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23rd October 2011 | IN PROCESS

Topological Clues to a Memorable 12-fold Systemic Pattern


- / -

Imagining a dynamic configuration of relevant "psychopter technologies"
Cognitive implication of  toroidal forms and dynamics
Cognitive implication of spherically symmetrical polyhedra
Cognitive implication of tesseract and related uniform polytopes
Cognitive implication of Mobius strip
Cognitive implication of drilled toroids
Cognitive implication of Klein bottle

Imagining psychopter operating "principles"
Imagining embodiment of psychopter control
Interlocking cycles enabling psychopter operation

Annex to Eliciting a 12-fold Pattern of Generic Operational Insights: Recognition of memory constraints on collective strategic comprehension (2011)


The main paper argues the need for Experimental clues to a memorable 12-fold systemic pattern (2011) using as a common framework the 12-fold pattern elaborated by Arthur M. Young  with a view to the design of a hypothetical "psychopter" -- a "winged self" (The Bell Notes: A Journey from Physics to Metaphysics, 1979). The possibility was introduced in an earlier paper (Engendering a Psychopter through Biomimicry and Technomimicry: Insights from the Process of Helicopter Development, 2011). Young was the designer of Bell Helicopter's first helicopter, the Model 30, and inventor of the stabilizer bar used on many of Bell's early helicopter designs. His approach to the design of a "psychopter" is inspired by his subsequent aspiration, through generalizing from those technical challenges.

The helicopter had been developed as a result of what subsequently became known as biomimicry (Janine Benyus (Biomimicry: innovation inspired by nature, 1997). The field had emerged in 1950 and was formalized in 2002 as the research network BIONIS: The Biomimetics Network for Industrial Sustainability. As previously argued, Young's subsequent initiative could be fruitfully understood as "technomimicry" -- seeking insight from previous technical discoveries as biomimicry had sought insights from nature. As an exercise in innovative thinking in quest of previously unrecognized capacities, Young's imputation of significance to the elements of the 12-fold pattern in the light of aerodynamics is seen as an indication of a method which can be used with other forms. Here the focus is on topological forms as a sources of clues to the nature of that 12-fold pattern and the relationships between its elements.

In the main paper, emphasis was placed on the manner in which memory and comprehension constrain the scope of any set of operational insights. Given the experiential nature of such insights -- as illustrated by the case of piloting a vehicle -- the complexity of the forms through which they may be variously identified and communicated are recognized as a potential barrier to individual and collective learning and to communication. The following is therefore best understood as a quest for cognitive clues which may be helpfully elicited through the subtleties of topological formalism -- and especially through visualizations offering a sense of the complexity that piloting a "psychopter" may well require. It is a quest for forms of greater complexity, commensurate with the cognitive challenge, with which psychosocial meanings may be associated -- if only for mnemonic purposes.

The main paper emphasizes (in an annex) that other 12-fold sets of clues may be usefully explored for mnemonic purposes, especially by those who are alienated by the forms below (Checklist of 12-fold Principles, Plans, Symbols and Concepts, 2011). .

Imagining a dynamic configuration of relevant "psychopter technologies"

Following Young's assumptions that the schema offers clues to the nature of the operation of the "psychopter" he had imagined, the framework can be tentatively used to interrelate "technologies" potentially indicative of its operation.

The approach taken below is to insert clues to such technologies into the 12 cells of the framework (see Adapting psychosocial insights from learning/action cycles in the main paper), such that the framework as a whole functions as a mental model -- a conceptual holding pattern, however crude that may prove to be. In this sense the framework is not a framework from Young, but rather one suggested by his reflections on the operation and learning challenges of the helicopter -- encouraged by his own mnmonic choices. It is intended as suggestive of "cognitive styles" that might be associated with the hypothetical technologies of the psychopter -- rather than being specifically related to the forms indicated. The preoccupation is not a matter of those forms but rather the patterns of cognitive operations suggested by those forms.

