Three-dimensional configuration of nonagon mandala
Meaningful configuration engendered only by tacit aesthetic entanglement
Ninefold configuration in practice and its comprehension constraints
Speculation on potential symbolic relevance of the concordian mandala
Transformational pathways between 2D and 3D in the light of graph theory and "magic"
Defective insight transmission system?
Portals for intercourse with an imagined reality
Configuring a confusion of concepts of consequence to contention, consensus and consummation?
Concord: The case for a "concordian mandala" was made previously (Con-quest Aesthetically Reframed via the Concordian Mandala, 2016). This was inspired by the social chaos purportedly addressed by the so-called Discordian Mandala. The mandala is described in the controversial Principia Discordia, elaborated by Greg Hill with Kerry Wendell, as the provocative doctrinal manifesto of Discordianism. It was originally published under the title Principia Discordia or How The West Was Lost (1965). The name was intended to signify The Principles of Strife.
In a period of remarkable global discord, the preculiarity of that mandala (and the paradoxical controversy with which it has been associated) is suitably matched by mention of it in the Wikipedia entry on Borromean rings -- recognized to be of fundamental significance as the 3D logo of the International Mathematical Union.
Discord: The Discordian Mandala is assumed here to be a valuable catalyst for new thinking -- given its explicit association with the discord characteristic of problematic psychosocial system dynamics. These can be usefully framed as "systemantics", variously documented by John Gall (2003, 1986, 1978) and separately summarized (Why Systems Fail and Problems Sprout Anew: commentary on the principles of 'Systemantics', 1980).
Rather than the thinking associated with conventional appreciations of concord, exemplified by the inappropriateness to the times of the Anthem of Europe, new insight could even be sought through the discordance of its reversal (Reversing the Anthem of Europe to Signal Distress: transcending crises of governance via reverse music and reverse speech? 2016). There is a case for new understanding of distress given the contradictions between the most fundamental human values, the situation of refugees, and the economic benefits deriving from the supply of arms to regions engendering them (potentially framed by a measure of Refugees Per Kiloton).
In that spirit, the earlier argument focused on the possibility of reframing the ingenious Discordian Mandala as a Concordian Mandala -- by interpreting the former through the variety of connotations of "con" (Considerable Conglomeration of "Cons" of Global Concern, 2012). Given the provocative ambiguity of those connotations, the concern is whether their suitable juxtaposition is suggestive of a means of transcending the constraints of the simplistic binary implications of "dis" versus "con".
Third dimension -- and more? In the following development of the argument, the initial concern is with the possibility of a more significant representation of the mandala in three dimensions rather than two (notably with respect to the configuration of the 5 nonagons of which it is composed). This may ensure the emergence of a pattern of pentagons -- interwoven as a set of Borromean rings.
Subsequently the argument here notes the variety of potential implications of such a configuration as a symbolic nexus, especially in the light of the importance of the nonagon to various schools of thought and practice. In addition to the extensively documented focus on the enneagram, this includes the explicit importance of the nonagon in the most complex symbol of the penultimate degree of Freemasonry -- with all that this is claimed to imply for global governance and leadership. This merits a degree of recognition as the historical driving force for the Holy War against Islam, through the crusading in relation to Jerusalem -- now meriting comparison with a "Christian jihad" in the best and most problematic senses.
Interpretation: Although the geometrical and graphic challenges of this argument are unambiguous, the interpretation of the significance of any such configuration is quite another matter. This is exemplified both by Discordianism and by Freemasonry -- of which each would typically claim to be the antithesis of the other. Both typically arouse the concern of conspiracy theorists in a world increasingly preoccupied with ever higher degrees of secrecy and the urgent need for collective confidence and mutual trust.
A particular form of detachment is therefore required in noting the truths upheld by such contrasting perspectives. This clearly follows from increasing recognition of the role of so-called "post-truth politics" (Post-truth politics: art of the lie, The Economist, 10 September 2016). It is no longer a question of whether any matter is credible, but rather whether some believe it to be so -- and are inspired thereby to act.
The paradoxical context is usefully exemplified by a principal advocate of Discordianism, namely the highly controversial Robert Anton Wilson. His stated commitment was to breaking down conditioned associations, to looking at the world in a new way, with many models recognized as models or maps, and no one model elevated to the truth. Unfortunately it could be said that Discordianism, as a model in its own right, appears to have failed to introduce a requisite degree of self-reflexivity into its own worldview.
Disconnectivity: As a point of departure, the exploitation here of the Discordian Mandala (in its relation to Borromean rings) is also suggestive of neglected insights into the valuable role of disconnectivity. This contrasts with the value so commonly and unquestionably associated with connectivity. Requisite variety in cybernetic terms may then be as readily recognized in terms of disconnectivity as of connectvity. It is delightfully highlighted by the separation of petals in a flower.
The quest for new comprehension of transcendent harmony and togetherness may thus depend to an unsuspected degree on ordering disconnectivity rather than seeking to order connectivity. As an example, such a focus on "apartness" would be consistent with the focus on "apartments" and the constitutional "separation of powers". The argument raises the question as to whether the hegemonic present-day role of the Pentagon derives in part from the cognitive confusion of 5-fold distinctions with those associated with 9-fold distinctions -- an instance of misplaced concreteness, given the illusory emergence of the 5-fold pattern in the the 3D rendering of the mandala.
In this light, the case for representation in 3D could be usefully framed by re-interpretation of the traditional adage that: truth is hidden in plane sight. Such representation, and the many other images in this argument, are of further value in that they are consistent with an increasing shift from dependence on text to the insight enabled by imagery.
The following images are the results of experiments in 3D visualization. The associated animations were created with X3D Edit (and can be viewed as such with H3DViewer as X3D files). The application allows the animations to be converted to the more common legacy format of VRML/WRL in which they can notably be viewed in browsers after installation of a Cortona3D plugin (for example).
|Screen shots of 3D dynamics: Model A
(interactive access variants: X3D format or VRML/WRL format)
|Top view (video)||Side view 1 (video)||Side view 2 (video)|
|Screen shots of 3D dynamics: Model B
(interactive access variants: X3D format or VRML/WRL format)
|Top view (video)||Side view 1 (video)||Side view 2 (video)|
|These experiments benefitted from guidance from Sergey Bederov of Cortona3D (who is not responsible for their deficiencies)|
Parameters: The particular challenge in the experiments is to determine the parameters which ensure that the rings do not touch. This is the condition of a Borromean ring configuration as suggested by depiction of the Discordian Mandala in 2D, as previously discussed (Con-quest Aesthetically Reframed via the Concordian Mandala, 2016). It is relatively easy to provide animations in which the rings do touch or intersect at some phase in the cycle. Such animations may well appear to have a greater degree of symmetry.
In adjusting parameters towards a Borromean configuration without such intersections, the following modifications were applied (to Model B):
The result can only be appreciated through observing the dynamics of the cycle of ring rotation. With various choices of parameters it is clear that the rings may touch/intersect at particular points in the cycle. Some parameters reduce this occurrence. As yet unclear are the parameters which would ensure maximal non-touching during the cycle -- a Borromean "window".
The parameters have been adjusted experimentally without mathematical support -- and in that sense the mandala is a "work in progress". The question is how the parameters might be best determined in terms of the geometry of the configuration, notably the degree of tilting. Given the aesthetic argument below, one possibility is that the tilting angle might be associated with phi. Another is some reation to the sine curve effect noted with respect to zonohedrification of a polyhedron with 9-fold symmetry (discussed below).
Also of interest is the possibility of associating the configuration with the constraints of tensegrity as explored by R. Buckminister Fuller and Stafford Beer.
|Critical clarification regarding achievement of the Borromean condition
by Sergey Bederov (Senior Developer of Cortona3D)
It can be shown that the Discordian Mandala (as indicated above) cannot be accurately reproduced using any spatial projection of regular nonagons. Let us analyze the topmost (yellow) nonagon in the image. Obviously the outer (top) sides are longer and thicker than the sides near the center of the mandala. One can try to achieve this effect using perspective, placing the top corner closer to the viewpoint than the centerward side. But then, the lowest side (forming the bottom side of the innermost pentagon) appears longer than the sides adjacent to it. Being farther, it should appear shorter; this contradiction shows that an accurate reproduction is not possible.
(X3D version in 3D)
|Model 1- revised
(X3D version in 3D)
There are the following deficiencies in these representations:
|Another approach is to abandon the need of regular planar nonagons and use ad-hoc nonagons specially designed for the mandala, as indicated in the images below from an X3D model in 3D. It contains five nonagons of special shape, and the initial viewpoint is, in contrast, very narrow-angle.
There are the following deficiencies:
|Model 3 (X3D version in 3D)|
Alternative representation of nonagons by circular coils: The clarification above emphasizes the need for a distinction between faithfully reproducing the Discordian Mandala in 3D and rendering a variant with some of its properties. The original suggests a rendering in 2D with a necessary distortion of a 3D
intuition -- according to the aesthetics of Cubism. Given the 3D animations already produced (illustrated above), it remains unclear whether the critical clarification
precludes some form of approximation to the Borromean
condition in a percentage of a rotational cycle. More specifically, is
there a condition of minimal touching/intersection (for a set of
parameter values) -- whether for all rings, or for some of them.
Presumably this could be determined mathematically, if that
was of interest to someone with the required skills.
The second model presented by Sergey Bederov suggests a further possibility that rather than rendering the 5 nonagons in simple circular form, they could instead each take the form of circular coils in which each of their 9 segments took the form of a curve. This recalls that of a sine curve in 2D, twisted in 3D over its whole length. The question is whether the parameters governing the rendering of such helical curves would offer the zigzag effect of the second Bederov model, thereby enabling the curves to interweave without touching over some portion of their rotational cycle.
One potential merit of this approach is that such a configuration is consistent with the justification of both the armature winding of electromagnetic coils (used in motors and dynamos) and the coiled coil so famously characteristic of the DNA molecule. Both have been variously depicted in 2D and 3D. The issue for the purpose here is to explore a 9-coiled circular helix -- effectively a torus with s single curve wound around it. More intriguing is the further possibility that the 5 nonagons might be wound together to form such a torus, as with a 5-stranded rope quoit -- with the strands separating during the course of the rotation (much as with the so-called "unzipping" of the molecule in DNA replication).
Exploration of the possibility of such a configuration is therefore inspired by the following thumbnails. Of particular interest is that on the right of a 9-coiled image of a torus knot (2,9) by Kazunori Horibe. With respect to the reference above to Buckminster Fuller, this image has been featured on a site concerned with fullerenes and beaded molecules.
|Illustrative depictions of a circular helix|
|Rope quoit||Double-spun helix||Circular helix||Torus knot|
|Deriving coordinates for a 9-coiled helix|
A preliminary attempt to determine the coordinates required for the extrusion of an approximation to a 9-coiled circular helix was based on the construction of a torus with 9-segments using Stella Polyhedron Navigator by rendering transparent most of the elements of a nonagonal prism.
