narby, The Cosmic Serpent: DNA and the origin of knowledge, 1998). As argued by Jamie Davies (A Closed Loop, Aeon, 26 September 2014):

The DNA helix gave 20th-century biology its symbol, but the more we learn, the more life circles back to an older image... The helix is too well-established an icon to be deposed any time soon. And yet, a simple loop would be a much more universal symbol of how life works at all of its scales and levels. Perhaps the Ouroboros, beloved of gnostics and alchemists, has been an ideal symbol waiting in the wings for centuries: there can surely be no more evocative symbol of feedback than a snake growing by devouring its own tail.

The Triple Helix approach has been recently extended to advocacy of Quadruple Helix and Quintuple Helix models. To the extent that such helices constitute a contemporary recognition of cognitive "serpents" between which communication has been inadequate, the ouroboros may well be understood as potentially made up of multiple strands between which osmosis of some kind is required (as further discussed below).

Möbius strip and Klein bottle: The intricacies of some variants of the ouroboros symbol notably anticipate the paradoxes of the Möbius strip. The torus itself is discussed extensively in relation to the ouroboros, the Möbius strip and the Klein bottle by Steven Rosen (Science, Paradox, and the Moebius Principle: the evolution of a "transcultural" approach to wholeness, 1994; Dreams, Death, Rebirth: a multimedia topological odyssey into alchemy's hidden dimensions, 2014). This concern includes recognition that the wholeness implied by the ouroboros is fundamentally characterized by paradox (Steven Rosen, Wholeness as the Body of Paradox, Journal of Mind and Behavior, 18, 1997).

Rosen is especially articulate with respect to the process of proprioception in relation to phenomenology and -- following the inspiration of David Bohm -- with the implications for proproceptive dialogue (Practicing Proproceptive Dialogue, Embodying Cyberspace). Proprioception is discused further below.

Möbius strip
		Discussed in: Psychosocial Work Cycle Beyond the plane of Möbius (2007) Visualization: quadrant systems / Möbius strips / Klein bottles (2007) Cognitive implication of Mobius strip (2011)
Klein bottle
		Discussed in: Intercourse with Globality through Enacting a Klein bottle: cognitive implication in a polysensorial "lens" (2009) Strategic Complexity -- Attracting Consensus: Klein is beautiful -- Sustaining identity in time (2011)
Animation by Konrad Polthier (Imaging Maths: inside the Klein bottle, +Plus Magazine, September 2003; [click for animated version or explore java applet]	Two-dimensional representation of the Klein bottle immersed in three-dimensional space (reproduced from Wikipedia)
Toroidal adaptation of Klein bottle (stretched image from Wikipedia)