As a design initiative, the focus on "structures" or "patterns" could be understood as a speculative exploration of the "geometrical" path indicated by Christopher Alexander (Harmony-Seeking Computations: a science of non-classical dynamics based on the progressive evolution of the larger whole, International Journal for Unconventional Computing (IJUC), 5, 2009) -- and discussed separately (Harmony-Comprehension and Wholeness-Engendering: eliciting psychosocial transformational principles from design, 2010).

The symbols, associated with the physical principles involved in the operation of a helicopter (according to Young) increase in complexity from the upper left portion of the original table to the lower right. To facilitate the following comments, the items added into that framework are presented separately below. The cognitive complexity of "psychopter operation" is then to be understood as increasing in complexity from the upper left to the lower right of the simplified table below. It should be stressed that the purpose is to give a sense of the possible set of cognitive operations as a whole -- recognizing that many arguments and insights could lead to amendment and refinement of the presentation and the items in it.

Inspiration for the development of a "psychopter" might be then be found in a potentially fruitful combination (or configuration) of the following "clues". These are organized as four clusters corresponding to the columns of the original framework -- and inserted into the adapted variant below:

"enabling structure"
(into which whole may be mapped)
"transformative dynamic"
(classes of
"imaginative implication"
(confrontation with possibility /
decision / risk / resonance)
"embodiment of insight "
(identification / cognitive dimension )
(continuous surface)
process container
(cognitive fusion)
[4a] Mobius strip
(unconventional / paradox)
propulsion / wings
"betwixt and between"
Spherically symmetrical polyhedra
(ordered structural integrity )
vector equilibirum pumping
(re-configuration of map )
Drilled toroids (truncated cube
imbue choice / possibility
(pattern of possible change)
(projection into extra dimension)
(inside-out / outside-in)
Klein  bottle
digestion / engender
(stepping into the mirror)
(laying down the path in walking)

Columns: In the discussion which follows, the table is usefully understood as composed of six sets derived from the first two columns of each row, and the last two columns of each row:

Rows: With respect to the organization of the rows, it is appropriate to note the potential cognitive implications of:

Cognitive implication: Given the use of each of these forms as templates on which cognitive significance can be variously "hung" (as "enabling structures"), their comprehension as "maps" then becomes an appropriately suggestive challenge -- especially when a "transformative dynamic" is in operation. It is for this reason that cognitively:

Cognitive implication of toroidal forms and dynamics

[1a] Enabling structure: Toroidal forms, as derived from the circle and intimately related to the sphere, are used here as a surface with which cognitive content can be associated -- onto which it can be variously mapped. Examples, as partially illustrated below, include:

Kalachakra Mandala
(from Wikipedia, enlargement)
(from Wikipedia)
Global Underground Map
(see enlargement)
Kalachakra Mandala Torus Global Underground Map

[1b] Transformative dynamic: Complementing its role as a mappable "surface", a toroidal form is significant both as a "container" for processes involving flow, and for the vortexes associated with certain flow patterns, as for example:

Animation of blame game
of Knight's moves on torus
Sphere / Torus
Toroidal design of fusion reactor
(from Wikipedia)
Animation of blame game of Knight's moves on torus Toroidal design of fusion reactor

Cognitive implication of spherically symmetrical polyhedra

[2a] Enabling structure: Spherical configurations of interlocking cycles, as mentioned above, give rise to the emergence of spherically symmpetrical polyhedra of strategic relevance (Configuring Strategic Dilemmas in Intersectoral Dialogue, 1992; Representation of Issue Arenas on Icosidodecahedral Net, 1992). The approach can be extended to the configuration of organizations (Towards Polyhedral Global Governance: complexifying oversimplistic strategic metaphors, 2008;  Polyhedral Empowerment of Networks through Symmetry: psycho-social implications for organization and global governance, 2008).