On the created torus, it was possible to change the colour of one of the
9-spheres sphere at each intersection (to black), shifting around the
torus in order to colour the path coordinates of a helix. The result was exported into a VRML variant and an X3D variant.
Basing the mandala on a toroidal helical coil: The above approaches can be framed as preliminary experiments which focused subsequent attention on the challenges of elaborating dynamic representations in terms of what is variously termed a toroidal coil or a toroidal helical spring. There is extensive literature on helical windings but relatively little on their toroidal form.
The further experiments in 3D dynamics, initially with guidance from Sergey Bederov, are described separately and more extensively in order to enable others to explore such possibilities (Visualization in 3D of Dynamics of Toroidal Helical Coils: in quest of optimum designs for a Concordian Mandala, 2016). The approach is partially indicated by the following images.
|Single helical coil winding||3-coil helical winding||5-coil helical winding|
Dynamics and graphic effects: Of further interest is the possibility of adjusting the dynamics of the configuration, possibly interactively, to render the cycle more or less rapid -- given the aethetic contrasts this offers. Some of these effects can already be achieved by downloading the relevant files and making adjustments to the code. Of particular interest are possibilities of:
The dynamic rendering of the mandala can, under certain conditions, be such that the 5 nonagons do not touch -- fulfilling the quest for a Borromean configuration. Other variations of the parameters may be such as to ensure they touch only occasionally, or that some always intersect. The argument here is that any such touching can be usefully reognized as a dysfunctional condition in which the processes associated with the distinct nonagons interfere with each other in some problematic manner.
The interest here is in a Borromean condition of the mandala in which the nonagons do not touch but are entangled through being looped together. It is in that condition that pentagonal configurations between the nonagons become especially apparent. The point to be stressed is that any pentagonal arrangement is an emergent perceptual effect for the observer. The nonagons are not connected in any such manner. There are no pentagons in the geometry of the set of nonagonal polygons. It is this particular arrangement that calls for further consideration.
Coherence through correspondence and proportion: The emergent pentagonal patterning suggests that the sense of order derives only from an aesthetic recognition of symmetry. It is not a feature of direct connectivity between the nonagons. Their relationship is then better understood in terms of correspondences. Understanding the latter has long been of interest to both the arts and the sciences (Theories of Correspondences -- and potential equivalences between them in correlative thinking, 2007).
Rather than logical connectivity, the coherence then derives from an aesthetic harmony of perceptual symmetry and proportion rather than any conventional logic of unified systemic coherence.
With respect to the patterns in nature and architecture appreciated for their aesthetics, considerable attention has been given to the cognitive role of the golden ratio (phi). Of some relevance to this argument is its association with the regular pentagon and -- potentially -- as the growth factor in the golden spiral. (*** ching papers Fibonacci???) The Fibonacci spiral: an approximation of the golden spiral
Interwoven threads: The nonagons can be usefully understood as "interweaving". The 5 9-phased threads pass over or under each other -- characteristic of what is recognized as warp and weft in weaving. Although that metaphor suggests that the threads touch, they are not connected as in a knot, nor do they intersect.
Rather than the conventional focus on connectivity, the weaving metaphor can be explored in terms of its significance for governance (Warp and Weft of Future Governance: ninefold interweaving of incommensurable threads of discourse, 2010; Warp and Weft: Governance through Alternation: world governance as a Gandhian challenge for the individual, 2002).
The weaving can also be usefully recognized in aesthetic patterns such as the Celtic Knot, or the so-called Endless Knot -- both valued for their symbolism. Although these terms use "knot", this refers to the pattern as a whole. As noted below, the threads of which they are composed are not knotted together. The mandala could then be considered a knot of this kind.
Bridging processes: An alternative metaphor is that for traffic, where there may be an overpass (a bridge) or an underpass (an arch). This metaphor is valuable because of the experiential engagement with either a bridge or an arch and the sense of separation offered in each case when traversed. This is a feature of traffic understood more generally (Being in the Flow on Strategic Highways and Byways: enabling sustainable self-governance through traffic signage, 2011). Whether as an arch or as pillars the experience may well characterize initiation rituals.
Fairly frequent use is made of the bridge metaphor to offer a particular sense of connectivity across what is otherwise perceived and experienced as an impassable gap. Of particular relevance to this argument is its use as the title of the Bridges Organization. This oversees the annual Bridges Conference on mathematical connections in art, music, architecture, education, and culture.
"Kiss-touch" connectivity: The tensegrity (tensional integrity) structures developed by Buckminster Fuller are of a form which is unique in addressing the challenge of using polarized (incompressible) elements. Without touching each other (as in conventional construction), these are configured and held in position by a continuous network of associative elements in tension. At most any "touching" is, as in geodesic domes, what was technically termed a "kiss touch" (by Fuller) -- being of no load bearing significance.
The last characteristic could be seen as a vital requirement in configuring polarized values, clashing cultures and other forms of binary thinking and relationship, into larger structures reconciling (but "respecting") their differences (From Networking to Tensegrity Organization, 1984). It thereby creates a free central space uncluttered by central "pillars" (as is normally the case). indicative of points of contact which do not affect each other through the transmission of stress or strain.
However, if Fuller's "thinking" is to be understood in dynamic, process terms, then the degree to which separate domains touch is indicative of a potential cognitive transition point from one to another -- potentially vital for the cognitive coherence of the pattern as a whole. Thinking then "moves" into a different loop essential to sustaining the whole -- a feedback loop to be understood in terms of knowledge cybernetics (Maurice Yolles, Knowledge Cybernetics: a new metaphor for social collectives, Intellect, 3, 2006, 1).
This argument is a feature of various separate discussions (Transforming Static Websites into Mobile "Wizdomes" enabling change through intertwining dynamic and configurative metaphors, 2007; Engaging with Globality through Dynamic Complexity, 2009). The underlying tensegrity principles have notably been used to describe cellular architecture (Donald E. Ingber, The Architecture of Life, Scientific American, 278, January 1998).
With respect to Value polarities as archetypal bonds (2007), the symbolic subtlety of such a bond was illustrated by the following much appreciated image.
of Man by Michelangelo (Sistine Chapel)
as an indicative representation of the subtlest relationship bond
(as discussed below with respect to the "kiss touch")
Of interest in the architecture of a geodesic dome, dependent on such a kiss-touch, is the emptiness of its centre -- with no supporting beam across it -- as has been appreciated in its symbolic value. This structural characteristic may be appropriately contrasted with the unfruitful "pillarization" of values and strategic intiatives, as with those of the European Community (as discussed in Challenge of "soullessness" -- beyond the "pillar-ization of Europe", 2004).
Requisite disconnectivity: There is of course widespread concern with the value of connectivity, assumed to be a prerequisite of harmonious global civilization. It is typically framed and promoted as fundamental in the quest for unity and unification in the sciences, the arts and religion. The focus of fundamental mathematics is primarily on connectivity although there is a degree of concern with disconnectivity (Xing-Ke Wang and Shang-Wang Tan. Ordering trees by algebraic connectivity Linear Algebra and its Applications, 2012).
Of some relevance to the development of this argument, as noted by Wolfram Mathworld:
Disconnectivities are mathematical entities which stand in the way of a space being contractible (i.e., shrunk to a point, where the shrinking takes place inside the space itself). When dealing with topological spaces, a disconnectivity is interpreted as a "hole" in the space. Disconnectivities in space are studied through the extension problem or the lifting problem. [see also disconnected space]
Disconnectivity in a wide variety of forms is conventionally regretted, if not deplored as being the antithesis of togetherness -- as epitomised by apartheid and alienation. Everything should supposedly be connected to everything.
Following the problematic religious practices of shunning, excommunication and disfellowship, it has become the focus of controversy in the case of disconnection as a practice in Scientology. This is the severance of all ties between a scientologist and any friend, colleague, or family member deemed to be antagonistic towards Scientology. Such disconnectivity could be explored in terms of confirmation bias -- perhaps to be understood as "conformation bias".
There is little recognition that requisite disconnectivity may be vital to connectivity of some "higher order" -- however that is to be understood. This is most obvious in electrical wiring and printed circuit boards where insulation between components is vital to the functioning of the whole. It is also evident in music. Appropiate disconnectvity is in fact vital to the operation of many systems in nature or in manufacturing processes. This is appreciated through recognition of the need for biodiversity, for example.
As a form of disconnection, far less understood are the implications of avoidance of issues and systemic neglect (Institutionalized Shunning of Overpopulation Challenge: incommunicability of fundamentally inconvenient truth, 2008; Vigorous Application of Derivative Thinking to Derivative Problems, 2013).
Potentially more challenging is the sense in which connectivity, as the cultivation of a form of universal intimacy conventionally understood, may effectively inhibit disconnectivity of another kind -- notably appreciated through the need for "space". This follows from provocative arguments that any "other" connected in this way is thereby reified to a degree in being disconnected as an "object" from the perceiver.
This disconnection can be explored in terms of extimacy (in contrast with intimacy) and the obverse (in contrast with reverse), as eloquently presented by Donald Kunze (Atlas of the Obverse, 2012). Extimacy (as defined by Jacques Lacan) is thus a poorly explored complement to intimacy, indicative of the existence, within the most intimate sphere of the I, of an "other", that which is external to the individual and with which the individual identifies.
The relation between obverse and reverse can be explored otherwise in terms of Chinese philosophy, as carefully clarified in relation to healing by Electra Peluffo (Cosmological Origins of the Obverse-Reverse Zang Fu Pairing in Chinese Medicine, Chinese Medicine, 2014, 5, 270 -276):
Chinese thought gives us a vision of the world based on the pairing among other concepts. And by using this word we want to describe becomingness or a process which starts from a duality in permanent interaction: opposition and complementariness as a result of a parallelism.
By contrast it is surprising to note that obverse and reverse are most frequently cited in western literature in relation to the two faces of coins. The particular relevance is that this is with respect to the primary token of mutual confidence -- the coin.
These various associations suggest that, in the quest for for unifying insight, the universal and the universe might be more meaningfully explored through a curious form of cognitive resonance between reverse and obverse. The health metaphor thus offers a way of reframing this quest (Remedies to Global Crisis: "Allopathic" or "Homeopathic"? Metaphorical complementarity of "conventional" and "alternative" models, 2009).
Preoccupation with diversity as a requisite complement to unity may be a form of distraction when some kind of contraction is essential. The may become evident through the inadequacy of "multiverse" as the new complement to universe (Transcending both scientific and poetic comprehension of multiverse, 2012).
Separation of "powers" and resources: Understanding of requisite separation and disconnection is widely recognized in terms of seemingly disparate domains. Constitutional law takes careful account of the need for separation of powers. Typically the state is divided into branches, each with separate and independent powers and areas of responsibility so that the powers of one branch are not in conflict with the powers associated with the other branches. Responsibilities are divided to limit any one branch from exercising the core functions of another. The intent is to prevent the concentration of power and provide for checks and balances. Typical branches include: executive, legislative, judicial, auditory, electoral, and prosecutory. Importance may also be attached to the separation of church and state -- "church" being understood as religion of any kind.
Particular attention may be given in system design to the isolation of resources, most notably to ensure that such systems fail safe in the event of any failure or as the result of any attack -- preventing any cumulative domino effect. Redundncy may then be values. Such separation is a vital preoccupation when exposed to any form of cyber-attack on information systems.
Understood in terms of any form of covert operation, requisite separation is clearly a valuable strategy for both cartels and organized crime. Arguably any covert collaboration between major corporations, may depend on such separation as a means of avoiding exposure to public criticism and investigation -- despite the connectivity offered by "old boy networks" and interlocking directorates (see Wikipedia List of people on multiple governing boards). Such possibilities enable plausible deniability ensuring that no concrete evidence of involvement can be legally proven (especially when covered by non-disclosure agreements or superinjunctions). This is especially important when there is recourse to so-called "dirty tricks".
Such protective measures have however been of considerable importance in the design of intelligence networks to ensure that detection of one cell does not jeopardise the continued operation of other cells and their personnel. Use may be made of a cutout -- a mutually trusted intermediary, method or channel of communication -- to facilitate the exchange of information between agents. Cutouts usually only know the source and destination of the information to be transmitted, but are unaware of the identities of any other persons involved in the espionage process. A captured cutout cannot therefore be used to identify members of an espionage cell.
Tacit bonds and soft power: A contrast can be recognized between well-defined and undefined contacts in a variety of domains:
The distinction invites the kind of encoding offered in Chinese culture by the distinction between yang and yin. This in turn highlights the unresolved dynamics between male and female, and the masculine exclusivity implied by fraternal bonding.
Requisite variety: As increasingly acknowledged with respect to biodiversity, the design considerations are of more general relevance from a cybernetic perspective in terms of requisite variety. Far less attention has been devoted to the requisite degree of socio-diversity (Sabine U. O'Hara, Valuing socio-diversity, International Journal of Social Economics, 22, 1995; P. Dansereau, Biodiversity, ecodiversity, sociodiversity: three aspects of diversity, Aspen Bibliography, 1997, 1535).
More controversially this may be recognized to be of relevance to the requisite variety of modes of intelligence, as argued by Edward de Bono (Six Thinking Hats, 1985; Six Action Shoes, 1991), or with respect to the 9 team roles identified by Meredith Belbin (Belbin Team Inventory).
Missing from such consideration is a means of creatively reframing the highly controversial issues associated with the perceived need for separation by gender (men, women, transgender, etc), ethnicity and race, culture and education, and otherwise -- as is so vigorously (and violently) argued in various contexts (Cultivating the Myth of Human Equality: ignoring complicity in the contradictions thereby engendered, 2016).
Although seemingly obscure, the domains in which the nonagon is held to be of relevance merit particular attention. As noted by Clemens Bechinger (A predilection for certain symmetries, Max-Planck Gesellschaft, 2010):
It is surprising that materials with 7-fold, 9-fold or 11-fold symmetry have not yet been observed in nature... This is all the more astonishing in view of the fact that patterns with any rotation symmetry can be drawn without difficulty on paper....The answer is of interest to us both from a fundamental point of view but also because it could be helpful for tailoring materials with novel properties for technical applications
Contexts in which ninefold configuration is recognized:
|Turing instabilities with 9-fold symmetry|
| In the paper by Alan Turing (The Chemical Basis of Morphogenesis, 1952) it was suggested that a system of chemical substances, reacting together and diffusing through tissue, would account for the generation of patterns in inn multi-cellular organisms. In this light a simplified computer model was derived, successively elaborated and used for the productio of artworks reminscent of electron microscope images of diatioms. These show a mixture of symmetries, with 3-fold symmetry imposed on the small scale Turing instabilities and 9-fold symmetry on the large scale. Differing short scale details are apparent within larger repeating patterns.
[The patterns have been described elsewhere, illustrated in colout and with animations: Psychedelic cells are fruit of Alan Turing's equations, New Scientist, 23 July 2014; Hypnotic Art Shows How Patterns Emerge From Randomness in Nature, Wired, 28 March 2014; These Psychedelic Images Find Order Amid Chaos, Smithsonian Magazine, 20 June 2014]
|Views of zonohedrified 9-gonal antiprism with 9-fold symmetry (9*2m)|
|Facetting diagram||Side view||Symmetry axes||Polar view|
|Stellation diagram||Unfolded net|
|72 faces (4 types)
144 edges (9 types)
74 vertices (5 types)
|Model and displays kindly developed by Robert Webb from Stella Polyhedron Navigator|
In the cybernetic analysis of the more complex and organized systems we recognize two distinct kinds of factors. There is the work component or components, which we shall designate X, and the governor, or controller, which we shall designate Y. Of course, the governor does work too (the strategic work), and we have simplified the relationships very greatly. There will be cases of a system made up of sub-systems, one controlling in some respects, not in others, and so on. Let us stay with the simpler case. Now, the processes that characterize X may, in the interaction with Y, be accelerated or in some way enhanced ( + ), or may be unaffected ( O ), or may be decreased ( - ). Similarly, the processes that Y undergoes. When the possibilities are cross-tabulated, it becomes evident that there are nine and only nine of these qualitatively different `coactions. [glossary]
As discussed separately (Cardioid Attractor Fundamental to Sustainability: 8 transactional games forming the heart of sustainable relationship, 2005) a detailed discussion of the application of this insight to a range of systems, notably in the natural environment (see table below) but also in the social environment. In the case of the different kinds of relations between animals in an ecosystem, the following patterns then emerge -- of which 8 of the 9 are non-neutral. Note that there are variations in the teminology of biological interaction, notably differing from Haskell's usage [more | more]. The dynamics of each of the 8 relationships might be described as a "game", however asymmetrical or predictable the outcome (as with the "cat-and-mouse" game of predation).
| Possible 9-fold context of Positive-Negative
(reproduced from Being Positive Avoiding Negativity, 2005)
|.||.||X = "Work component"|
Implications of polygonal configurations for comprehension of complexity: Of potential relevance to this argument is the attention given to developing an appreciation of nine-foldness in children, as described by Sheryl Nichols Morris (Nine: Observing Nine with Young Children (SNAP: Scaffolding for Numerical Synapses: Awakening Curiosity in the Numbers One to Ten. 2013).
The challenge of acquiring such insight -- beyond the binary -- is remarkably articulated using a triangle, from the mathematical perspective of q-analysis by Ron Atkin (Multidimensional Man; can man live in 3-dimensional space?, 1981). Atkin illustrates the challenge of comprehension in relation to experience constrained "within" the geometry of a triangle -- especially with regard to the perspective necessary to comprehend the geometry of the triangle as a whole.
The cognitive significance of this approach is well-illustrated by visual sensitivity to colours resulting from the three primary hues (red, green and blue). These may be represented on a simple triangle. Here the vertices (O-simplexes) represent the primary hues, the sides are twofold combinations (1-simplexes), and the combination of the three hues makes the central white (2-simplex).
|Challenge of recognizing a higher-order perspective
schematic used by Ron Atkin
As separately summarized, Atkin's argument can be explored in terms of "failure to see the central point" (Comprehension: Social organization determined by incommunicability of insights). This is a consequence of being trapped in a form of tunnel vision "within" the lines (of argument) configured as communication traffic around the point (especially when the lines are constructed as cylinders). This inhibits the acquisition of any sense of perspective. The argument is developed separately in more general terms (Metaphorical Geometry in Quest of Globality -- in response to global governance challenges, 2009).
Comprehension of Numbers Challenging Global Civilization: number games people play for survival, 2014). As indicated above, the latter notes key constraints on comprehension and preferences for pattern recognition.
Atkin's approach could also be taken further in the light of the polygon circumscription discussed below. What are the challenges of seeing the whole through configurations of 4, 5, 6, 8, or 9, points, for example? In terms of integrative comprehension, how is the "white" framed by 3 distinct from that framed by 9? In each case there is an ever more complex challenge framed by the inability of seeing the forest for the trees.
Enneagon construction and the vesica piscis: Considerable symbolic importance has been attributed over centuries to the twofold vesica piscis and to its threefold variant the triquetra. The construction of a nonagon, in the light of the these figures, was first described by Albrect Durer as shown below.
(reproduced from The Polygons of Albrecht Dürer (1525) by G.H. Hughes)
|"You can construct a nine-sided figure based on a triangle. Draw a large circle with center a. Then without changing the opening of the compass draw three "fish bladders" whose upper end on the periphery you will mark b. Mark the other c and d. Within the upper fish bladder draw a vertical line ba and divide this line with two points 1 and 2 into three equal parts. Point 2 should be closest to a. Then draw a horizontal line through point 2 at right angles to the vertical line ba. Where the horizontal line crosses the fish bladder mark points e and f. Then place one leg of the compass on center a and the other on point e and draw a circle through f. Line ef will then represent one of nine sides which will compose a nonagon inside this smaller circle...."|
Of potential relevance to the further development of this argument is related insight into Villarceau circles derived from a torus.
Framing the challenge of ninefold comprehension through nesting regular polygons: The question of the relationship between polygons is of continuing interest to mathematics (Nested Regular Polygons, Wolfram Demonstrations Project; Shiqing Zhang and Zhifu Xie Nested regular polygon solutions of 2N-body problem, Physics Letters A, 281, 2001, 2-3, pp. 149-154; S. J. Dilworth and S. R. Mane, On a problem of Croft on optimally nested regular polygons. Journal of Geometry, 99, 2010, 1-2).
Of potential interest to this argument is the approach of John Derbyshire (Nested Polygons) who describes a procedure with respect to the polygon-inscribing constant [Kepler-Bouwkamp constant] (approximately 10/87) and its reciprocal the polygon circumscribing constant (8.7000366252), as illustrated by the following. Derbyshire notes the relevance of an article by Roger S. Pinkham (Mathematics and Modern Technology, American Mathematical Monthly, 1996).
The original static rendering of polygon circumscription has been animated experimentally (and arbitrarily) to suggest other dimensions to the challenge of comprehension. Note that the animation lends itself to various developments and technical improvements. In addition to the choice of direction of rotation (counterclockwise for red, odd-sided polygons; clockwise for blue, even-sided), line thickness is increased with the number of sides. Other effects suggestive of significance to comprehension could be offered by changes of relative rotation rate. (Speed of rotation can be controlled by right-click within the animation).
|Nesting of regular polygons|
(reproduced from Wikipedia)
(animation developed from Wikipedia)
The animation invites reflection in terms of the following:
Symbolism has a fundamental cognitive role, notably with respect to collective psychosocial organization as a catalyst to the imagination in enabling belief in subtlety of many kinds (The Book of Symbols: reflections on archetypal images, Archive for Research in Archetypal Symbolism, 2010; Hans Biedermann and James Hulbert, Dictionary of Symbolism: cultural icons and the meanings behind them, 1994; Albert Gallatin Mackey, The Symbolism of Freemasonry: illustrating and explaining its science and philosophy, its legends, myths and symbols, 2012; Alfred North Whitehead, Symbolism: its meaning and effect, 2014).