The approach can be further extended to human values (In Quest of Engaging Values: context of the Human Values and Wisdom Project, 2008; Coherent Value Frameworks: pillar-ization, polarization and polyhedral frames of reference, 2008; Psychodynamics of Collective Engagement with Polyhedral Value Configurations, 2008). The argument has also been presented in poster form In Quest of a Strategic Pattern Language: a new architecture of values [also available in PDF].

The cuboctahedron was a central focus of the geometrical and design reflections of R. Buckminster Fuller (Synergetics: Explorations in the Geometry of Thinking, 1975; Synergetics 2: Further Explorations in the Geometry of Thinking, 1979). He named it the vector equilbirum.

Rhombicosidodecahedral representation
of articles of
Universal Declaration of Human Rights
Icosidodecahedral representation of issue arenas
on a net for Earth Summit (1992)
(see Spherical Representation of Icosidodecahedral Net of Strategies)
Arrangement of 12 Archimedean polyhedra
as a cuboctahedron, around a truncated tetrahedron
(from Keith Critchlow, Order in Space, 1969, p. 39).
[see interactive enlargement and commentary]
Rhombicosidodecahedral representation of articles of Universal Declaration of Human Rights Arrangement of 12 Archimedean polyhedra as a cuboctahedron, around a truncated tetrahedron

[2b] Transformative dynamic:

(Dynamic Exploration of Value Configurations: Polyhedral animation of conventional value frameworks, 2008)

The argument is summarized in a final paper (Embodying Values Dynamically through Alternation: integrating sets of polarized static values through indicative metaphor, 2008).

System duality represented
by transformation between polyhedra

Icosahedron morphing into Dodecahedron
(made with Stella Polyhedron Navigator)
Bucky pumping
Morphing icosahedron  

(Vector Equilibrium and its Transformation Pathways, 1980)


Cognitive implication of tesseract and related uniform polytopes

Related uniform polytopes

[3a] Enabling structure:

Representation of 4D tesseract
(reproduced from Wikipedia)
Representation of 4D rectified tesseract
(reproduced from Wikipedia)
Representation of 4D truncated tesseract
(reproduced from Wikipedia)
Representation of 4D tesseract Representation of 4D rectified tesseract Representation of 4D truncated tesseract

[3b] Transformative dynamic:

A 3D projection of a 4D tesseract
performing a simple rotation about a plane which bisects the figure
from front-left to back-right and top to bottom
(reproduced from Wikipedia)
A 3D projection of a 4D tesseract
performing a double rotation
about two orthogonal planes
(reproduced from Wikipedia)
3D projection of a 4D tesseract 3D projection of a 4D tesseract

Cognitive implication of Mobius strip

More provocatively it could be hypothesized that special new demands might be placed on the "pilot" of such a vehicle -- as was discovered in the case of the helicopter

Imagining embodiment of psychopter control

imagination au pouvoir

With respect to the last two columns of the above table

: As an apparently simple, readily comprehensible, topological form, the Möbius strip is mentioned here in anticipation of its further significance below. It is relevant to the construction of a Klein bottle as a much more complex form.

[4a] Imaginative "Initiation":


[4b] Embodiment:

Cognitive implication of drilled toroids

[5a] Imaginative "Initiation"

Fig. 17: Drilled truncated cube -- a polyhedron approximating a torus, with 64 edges
(Variously rotated with selected faces coloured or not)
Fig. 17a Fig. 17b
Drilled truncated cube -- a polyhedron approximating a torus, with 64 edges Drilled truncated cube -- a polyhedron approximating a torus, with 64 edges
Fig. 17c Fig. 17d
Drilled truncated cube -- a polyhedron approximating a torus, with 64 edges Drilled truncated cube -- a polyhedron approximating a torus, with 64 edges

[5b] Embodiment:

Fig. 11: Selected images of the Yi-globe of József Drasny
reproduced with permission from The Image of the Cosmos in the I Ching: the Yi-globe (2007)