The potential symbolic relevance of the mandala above is presented in an annex (Speculation on Potential Symbolic Relevance of the Concordian Mandala, 2016) as distinctive articulations in the light of contrasting insights, perspectives and projections. All frame the challenge of comprehending insight of a higher order. Thematic headings in the annex are:
Masonic nonagonal icon
|Wu Xing and Hygieia
Lord of the Rings
Of particular relevance to this argument is the extensive review which emerged from the Rodin Coil Project (Vortices, Portals and Dimensional Doorways: a study of the number nine in science, myth and mysticism, 1998). This offers many additional examples, most notably from mythology. However the origin of this review is itself illustrative of the deprecation by which many such examples are framed, from whatever belief system they originate. The point to be emphasized is that it is not the validity of such examples (according to some particular criteria) which is of relevance but rather the fact that some hold them to be credible -- irrespective of views to the contrary.
The point is further emphasized by the significance long claimed (and denied) for Freemasonry from a variety of perspectives, most notably with respect to global governance. It is in this sense that the nonagonal Camp symbol of the penultimate degree (as presented below) is the basis for more extensive commentary.
|Variant images of the Camp symbol of Freemsonry|
|A number of images of the symbol are presented by Eric Ginette
(The Long Journey of the Masonic Camp, The Northern Light, August 2015).
|In Hoc Signo Vinces|
|We Shall have World Government... by Conquest or Consent.
Declaration by James Warburg to U.S. Senate Committee on Foreign Relations, 17 February 1950
(see commentary by Global Research)
Embodiment of value in gemstones: It is curious that the highest values are ambiguously associated with the capacity of certain stones to focus, colour and reflect the light -- to sparkle elusively. When transparent, the stones are cut and polished into the form of polyhedra to enhance the inherent properties of their crystal structure (Summary of Gemstone Faceting and Crystals, 2002). They are then typically set into the most significant symbolic artefacts -- rings, necklaces, crowns.
The relationship in practice between such jewels and psychosocial processes and their ordering remains elusive. Emblematically they may well offer the ultimate metaphor for insight into the potential of transcendent relationships, as separately discussed (Patterning Archetypal Templates of Emergent Order: implications of diamond faceting for enlightening dialogue, 2002).
It is intriguing that Freemasonry should ensure that its penultimate degree is the 32nd, given the number of crystal classes (Stephen A. Nelson, External Symmetry of Crystals, 32 Crystal Classes, 2013). The 7 crystal systems consist of 32 crystal classes (corresponding to the 32 crystallographic point groups).
A point groups is a set of symmetry operations, like rotations or reflections, that leave a central point fixed while moving other directions and faces of the crystal to the positions of features of the same kind. The group must also be consistent with maintenance of the three-dimensional translational symmetry that defines crystallinity. The macroscopic properties of a crystal would look exactly the same before and after any of the operations in its point group.
Clearly such distinctions allow themselves to be interpreted in terms of degrees of insight and the cognitive modalities they enable. This is the case with respect to the Hindu system of 7 chakras, for example.These are successively associated with ruby, pearl, red coral, emerald, yellow saphire, diamond, and blue saphire. However these contrast with the traditions of China regarding jewels of such significance. Arguably degrees of symmetry can be fruitfully understood in terms of strategic nimbleness and the ablity to "turn on a dime" under complex constraints. For those exposed to this skill it may be readily framed in terms of "wriggling", "trickiness" or "slipperyness".
Philosopher's stone: Metaphorically the capacity of a stone to embody or transform value in some way is central to the sense of the philosopher's stone as a focus of aspiration in relation to comprehension of the seemingly chaotic condition of the world (In Search of the Philosopher's Stone). In the alchemical tradtion it is understood as having the capacity to transform base metals into gold -- however these are to be understood metaphorically. It may be understood as a fiery jewel capable of combining all oppositions into dynamic harmony. Numerous synonyms have been used to make oblique reference to the stone -- including Melancholia, central to the image discussed below. A more accessible variant can be recognized in metaphorical use of the Rosetta stone.
Curiously "stone" has been variously understood metaphorically, as discussed separately (Transforming and Interweaving the Ways of Being Stoned: imagination, promise, rocks, memorials, petrification, 2012). Given its topological focus, there is relevance to the recognition of the current significance of the alchemical process as offered by Steven M. Rosen (Dreams, Death, Rebirth: a multimedia topological odyssey into alchemy's hidden dimensions, 2014).
Role of number: The implications are usefully framed from a cognitive perspective by George Lakoff and Mark Johnson (Metaphors We Live By, 1980) and subsdquently by George Lakoff and Rafael Nüñez (Where Mathematics Comes From: how the embodied mind brings mathematics into being, 2000).
It is in the light of the latter arguments that number-related issues with respect to recognition of order merit consideration (Representation, Comprehension and Communication of Sets: the role of number, International Classification, 5, 1978; Comprehension of Numbers Challenging Global Civilization: number games people play for survival, 2014). As indicated above, the latter notes key constraints on comprehension and preferences for pattern recognition.
Portrayal of melancholy by Albrecht D¨rer: Given the aesthetic argument above, it is appropriate to note recognition of melancholy in relation to comprehension of the philsopher's stone as articulated, for example, by Barbara R. Barry ('Sehnsucht' and Melancholy: explorations of time and structure in Schubert's Winterreise -- in: The Philosopher's Stone: essays in the transformation of musical structure, 2000) and otherwise as the theme of an exhibition (Ophelia: Sehnsucht, Melancholy and Desire for Death).
Given the religious dimensions of the crisis of the times, melancholy can be considered as a central motif (Alina Feld Melancholy and the Otherness of God: a study in the genealogy, hermeneutics, and therapeutics of depression, 2011).
In the light of the above references, melancholy is usefully recognized as a theme of considerable significance to the discord and distress of the present day. It is therefore relevant to note the remarkable degree of attention over centuries, notably by Freemasonry, to an image by Albrect Dürer, entitled Melencolia I, elaborated in 1514 in the midst of the Holy Roman Empire. The work has been the subject of more modern interpretation than almost any other print. The image is notable for featuring both a (3D) polyhedron and a (2D) magic square. As remarked by John Read (Interpretation of this Drawing):
There is no evidence that Durer had an acquaintance with alchemy so intimate as, say, that of Chaucer in an earlier age; nevertheless, alchemy formed an essential ingredient of the cultural background of his times. To contemporary men of culture in general, as to the alchemist in particular, the "Melencolia" must have appeared as a rich repository of the pictorial symbolism of alchemy.
The form of the polyhedron which has been the focus of so much commentary is curiously reminiscent of the 2D depiction of the Discordian Mandala composed of 5 distorted nonagons configured into a semblance of pentagons. The 6 similarly distorted pentagons of which the Durer solid is primarily composed appear to be the result of stretching a cube and then truncating it to yield 2 bounding triangular faces. As variously illustrated below, the 12 vertices lie on the circumsphere of vertices (incidentally recalling the 12 stars of the current Flag of Europe). The 18 edges form 9 pairs, with each pair on one of 9 great circles.
|Views of Durer solid|
|With circumsphere||18 edge-pairs on 9 great circles||Faces transparent|
|Model kindly developed by Robert Webb for use in Stella Polyhedron Navigator|
|Animations of selected morphings of Durer solid to and from its dual|
|Expansion||Sizing||Tilting to rectify|
|Animations made using Stella Polyhedron Navigator|
|Animation of Durer solid unfolding||Animation of dual of Durer solid unfolding|
|Animations made using Stella Polyhedron Navigator|
Dimensional transition between 3D and 2D: With respect to the argument here, curiously absent is the nature of the cognitive transition from 2D to 3D. The masonic symbols noted above in 2D are depicted on aprons and sashes, and may be reflected in carpetry for ritual purposes. Much is made of masonic architecture as a context for collective ritual. Secret though it may be, the transition merits consideration in a global context (Metaphorical Geometry in Quest of Globality -- in response to global governance challenges, 2009).
Freemasonry has traditionally attached considerable significance to polyhedra, most notably the 5 Platonic solids (Lee Miller, A View of Contemplative Geometry in Freemasonry, 2002). As Miller notes:
Freemasonry is long on ritual, history and tradition but is unfortunately very short on footnotes. Regarding its basis, "Geometry", we can find mentions of Euclid and Platonic Solids, but little more that would lead us to a specific person, book or similar source. One may only guess at the sources the earlier ritual writers had in hand when they compiled their Work.
Given the references above to the Catholic Church, of ironic relevance to this argument is the widely remarked tendency of Pope Francis to refer to polyhedra. As indicated by Leonardo De Chirico (Is Unity Like a Sphere or a Polyhedron? Vatican Files: evangelical theological perspectives on Roman Catholicism, 18 September 2014):
Pope Francis does not like spheres: he likes polyhedrons. In various recent speeches and in different contexts he used the image of the polyhedron to illustrate what he has in mind when he thinks of unity, i.e. Christian unity and the unity of mankind....
Francis first began talking about the polyhedron in the context of globalization. In a message to a festival on the Social Doctrine of the Church, which addressed the issue, he said: I would like to translate the theme into an image: the sphere and the polyhedron. Take the sphere to represent homologation, as a kind of globalization: it is smooth, without facets, and equal to itself in all its parts. The polyhedron has a form similar to the sphere, but it is multifaceted. I like to imagine humanity as a polyhedron, in which the multiple forms, in expressing themselves, constitute the elements that compose the one human family in a plurality. And this is true globalization. The other globalization -- that of the sphere -- is an homologation (Dec 6th, 2013).
Of particular relevance to the current distress of Europe, the theme figured in the Pope's subsequent Address to the Council of Europe (Vatican Radio, 25 November 2014):
The task of globalizing Europe's multipolarity cannot be conceived by appealing to the image of a sphere -- in which all is equal and ordered, but proves reductive inasmuch as every point is equidistant from the centre -- but rather, by the image of a polyhedron, in which the harmonic unity of the whole preserves the particularity of each of the parts. Today Europe is multipolar in its relationships and its intentions; it is impossible to imagine or to build Europe without fully taking into account this multipolar reality.
Extensive comment is provided on the Durer image by Ernst Theodor Mayer (Melencolia I) in a multilingual European encylopedia of Freemasonry (Freimaurer-Wiki). Appropriate to this argument, such commentary has been presented in a variety of forms on the 500th anniversary of its elaboration, notably by Günter M. Ziegler (Adventures in Numberland Dürer's polyhedron: 5 theories that explain Melencolia's crazy cube, The Guardian, 3 December 2014; Dürer-Polyeder: Ein mysteriöser Körper wird 500, Spiegel Online, 12 December 2014; Freemasonry Matters, 22 July 2016).