Yi-globe of József Drasny Yi-globe of József Drasny
Yi-globe of József Drasny Yi-globe of József Drasny
These images point to the possibility of a correspondence between the spherical organization of conditions of change
and the more familiar understanding of how the Earth, as a globe, is exposed to light and darkness

Cognitive implication of Klein bottle

[6a] Imaginative "Initiation"

Representations of the Klein bottle
Konrad Polthier (Imaging Maths: inside the Klein bottle,
+Plus Magazine
, September 2003;
View Polthier's animated version (997K) or explore his java applet
Screenshot from video of formation of Klein bottle
developed by University of Hannover
(available with others on YouTube)
Klein bottle

[6b] Embodiment:




Fractal organization (Mandelbrot set):

Design Principles -- 15 Alexander

Imagining psychopter operating "principles"

Imagining embodiment of psychopter control

Description of how to -- without necessarily knowing how to embody those principles in practice -- bicycle description, dynamic equivalent of spiral staircase

Embaie Ferrow (The Five-Point Peace Plan: a Weyanized version of Necker's Cube, 2004)

*** fractal 12 ... 144

Clare Graves. Human Nature Prepares for a Momentous Leap. The Futurist, April 1974, pp. 72-87 [text]


Christopher Alexander:

Ron Atkin. Multidimensional Man; can man live in 3-dimensional space? Penguin, 1981 [summary]

Alick Bartholomew. Hidden Nature: the startling insights of Viktor Schauberger. Adventures Unlimited Press, 2005

Gregory Bateson. Mind and Nature: a necessary unity. Hampton Press, 1979

Susan Baur. The Dinosaur Man: tales of madness and enchantment from the back ward. Perennial, 1992 [review]

Stafford Beer:

Janine Benyus. Biomimicry: innovation inspired by nature. William Morrow / Quill, 1998

Jayne Bolsover. Dancing with Dinosaurs. Primrose Education Resources, 2001 [text]

Thomas Homer-Dixon. The Upside of Down: catastrophe, creativity, and the renewal of civilization. Island Press, 2006

Kenneth Boulding:

Nicholas Carr. The Shallows: what the Internet is doing to our brains. Atlantic Books, 2010

Michael Caley. Mindscapes: the epistemology of Magoroh Maruyama. Routledge, 1994

Jane Cobbald. Viktor Schauberger: a life of learning from nature. Floris Books, 2009

Mihaly Csikszentmihalyi:

Mihaly Csikszentmihalyi, Howard Gardner and William Damon. Good Business: leadership, flow, and the making of meaning. Basic Books, 2002

Erik Davis. Technognosis: myth, magic and mysticism in the age of information. Harmony Books, 1998

Luc de Brabandère:

Joël de Rosnay. The Macroscope. Harper and Row, 1979 [text]

William Easum. Dancing with Dinosaurs: Ministry in a Hostile and Hurting World. Abingdon Press, 1993

R. Buckminster Fuller with E. J. Applewhite:

Clare Graves. Human Nature Prepares for a Momentous Leap. The Futurist, April 1974, pp. 72-87 [text]

Yitzkak Hayut-Man. The Twelve fold Pattern Enfolded in Parashat Bereshit Genesis., 20 May 2010 [text]

Mark Patrick Hederman. Dancing with Dinosaurs: a spirituality for the 21st Century. Columba Press, 2011 [introduction]

Chris Hedges. Dancing with Dinosaurs. The New Humanist, 122, 2, March/April 2007 [text]

Thomas Homer-Dixon. The Upside of Down: catastrophe, creativity, and the renewal of civilization. 2006 [summary]

Antony Jay. Corporation Man. Jonathan Cape, 1972

Lars Johansson, Curt Hallberg and Morten Ovesen. Self Organizing Fluid Technique in the Footsteps of Viktor Shauberger. Vortex World, 1997 [text]

Harald Jockusch and Andreas Dress. From Sphere to Torus: a topological view of the metazoan body plan, Bulletin of Mathematical Biology, 2003, 65, 1,pp.57-65[text]

Rosabeth Moss Kanter. When Giants Learn To Dance. Free Press, 1990

James Patrick Kelly. Think Like a Dinosaur. Asimov's Science Fiction, June 1995 [comment]

Leopold Kohr:

J. G. Leishman. Principles of Helicopter Aerodynamics. Cambridge University Press, 2000

Geshe Gedün Lodrö. Calm Abiding and Special Insight: achieving spiritual transformation through meditation. Snow Lion Publications, 1998

Dudley Lynch and Paul L. Kordis. The Strategy of the Dolphin: scoring a win in a chaotic world. William Morrow, 1989

Magoroh Maruyama:

Stefan Marinov. Eppur Si Muove: axiomatics, fundamentals and experimental verification of the absolute space-time theory. Est-Ovest, 1981

Stanley S. McGowen. Helicopters: an illustrated history of their impact. ABC-CLIO, 2005

Steve McIntosh. Integral Consciousness and the Future of Evolution. Paragon Press, 2007

Ernest G. McClain:

Donald Michael. On Learning to Plan and Planning to Learn. Jossey-Bass, 1973

Paul Moxnes. Deep Roles: Twelve Primordial Roles of Mind and Organization. Human Relations, 1999, 52 (11), pp. 1427 -1444 [abstract]

Nathaniel Alexander Nicholson. E Pur Si Muove: essays. General Books, 2009

Etienne Oehmichen:

Steve Palmquist. The Circle of Twelvefold Systematization. In: The Geometry of Logic: a treatise on the perspectival transformations of logic and life through the systematic mapping of geometrical symbolism. 1986 [text]

Monique Pommier. The Twelvefold Archetype of the Human Soul: insights into the cosmic ground of philosophical imagination. Xlibris Corporation, 2009

William Powers. Hamlet's BlackBerry: building a good life in the Digital Age. Harper Perennial, 2011

Robert D. Romanyshyn. Technology as Symptom and Dream.Routledge, Chapman and Hall, 1989

Larry Rosen:

Frank Ross Jr. Flying Windmills: the story of the helicopter. Lothrop Lee and Shepard, 1953 [extract]

John Ralston Saul. The Unconscious Civilization. Simon and Schuster, 1995

Viktor Schauberger:

Sarah A. Sheard:

Henryk Skolimowski. The Participatory Mind: a new theory of knowledge and of the universe. 1995 [review]

Susan G. Sterrett. Wittgenstein Flies a Kite: a story of models of wings and models of the world. Pi Press, 2005

Rene Thom. Structural Stability and Morphogenesis:an outline of a general theory of models. Benjamin, 1975

Sherry Turkle:

Dmitri Tymoczko:

Ludwig von Bertalanffy:

M. M. Weil, and L. D. Rosen. The Psychological Impact of Technology From a Global Perspective: a study of technological sophistication and technophobia in university students from 23 countries. Computers in Human Behavior, 1, 1995, 1, pp. 95-133 [text]

R. Bruce Williams. Twelve Roles of Facilitators for School Change. Sage, 2008

Peter Woit. Not Even Wrong: the failure of string theory and the search for unity in physical law. Basic Books, 2007

Shing-Tung Yau and Steve Nadis. The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions. Basic Books, 2010 ("In fact, the Standard Model has four "global" symmetries relating to matter particles and chrage conversation that are not gauged" (p. 204)

Maurice I. Yolles:

Maurice Yolles, B.R. Frieden and G. Kemp. Toward a formal theory of socioculture: a yin-yang information-based theory of social change. Kybernetes, 37, 2008, 7, pp. 850 - 909 [abstract]

Arthur M. Young:

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