Part of the puzzle lies in the form of the polyhedron as a whole, since it is only partially visible -- therefore inviting much speculation, as well as assumptions regarding preferred conclusions. One such, naming it as a truncated rhombohedron, is described mathematically as a truncated triangular trapezohedron (Dürer's Solid, Wolfram MathWorld). A detailed interpretation of its symbolic significance in the light of its construction is presented by Yvo Jacquier (The Polyhedron of Melecolia I: "polyhedron of Dürer", 2013).
|Relationship between polyhedra and magic squares of interest to Freemasonry
(reproduced from the entry by Ernst Theodor Mayer on Dürer's Melencolia (1514) in the Freimaurer-Wiki)
Other possibilities are noted by Hans Weitzel (A further hypothesis on the polyhedron of A. Dürer's engraving Melencolia I, Historia Mathematica, 31, 2004). Preferring closure, despite its partial visibility, it is curious that few seem to consider what other forms it might take -- although this might be considered as deliberately inviting further speculation regarding its relation to other elements of the image. As rermarked by one renowned polyhedral specialist, George W. Hart as:
I cannot decide if the lower vertex is also truncated so the solid rests on a triangular face, or if the lower vertex symbolically penetrates the earth, but no other writers seem to allow for that possibility. Panofsky accurately describes it simply as a "truncated rhomboid". One might speculate that the cube represents masculinity and truncating one in an upright position may have some Freudian symbolism. (Durer's Polyhedra)
Magic graphs, magic squares, magic cubes, etc: Of particular relevance here is the extensive commentary on the relation of the assumed form of the 3D polyhedron to the 2D magic square in the original image. A magic square is an arrangement of distinct numbers (i.e., each number is used once), usually integers, in a square grid, where the numbers in each row, and in each column, and the numbers in the main and secondary diagonals, all add up to the same number, called the "magic constant". (S. S. Block and S. A. Tavares, Before Sudoku: The World of Magic Squares, 2009). In addition to commenting on the polyhedron, Wolfram MathWorld provides entries on both Dürer's Magic Square and the Dürer Graph with which it is associated (as does Wikipedia: Dürer's Magic Square, Dürer Graph).
Curiously such arrangements have been interpreted as being of considerable significance by Freemasonry, as indicated by a Mason of the 32nd degree (William Steve Burkle, Speculation on the Symbol of the Square and Compasses: the Freemasons' magic square, Pietre-Stones: review of Freemasonry).
The importance to Freemasonry of magic squares is discussed separately with the use of animations in 2D (Salvation Enabled by Systemic Comprehension: via aesthetics of magic squares? 2015). This notes their perceived relevance for Benjamin Franklin, a Freemason and one of the Founding Fathers of the US, instrumental in the formulation of its Constitution. A focus is given by mathematics to Franklin's magic squares, and his widely remarked skill in their construction as noted by Paul Pasles (Franklin Squares 2006; Benjamin Franklin's Numbers: an unsung mathematical odyssey, Princeton University Press, 2007) and by Maya Mohsin Ahmed (Unraveling the secret of Benjamin Franklin: constructing Franklin squares of higher order, 23 September 2015).
|Franklin's 8x8 magic squares: animations of movement of selected bent diagonals|
|Vertical movement||Combined movement||Horizontal movement|
Franklin called his 16x16 magic square the most magically magical of any magic square ever made by a magician -- with which many mathematicians and mystics would now be held to agree (Peter Loly, Franklin Squares: a chapter in the scientific studies of magical squares, University of Manitoba, 2006; William H. Richardson, Ben Franklin's Amazing Magic Square [including animation], Wichita State University)
Especially intriguing with respect to magic squares are the implications of the patterns to those of a Knight's move, so significant to chess and go, as discussed separately (Insights from Knight's move thinking, 2012; Stratagems and ploys characteristic of Knight's move thinking, 2012). Given its relation to creativity and strategic surprise, such capacity can be readily assumed to be associated with the degrees of strategic secrecy of Freemasonry.
Given the traditional interest of non-western cultures in magic squares, it is appropriate to explore their relevance for subler orderings of insights (9-fold Magic Square Pattern of Tao Te Ching Insights: experimentally associated with the 81 insights of the T'ai Hsüan Ching 2006). The latter includes discussion of: Pan-magic square presentation, Bimagic squares, Most-perfect magic squares, Magic cubes, Magic hypercubes as well as reference to the Possible psychological implications of magic square ordering (2003). ***
With respect to the Durer image, discussion also focuses on magic cubes, notably the 4x4 Magic cube associated with the Durer magic square. (Dürer's square, Complex Projective 4-Space, 8 September 2012). This necessarily incorporates the integers from 1 to 64 and has a magic constant of 130. Especially intriguing is the evident worldwide attraction of Rubik's 3x3x3 cube.
With respect to Europe, it is a delightful coincidence that this incorporates the integers 1 to 27, namely the number of member countries of the European Union (now that the UK has designed itself out). Equally delightful is that prior to May 2004, the EU had 15 member countries -- the magic constant of a 3x3 magic square. With respect to Durer's 4x4 magic square, with its magic constant of 34, it is also a nice irony that Brexit has resulted in calls for 34 separate referendums across Europe.
Simple graphs, pattern recognition and symbolism: Far less commonly associated with symbolism and psychosocial significance, the simple graphs (of a set of vertices and edge), as recognized by graph theory, are typically associated with fundamental processes and patterns -- potentially of cognitive significance. Unfortunately these are readily confused with graphs of a function as a somewhat distinct preoccupation. Both lend themselves to pattern recognition.
Further confusion is created by the capacity of various software applications to render functions in graphical form, for example: the Graph Gallery of network graphs developed using NodeXL; the R Graph Gallery; the SAS Graphics Samples Output Gallery; and the MATLAB Plot Gallery. These all emphasize the presentation of data in "graphic" form, typically in conjunction with statistical analysis. Little effort is made to detect systematically the variety of "graphs" through which the operation of social media networks might be better understood, for example.
In any quest for clarification, or an understanding of the relevance to implied patterns of psychosocial dynamics, the degree of confusion is unfortunately compounded by the literature on the "classification of graphs". Some clarity is evident in the distinctions made in the case of functions (see eight basic types of functions: linear, quadratic, power, polynomial, rational, exponential, logarithmic, sinusoidal).
The literature on "classification of graphs" is however very extensive and with many issues seemingly unresolved (Kaspar Riesen and Horst Bunke, Graph Classification and Clustering Based on Vector Space Embedding, World Scientific, 2010; Yun Fu and Yunqian Ma, Graph Embedding for Pattern Analysis, Springer, 2012). There is little sense of the types of classification, of any classification of such types, or of how graphs of functions relate to those of graph theory.
Curiously however, some graphs are rendered in forms which are highly reminiscent of common symbols. Missing is the systemic development of the association between "graphs" and social processes, especially those with which many are familiar. Such associations are seemingly hidden in the extensive literature. Arguably the significance of many graphs is then clouded in a form of secrecy reminiscent of that associated with the interpretation of masonic and other symbols.
For the purpose of this argument, the focus is on what could be variously termed minimum, basic, fundamental, or simple graphs -- for which there may be extensive listings, but possibly without relevant images (Common Graphs. Sage).
|Indicative selection of simple graphs
(reproduced from Wikipedia: Gallery of named graphs)
Fundamental to such undirected graphs is the cycle basis, namely a set of simple cycles that forms a basis of the cycle space of the graph. That is a minimal set of cycles that allows every Eulerian path through the graph to be expressed as a symmetric difference of basis cycles. This set of subgraphs can be described algebraically. A Hamiltonian path (or traceable path) is a path in such a graph graph that visits each vertex exactly once. A Hamiltonian cycle is a path that constitutes a cycle. Of relevance to this argument is that the theory of Hamiltonian paths was first developed with respect to platonic polyhedra when represented as graphs.
Curiously, although there are a variety of studies on "learning cycles" in relation to graph theory, the sense in which traversing any such pathway cognitively could be associated with a learning process (or a habitual pattern) would seem to absent or obscured -- even with respect to process thinking, as discussed separately (Learning through Hamiltonian cycles and pathways, 2011). The issue of the classification of graphs may be less important than communicating insight into the transformation between classes.
Given the promotion of sudoku (and brain games) as a useful preventive exercise against ageing of the brain, and its relation to magic squares, provocatively the point could be made otherwise (Jeremy Grabbe, Sudoku and Working Memory Performance for Older Adults, 2011; Marcel Danesi, The Puzzle Instinct: the meaning of puzzles in human life, 2004). As a preventive measure, is there a case for engaging in a wider range of pathways -- rather than those characteristic of such games?
A remarkable exception is the study by Burkard Polster (The Mathematics of Juggling, 2006) which fruitfully integrates perspectives regarding polyhedra (and their great circles), Hamiltonian cycles and distinctive complex juggling patterns (with an annex of stereograms of more complex Hamiltonian cycles). It includes the following graph.
symmetric drawing of 3-ball state graph
(reproduced from Polster, 2006, p. 62)
In cognitive terms, this suggests a perspective relating to the art of unicycling and bicycling, in contrast to that of tricycling and quadcycling, whilst raising questions regarding "pentacycling" or "nonacycling" in a space of higher dimensionality.
As noted by Wikipedia, a magic graph is a graph whose edges are labelled by positive integers, so that the sum over the edges incident with any vertex is the same, independent of the choice of vertex. Further constraints identify graphs as supermagic, vertex magic or total magic, although there are currently various labelling conventions and terminologies (see images in K. Jeya Daisy and P. Selvagopal, Z4 Magic Labeling of Simple Graphs, American Journal of Mathematical Science and Applications, 2014).
The selection above includes the Durer graph which is mapped in relation to the Durer solid in the image discussed above. Of similar interest is the polyhedron formed from the Herschel graph also indicated above.
Possible variant of the Pentagramma Mirificum: Any quest for a configuration beyond discord can be usefully inspired by the discovery of the 5-fold spherical geometry of the so-called Pentagramma Mirificum. This originally proved to be fundamental to navigation of the globe, as separately discussed (Global Psychosocial Implication in the Pentagramma Mirificum: Clues from spherical geometry to "getting around" and circumnavigating imaginatively, 2015; Beyond dispute in 5-dimensional space: Pentagramma Mirificum? 2015).
Challenging speculative analysis: The previous sections, and the annex, note many patterns with which significance is variously associated -- and possibly deemed to be of the most profound importance by some. Such significance may well be deprecated by others. There is little capacity to reconcile such contrasting appreciations -- even within disciplines..
This difficulty is all the greater in that recognition of significance is typically associated with exclusive perspectives, readily perceived by others as dubious mystification -- or just "plane wrong". This is especially the case with respect to elitist goups, such as Freemasonry, and those attributing particular significance to information that they consider should be treated as secret -- as with intelligence agencies and their corporate analogues.
The issue is further complicated by a proactive initiatives through which the nature of questionable secret agendas is deliberately and cynically disguised by misinformation and disinformation -- possibly to the point of staging incidents and dramas as a focus of media attention. Speculation may thus be deliberately evoked in order to challenge its credibility as ill-informed -- at the same time calling into question any more informed speculation rendered indistinguishable by the process.
In the case of hierarchies and degrees of secrecy, mystification may be cultivated in order to disguise the relative poverty of the insight to which allusion is made by those upheld as being more deeply informed. Speculation -- however futile -- may well be tacitly encouraged as part of the process of upholding authority. Alluding to mystery in this way may bear comparison with a classic definition of a mini-skirt: long enough to cover the subject, but short enough to be interesting.
The most significant feature of the process of cultivating secrecy and speculation may follow from the insight of Marshall McLuhan: the medium is the message. More fundamental insight may well not be the issue, rather it is sustaining the pretence that it can be acquired which is systematically cultivated -- a form of Ponzi scheme with regard to intangible values.
Potentially more problematic is the degree to which such lack of transparency may disguise the inability of those who cultivate it to engage effectively in appropriate forms of self-criticism and self-reflexivity -- thereby inhibiting their own learning and reinforcing patterns unfit for evolving circumstances.
The paradox has been partially illustrated by the notorious "poem" of Donald Rumsfeld, presented at a press briefing whilst US Secretary of Defense during the intervention in Afghanistan (The Unknown, 12 February 2002), and discussed separately (Unknown Undoing: challenge of incomprehensibility of systemic neglect, 2008). The insight has been used in the analysis by Nathan Freier (Known Unknowns: unconventional 'strategic shocks' in defense strategy development, Strategic Studies Institute, U.S. Army War College, 2008).
|As we know:
There are known knowns.
There are things we know we know.
We also know: There are known unknowns.
That is to say: We know there are some things
We do not know.
But there are also unknown unknowns,
The ones we don't know
We don't know.
The theme is a feature of discussion by risk analyst Nassim Nicholas Taleb (The Black Swan: the impact of the highly improbable, 2007).
Institutional implications explored by conspiracy theorists: Much has been made of the masonic associations of the Founding Fathers of the USA, as well as the of the secretive influence of Freemasonry on governance worldwide in recent centuries. Commentary is rendered problematic in the light of recent surveys and studies of Freemasonry (Study on the Present Conditions of Freemasonry in the World, QSA Member Forum and Masonic Research, 2014; Gabrielle Pickar, Are The Freemasons Becoming Less Secret to Combat Declining Membership? Top Secret Writers, 13 April 2015; Social Issues Research Centre, The Future of Freemasonry, 2012). The latter introduces publication of a report of that title commissioned by Freemasons, and providing a focus for concerns at that time (Tom de Castella, Would you want to be a Freemason? BBC News, 9 March 2012).
Whether appreciated or deprecated (as noted by the entry on masonic conspiracy theories in Wikipedia), such speculation has been nourished by the architecture and symbolism seemingly cultivated as a consequence. The inherent secrecy, acknowledged as an overriding characteristic of Freemasonry, renders any discussion of its significance inherently questionable and problematic. This is also the case with respect to the influence of the Catholic Church (as exemplified by the entry on Vatican conspiracy theories in Wikipedia).
Potentially of particular relevance is the powerful role attributed to the Pentagon of the US Department of Defense (Stephen F. Vogel, The Pentagon: A History -- The Untold Story of the Wartime Race to Build the Pentagon and to Restore It Sixty Years Later, 2007). As with Freemasonry, the extent of the influence of the Pentagon and its associated agencies is deliberately shrouded in secrecy. This invites speculation of every kind and quality, including suspicions regarding secret modes of governance, as noted in the Wikipedia entry on Shadow government (conspiracy).
With respect to the construction of Washington DC itself, commentary includes Christopher Hodapp (Solomon's Builders: Freemasons, Founding Fathers and the Secrets of Washington DC, 2006). As argued, for example, by Alan Butler (Washington DC's Chamber of Secrets):
A Freemasonic connection to all the Megalithic lines of Washington DC cannot be denied. Many of the Presidents involved in various critical stages of the development of the city were high-ranking Freemasons. These include George Washington, Andrew Johnson and Franklin D Roosevelt. There were others but these three men seem to have had the most influence in the development of this system of Megalithic measures. In reality the later stages of the Megalithic adventure in Washington DC is, in itself, an allegorical story of Scottish Rite Freemasonry. In this branch of the Craft there are 33 degrees, or stages of initiation. Many Scottish Rite Masons achieve the 32nd degree but very few get to be genuine, full-status 33 degree Freemasons. Such a thing did not exist at the time of George Washington but both Andrew Johnson and Franklin D Roosevelt achieved this distinction. The 32nd degree of Scottish Rite Freemasonry, like all the others, has a symbol, which in the case of the 32nd degree is a pentagon.... It can be seen that within the 32 degree pentagon is a triangle, and this triangle is the sacred symbol of the 33rd degree of Scottish Rite Freemasonry.
Given the secrecy cultivated and suspected in relation to any such interpretations, it is not to be expected that much clarity will soon emerge. That said, any such interpretations play into preoccupation with the manner in which Washington DC has come to define itself as the nexus of global democracy, notably as the focus of major institutions of global governance such as the World Bank and the IMF arising from the Washington Consensus.
Bodies promoting the neoliberal agenda -- as with the World Economic Forum, the Mont Pelerin Society and the Club of Rome -- are naturally challenged to respond to accusations that significant numbers of their influential membership are Freemasons. Unfortunately for Freemasonry, it has become identified with an establishment which is now called into question by populist movements currently exemplified by the successful campaigning of Donald Trump.
Concerns are now raised with regard to the controversial extension of that consensus through the highly secretive elaboration of the Transatlantic Trade and Investment Partnership (TTIP) and the related Trans-Pacific Partnership (TPP), Trade in Services Agreement (TISA), as variously argued (Larry Elliott, The new Washington consensus, The Guardian, 5 October 2014; Wan Fayhsal, The Geopolitics of the Trans-Pacific Partnership Agreement: TPPA, Global Research, 15 January 2016; Jérémie Cohen-Setto, The Washington Consensus and neoliberalism, Bruegel, 3 June 2016).
Commentary, notably that of Antonio Gramsci (1975), readily associates the agenda of neoliberalism with that of Freemasonry (Dieter Plehwe, et al, Neoliberal Hegemony: a global critique, 2007; Carlos Alberto Torres, Neoliberalism as a new historical bloc: a Gramscian analysis of neoliberalism's common sense in education, 2013). Neoliberal agendas could even be explored as a "makeover" of those of Freemasonry.
Questionable parallels: It is therefore curiously appropriate to note the formal parallels between the degrees of secrecy cultivated within Freemasonry and those cultivated by the Pentagon in terms of degrees of security clearance. Both are understood to be protective of ever higher levels of insight purportedly vital to the highest values essential to governance and sustaining civilization. Members of both are sworn to secrecy, potentially irrespective of conventional considerations of legality and complicity in wrong-doing -- and with the threat of the severest sanctions in the event of disclosure.
The implication is one of ever higher levels of intelligent insight into the complexity of governance and order -- whatever integrative understanding and "oversight" is held to be associated with "global". However, as a consequence, the secrecy is intimately associated with strategic mystification and duplicity in relation to the levels of confidence implied by the succession of "initiations" into the insight which these patterns represent. The patterns are of course echoes of those traditionally cultivated in secret societies and religious priesthoods and are typically indistinguishable from the self-interest of those justifying such systems. Provocatively it might be asked whether the mindset of Freemasonry could be said to have had a significant role in "grooming" the thinking of the Pentagon.
The 32nd degree of Freemasonry is explicitly associated with a "Royal Secret" (I. Edward Clark, The Antiquity Of Freemasonry and the Royal Secret, 2010; Arturo de Hoyos, Freemasonry's Royal Secret: the Francken Manuscript, 2014). It is in this sense that there is some irony to the official claim that the recent disclosure of secret US diplomatic cables did not include the "Crown Jewels" associated with that secrecy (Barbara Starr, Official: Snowden did not get 'crown jewels', CNN, 22 July 2013).
The explicit focus now attributed to the Pentagon, as the primary instrument of governance of the world's acclaimed superpower, then raises the interesting question as to the roles associated with the other symbols configured in the iconic Camp symbol. Whilst that of the triangle is central to the iconography of the dollar bill as the world's default currency, the significance of the heptagon and the nonagon would seem to have been "lost". This is in the best traditions of the appealing mystification of the fiction of Dan Brown (The Lost Symbol, 2009; Washington DC and its freemasons braced for new Dan Brown book, The Guardian, 13 September 2009).
Symbolic absences? Potentially more significant is the apparent absence of the nonagon in the organization of contemporary governance. A curious exception, potentially trivial, is the past role of "nonagonal games" -- succeeding that of the heptagonal games (H. Jameson Swarts, Thirteenth Annual Nonagonal Games (Formerly Heptagonal Games), 1947).
Of particular relevance, however, is the current organization of the Roman Curia, the central governing body of the Catholic Church, into 9 congregations, under 3 secretariats (Secretariat of State, Secretariat for the Economy, Secretariat for Communications). Consisting of selected groups of cardinals, the nine are:
It is however noteworthy that US Supreme Court normally consists of the Chief Justice of the United States and eight associate justices -- currently the focus of extreme controversy (The future of the US Supreme Court is at stake no matter who wins in November, The Conversation, 20 September 2016; Presidential Election Will Shape Supreme Court, and National Policies, for Years to Come, The Wall Street Journal, 22 July 2016; The Supreme Court reminds everyone why it's the most important issue in the 2016 election, The Washington Post, 23 June 2016).
There is however the possibility that the heptagon is typically implied by the 7-fold representation in any Council of the Wise -- and possibly in formulation of any associated 7-fold declaration of strategic principles. Potentially more intriguing in relation to Freemasonry is that its 33rd degree (allegedly the highest), can be understood in relation to the focal triangle of its symbolism as 3x11, thereby embodying the hendecagon -- in anticipation of the dodecagonal organization which is so evident otherwise.
Any "absence" is evident to an even higher degree through the cultivation elsewhere of 12-fold patterns of governance (Checklist of 12-fold Principles, Plans, Symbols and Concepts : web resources, 2011). However the dodecagon does not figure in the Camp symbol, despite its fundamental importance to traditional understandings of governance. These remain an inspiration for civilization (as with the Greek Dodekatheon and the Roman Dii Consentes).
Encampment: Do such "absences" suggest that the current focus on the Pentagon as an instrument of governance is in effect an instance of the reification fallacy of misplaced concreteness -- lacking as it does the subtler framing implied by the heptagon and the nonagon in the Camp symbol. This is curiously ironic given the worldwide "encampment" of a network of the 1000-plus military camps and bases by the Pentagon in preparation for the various campaigns envisaged (Wikipedia, List of US Military Bases; Jules Dufour, The Worldwide Network of US Military Bases: the global deployment of US military personnel, Global Research, 1 July 2007). Or is the organization of the US Supreme Court to be considered as just such a framing -- despite the disputes between the Pentagon and the US Department of Justice?
The camp-focus of the Pentagon merits comparison with that of Freemasonry through its Camp icon and the military mindset that engendered and cultivated. Provocatively this may be extended to the campus preoccupation of academia. Given the selective framing of enemies by all three, "camp-us" can be usefully understood as configured in response to "camp-them" (Us and Them: Relating to Challenging Others -- patterns in the shadow dance between "good" and "evil", 2009; Changing Patterns using Transformation Pathways: exploring "camp-us" inspiration by an alien world view as a metaphor 2015). A "camp-us" encampment is reminiscent of the process of circling the wagons to protect the livestock.
The question is whether military terminology and "holy war" offer a dynamic of subtlety appropriate to the times (Enhancing Sustainable Development Strategies through Avoidance of Military Metaphors, 1998). Use of "crusade" has notably characterized the questionable framing of response to Islamic jihad (Jay Rubenstein, Crusade vs. Jihad: Which Is Worse? The Huffington Post, 19 Decedmber 2011; James M. Arlandson, Islamic Crusades vs. Christian Crusades: who should own the "Kingdom of Heaven"? Answering Islam).
As a distinctive feature of the Camp icon, there is some irony to adoption of the "tent" metaphor with respect to the multiplicity of polico-religious gatherings. A case has been made for a "big tent" to encompass the diversity of political and religious perspectives. This could itself be usefully reframed in terms of the quality of thinking from astrophysics (Global Brane Comprehension Enabling a Higher Dimensional Big Tent? Strategic implication in encompassing nothing and coming to naught, 2011).
Given the widespread importance associated with 10-fold patterns of organization -- a (facile) binary reflection of pentagonal organization -- is it possible that the "gearbox" of governance has lost any functional coherence as an insight "transmission system"? Irrespective of the case of the heptagon and the nonagon, what of the hendecagon as a "transitional gear" to dodecagonal organization and the enlightenment with which it is so widely held to be associated?
The interfaces between these modalities would seem to be fuzzy and reflective of institutional practices potentially "unfit for purpose" -- as may be so perceived by the future.
This argument has used the Eastern "mandala" as a way of framing what might be recognized as some combination of meanings, paralled by the Western rose window (M. Brauen, The Mandala: sacred circle in Tibetan Buddhism, 1997; Giuseppe Tucci, The Theory and Practice of the Mandala, 1973; Painton Cowen, The Rose Window, 2005). In cognitive terms these may be variously associated with a sense of portal, gateway or doorway (Aldous Huxley, The Doors of Perception, 1954; Vortices, Portals and Dimensional Doorways, Rodin Coil Project, 1998; Philip Gardiner, Gateways to the Otherworld, 2007). Naming a popular computer operating system and interface as Windows is consistent with this pattern.
Sphincters and orifices: The operation of such a device may be imagined in mechanical terms as an annular valve, or physiologically as with the mitral annulus or some kind of sphincter, like that of the iris.
The sphincter is a circular muscle that normally maintains constriction of a natural body passage or orifice and which relaxes as required by normal physiological functioning. There are over 60 types in the human body, some microscopically small (Robert W. Lam and Shinya Fukuda, Sphincters: properties, types and applications, 2012).
Portals of the imagination: Through science fiction and fantasy the collective imagination has cultivated the sense of a portal -- now variously rendered ever more fascinatingly attractive (if not hypnotic) through special effects. This is understood to be a technological or magical doorway that connects two distant locations separated by spacetime. It usually consists of two or more gateways, with an object entering one gateway leaving via the other instantaneously.
One example is the stargate of the TV series (the theme of the Stargate Wiki; notably evoking the creativity of artists such as Silvia Hartmann, Stargate Symbols). Another is that of the imaginative mechanism presented in the movie Contact (1997), as shown below --imagined in the light of science fiction novel by astrophysicist Carl Sagan (Contact, 1985).
|Images associated with the Contact movie (1997)|
(distributed by SciFi Geeks)
As described by John DeNardo (Portals in Science Fiction and Fantasy, Kirkus Reviews, 20 November 2013):
The most obvious definition of portals -- doorways to other places -- is too simplistic to convey the true nature of them. Sometimes those places are real, but far away. Sometimes they are fantasy worlds that shouldn't exist but do. Sometimes they are physical, sometimes metaphorical, mere plot devices to advance the story. They could take many forms, from holes in the ground, to mirrors, to large constructs big enough to fly a starship through. Sometimes they aren't about traveling distance at all, but instead are about traveling through time.
The numerous references to one design of such portals (or the doorways commonly depicted in the space ships of science fiction) can be traced through terms such as "irising doorways" ("irising portal", "irising sphincter"), but curiously there would seem to be little indication as to how these might be designed in practice -- other than for their use in theatre. Notable exceptions include: the work of Carolina Rodriguez (The Swivel Diaphragm: a geometrical examination of an alternative retractable ring structure in architecture, 2006); the irising window of the Computer Science Department (University California San Diego); and some work on irising apertures (T. Galal, Solar Central Receiver with Irising Aperture, 2010).
Curiously a report on NASA-sponsored research has recently noted a newly discovered portal phenomenon (as previously imagined by science fiction), namely places where the magnetic field of Earth connects to the magnetic field of the Sun, creating an uninterrupted path leading from our own planet to the sun's atmosphere 93 million miles away (Hidden Portals in Earth's Magnetic Field, NASA Themis, 2 July 2012) :
Just one problem: Finding them. Magnetic portals are invisible, unstable, and elusive. They open and close without warning... Portals form via the process of magnetic reconnection. Mingling lines of magnetic force from the sun and Earth criss-cross and join to create the openings. "X-points" are where the criss-cross takes place. The sudden joining of magnetic fields can propel jets of charged particles from the X-point, creating an "electron diffusion region."
Singularity as portal: Much is currently made of a singularity, whether as a phenomenon of astrophysics (possibly associated with a wormhole), a problematic technological development, or as an opportunity for transportation to other regions of the universe -- a portal. It may refer to civilizational collapse and end times scenarios -- offering a distinctive sense of portal.
Possibly more intriguing is the sense of a cognitive singularity of some kind with which the technological variant may be confused, as discussed separately (Emerging Memetic Singularity in the Global Knowledge Society, 2009). Related terms of relevance might include a "memetic singularity", a "semantic singularity", or a "meta-systemic transition".
Magical mirrors: A portal may be imagined as having qualities of a mirror -- with the possibility of stepping into and through it (Stepping into, or through, the Mirror: embodying alternative scenario patterns, 2008; Lewis Carroll, Through the Looking-Glass, 1871).
Countries, communities and organizations cultivate the legendary posture of inquiring: Mirror, mirror, on the wall, who is the fairest of them all? Expectant inquiry about the future has variously focused on a "mirror" -- whether magical or metaphorical -- and the "speculation" to which it gives rise. This is notably evident in the sophisticated strategic philosophy underlying the BaGua mirror of Chinese culture (Keith Rudell, Feng Shui and the Ba Gua Mirror).
The implications of mirroring have long been an inspiration for introspection across cultures (Bill Halpin, Engaging Emptiness: Stepping into the Mirror, 2000; Paul Demiéville, The Mirror of the Mind, 1991). This exploration of collective self-image might be understood as an essential feature of modern public relations, self-promotion and their "cosmetic" requirements.
To a curious extent, investment in the technology of portal design (as imagined) has been anticipated in the widespread deployment of the security gates by which people are obliged to be scanned at the entry to new environments and for transportation elsewhere. They offer a suggestive metaphor regarding the cognitive implications of passage from one degree of initiation to another, especally since their design may be reminiscent of the pillars through which people symbolically pass in the course of initiation rituals.
Chakras: The form of any such "mandala" figures notably in depiction of the Hindu sequence of seven basic chakras variously understood as psychic centres of consciousness. It is claimed that these may be successively "awakened" through the meditative practice of kundalini yoga -- from a condition in which energy is held to be "coiled up" at the base of the spine, through to more evolved conditions of awareness.
|(images by Morgan Phoenix; reproduced from Wikipedia)|
The form of these traditional patterns is curiously reminiscent of the patterns of polygons discussed above -- with the emphasis here on those with an even number of sides (complementing those presented earlier with an odd number of sides).
Strange attractors of overriding fascination: As stressed in relation to the portals of science fiction, such patterns have long offered a strange fascination through the imaginative possibilities they evoke. These relate to the sense of a mysterious doorway to a different "place", "time" or mode of understanding. They offer a sense of potential in ways whose nature has yet to be explored. Its characteristics include an unusual degree of integration -- cultivated otherwise in the quest for unitary patterns. Often described metaphorically in terms to configurations of petals in the case of chakras, aesthetically they offer a sense associated with flowers (Flowering of Civilization -- Deflowering of Culture, 2014; Keith Critchlow, The Hidden Geometry of Flowers: living rhythms, form and number, 2011).
Controversially, perhaps of fundamental necessity, this fascination is consistent with the overriding fascination with sexual attraction and the unconstrained impulse to intercourse -- with "the birds and the bees". The association is allusively elaborated in poetic and other references to "secret garden". Through the sense of consummation offered by intercourse it offers imagined forms of transformation of which the portals of science fiction could be readily understood to be surrogates -- perhaps to be deprecated as the palest of surrogates. The association of the pattern of chakras with the controversial practices of tantric yoga is consistent with this fundamental association (Emine Saner, Take me to the cosmic vagina: inside Tibet's secret tantric temple, The Guardian, 10 November 2015).
Although readily avoided, the preoccupation with sexual attraction and the promise it offers is a major factor in commercial advertising. Most curious, however, is the degree to which this experiential factor is systematically ignored in the framing of policy, despite the requisite scientific expertise, as discussed separately in a review of The Royal Society report People and the Planet (Scientific Gerrymandering of Boundaries of Overpopulation Debate, 2012). As with academia, there is every reason to argue that neither the Abrahamic religons nor Freemasonry have any meaningful understanding of sexuality -- other than at some deniable subconscious level over which they seek unquestionable dominance. They are essentially sexless in any experiential sense, despite the potentially fruitful cognitive implications embodied in their respective symbolism.
Intercourse: "Intercourse" with reality in whatever form then lends itself to further significance through its consequences ("Human Intercourse": "Intercourse with Nature" and "Intercourse with the Other", 2007). These can be variously understood both in terms of "birth" and as "rebirth" (Varieties of Rebirth -- distinguishing ways of being "born again", 2004). The latter distinguishes:
In a number of these cases, such rebirth may be framed as a form of initiation into an order of understanding distinguished metaphorically as "higher", however that is to be understood. More challenging is the capacity to recognize the nature of such understanding -- and to engage with it -- given the complexity with which it is purportedly associated (Engaging with Insight of a Higher Order: reconciling complexity and simplexity through memorable metaphor, 2014).
Questionable answers: It is especially intriguing that engaging with any kind of portal tends to hold the attractive implication that intractable existential questions may be thereby fruitfully answered otherwise -- whatever that may signify. With respect to any degree of rebirth, this calls into question the very nature of questions and the answers deemed fruitful.
Especially with regard to any sense of identity, intercourse of any form is then to be understood as potentially reframing the classical WH-questions: when, where, which, what, who, why and how. This is as much the case with regard to the portals imaginatively offered by science fiction and fantasy as it is with regard to sexual consummation and the sense of transcendence it may be imagined to offer. Presumably, according to degree, such intercourse reframes a sense of direction and orientation -- and purpose. Its implications are necessarily incommunicable and "secret".
Arguably the transformation offered by sexual intercourse, through the existential questions it raises, constitutes a secular equivalent to the quest for the Holy Grail, as can be variously explored (Interrelating Cognitive Catastrophes in a Grail-chalice Proto-model: implications of WH-questions for self-reflexivity and dialogue, 2006). In a period in which there is reference to the need for "economic viagra" to ensure "sustainability" (The Viagra economy, The Economist, 22 September 2005), the metaphor suggests even further exploration (In Quest of Sustainability as Holy Grail of Global Governance, 2011).
Degrees of masonic intercourse with reality: The influence, degrees and claims of Freemasonry offer an example variously characteristic of rebirth in other belief systems. Given its secrecy it is necessarily a focus of conspiracy theorists as noted above. In a period when the relation between the sexes has long been vigorously called into question as a philosophy justifying masculine exclusivity and dominance merits particular attention, as in the case of the Catholic Church (Freemasonry and women; Ordination of women in the Catholic Church).
The difficulty extends in various ways, and to varying degrees, to the relation between "whites" and "people of colour", and to those exploring distinctive gender identities. The strategic dependence on the posture of engendering and confronting enemies can be understoodmore generally as a profound difficulty in engaging creatively with "otherness" -- despite claims to the contrary (Reframing the Dynamics of Engaging with Otherness, 2011; Us and Them: Relating to Challenging Others, 2009).
These questionable attitudes are of course evident and highly influential in the other Abrahamic religions. Together they have positioned women most problematically -- contrary to the principles of human rights they otherwise claim to promote and uphold -- and have actively inhibited any constraint on the overpopulation which engenders so much suffering (Indifference to the Suffering of Others: occupying the moral and ethical high ground through doublespeak, 2013).
The use of the frock-like vestments of the male priesthood of various religions -- explicitly recognized in terms of "defrocking" -- offers one point of departure, as with the cover-up of sexual above by clergy. The use of decorative "aprons" by Freemasonry, to distinguish degrees of insight, offers another. In a period of extensive debate on the implications of cross-dressing, the implication is both cases call for radical reassessment from the perspective of such as Freud, Jung or Lacan (Ph. D. G. G. Bolich, Transgender History and Geography: crossdressing in context, 2007).
It is especially curious that the aprons can be recognized as decorated with symbols suggestive of particular modes of intercourse with the reality of the world -- and that, as aprons, they should be worn in a position associated with the male organ primarily preoccupied with physical intercourse. The position is reminiscent of the traditional biblical fig leaf -- itself with controversial implications, given complicity in so-called cover-ups and issues of impunity (Hillsborough: Police Watchdog Pursues Freemason 'Line of Inquiry', Sky News, 25 August 2016).
The question here is far less a matter of prudery than of the subtle cognitive dimensions to which prudery is an embarrassed simplistic response. Given the suspicions of conspiracy theorists, there is a sense in which the degrees of Freemasonry can then be held to distinguish ways in which its members may engage in "fucking the world", however this may be more euphemistically expressed. The same can be imagined with regardto the degrees of secrecy considered essential by the intelligencee services to thier role in safeguarding democracy.
This could be explored more appropriately in symbolic terms, as discussed separately (Reframing the Dynamics of Engaging with Otherness: triadic correspondences between Topology, Kama Sutra and I Ching, 2011).
Operative Freemasonry: The need for the development of "operacy", as the "skill of doing", has been promoted by Edward de Bono (Judgment, Recognition and Operacy, Extensor). As noted in the Routledge Encyclopaedia of Educational Thinkers, 2016, p. 540):
In his work related to education, de Bono places a great deal of emphasis on "operacy", a term mwhich he uses to refer to the practical skill of thinking for action and which he advocates as a thrid "gateway" for success in formal education (together with literacy and numeracy).
A fundamental distinction is made between Speculative and Operative Freemasonry -- however the latter may be associated with "operacy". For those concerned with the undue influence of Freemasonry on governance, "speculation" could indeed be discounted as merely a matter of opinion, however it is related to degrees of insight. There is of course the irony that in so specifically promoting speculation, Freemasonry has invited such a problematic degree of speculation regarding its own agenda and influence.
Of far greater concern is the implication of "operative" beyond the "good works" with which Freemasonry commonly claims to be associated in its relation with the public -- much as may be claimed by major corporations as a manner of distracting from more dubious practices. This is especially the case given the importance attached to secrecy. Those who have the power to deceive necessarily have the greatest difficulty in providing credible proof that they have not -- protected as they are by plausible deniability. The degrees of Speculative Freemasonry may well be designed to camouflage the degrees of Operative Freemasonry -- as a predictable form of dissimulation for the unvigilant.
In a period in which there is general concern with secretive "operators" and their skills in influence peddling in governance (as exemplified by lobbyists), the focus of conspiracy theorists is naturally on the subtler forms of confidence trickery which may be enabled by the secret skills associated which the higher degrees of Freemasonry. How do Freemasons "work their will" on society in their intercourse with it -- framed as they have long been by a masculine mindset?
How might these skills serve in the defensive denial of complicity in wrong doing -- a pattern of techniques remarkably identified in the forms and degrees of cover-up by Wikipedia, as discussed separately (Identifying patterns of moves with respect to "intelligence", 2013)? Curiously the process relates to the preoccupation with covering up loopholes and lacunae -- or with their exploitation.
Such questions are especially relevant in a period in which there is concern with lack of transparency and ethical conduct in institutuons such as the European Commission -- whether with respect to lobbyists or as exemplified by the current cases of Jean-Claude Juncker, Neelie Kroes and José Manuel Barroso (Commission bumps into its own revolving door, Politico, 23 September 2016). Justifying the suspicions of conspiracy theorists, it is is of course impossible to ask how many of those involved are subject to questionable pressures from Freemasonry, the Caholic Church, or otherwise.
Given the cultivation of complex patterns of secrecy by both the intelligence services and by Freemasonry, it is appropriate to ask whether the skills of covert operations by the former (and the use of false flags) has been significantly adopted by the latter (from which they may well have derived).
To what extent have interpretations of the goals and values of democracy been cynically promoted as false flags-- as "humanitarian shields" -- primarily to distract collective attention in order to disguise the allocation of resources otherwise? The trinity of "Liberty-Equality-Fraternity" is upheld as the ultimate nexus of core values (Democratic Values: Liberty -- Equality -- Justice, American Government Online) -- deriving from the tripartite preoccupation institutionalised by France: Liberté, Egalité, Fraternité and adopted by many other nations as an ideal. As might be suspected it is alsothe motto of the Grand Orient de France and the Grande Loge de France, features widely in masonic ritual and symbolism. It was also significant in the elaboration of the the First Article of the Universal Declaration of Human Rights.
Could the motto be explored as an inherently unquestionable false flag? The evidence is most striking with respect to the treatment of women by those who promote that trinity of values -- "equality?", "liberty?, "fraternity?" -- as separately argued (Cultivating the Myth of Human Equality: ignoring complicity in the contradictions thereby engendered, 2016; Randall C. Pendland, Separate but Equal Brotherhood, Order of Former Freemasons, 2003; Harry A. Gaylord, The Slavery of Freemasonry, Sun and Shield, 31 July 2007).
The point might be explored through reframing the economic policy trilemma, also known as the impossible trinity, whereby a country must choose between free capital mobility, exchange-rate management and monetary autonomy -- the three together being incompatible, although any two are viable (The Mundell-Fleming Trilemma: two out of three ain't bad, The Economist, 27 August 2016).
Is the "fraternity" so fundamental to Freemasonry essentially abusive -- as may be the perspective of the future? Could the key international bodies, in which the proportion of masonic membership is suspected to be significant, indeed be engaged in such dynamics -- potentially matched by the corresponding initiatives of the Catholic Church as has also been suspected? If neoliberalism could indeed be explored as a "makeover" of those of Freemasonry, is there a case for recognizing the emergence of a dubious "neoegalitarianism" or a "neofraternalism" (as already debated)?
Holes and orifices: However the mandala is to be speculatively understood as a mirroring portal -- a cognitive sphincter protective of a complex orifice -- the "hole" which is the focus of any form of intercourse merits new thinking. The centro-symmetrical nature of the Camp and the symbols of lower degree could also call for reflection in terms of sexual dysfunction -- consequent on confrontation with a form of vagina dentata.
The neglected significance of a "hole" has been remarkably discussed by Roberto Casati and Achille C. Varzi (Holes and Other Superficialities, 1994) -- with respect to the borderlines of metaphysics, everyday geometry, and the theory of perception (as they summarize in the entry on holes in the Stanford Encyclopedia of Philsophy).
The authors seek to answer two basic questions: Do holes really exist? And if so, what are they? Philosophers would typically like to expel holes from their ontological inventory. Arguing in favour of the "existence" of such absences as full-fledged cognitive entities, the authors examine the ontology of holes, their geometry, their part-whole relations, their identity, their causal role, and the ways they are perceived. In cylindrical form holes are centred on an implied axis, with which a "point" can only be associated dynamically -- or as a succession of "points".
So framed, what is the strange attraction of the hole which is such a powerful catalyst for intercourse?
The significance can be partly explored through the arguments regarding the curious potential of "nothing" and the "missing", as presented by Terrence Deacon (Incomplete Nature: how mind emerged from matter, 2011), under the heading Nothing Matters:
When Western scholars finally understood how operations involving zero could be woven into the fabric of mathematics, they gained access to unprecedented and powerful new tools for modeling the structure and dynamics of the physical world. By analogy, developing a scientific methodology that enables us to incorporate a fundamental role for the possibilities not actualized -- constraints -- in explaining physical events could provide a powerful new tool for precisely analyzing a part of the world that has previously been shrouded in paradox and mystery. The mathematical revolution that followed an understanding of the null quantity in this way may presage a similarly radical expansion of the sciences that are most intimately associated with human existence. (pp. 540-541)
Scoring a goal and making a point: The powerful attraction of a hole could be considered to have extraordinary implications of relevance to this argument. This is most evident in the preoccupation with "scoring", whether understood metaphorically in regard to intercourse, or as the notable characteristic of many ball-games.
As noted above, it is remarkable that both the football and the golf ball have designs based on spherical polyhedra. More remarkable, ironically, is the extent to which such symbolic artefacts are kicked, thrown or hit in the competitive dynamics of scoring -- so extensively the focus of global media coverage and gambling.
With respect to the symbolism of Freemasonry and other belief systems, the central point of a symbol (whether explicit or inferred) has special significance as an origin. This is consistent with the significance associated with "making a point", as discussed separately (Cognitive Realignment: making points and aligning a target, 2009; Comprehension framed by "Point", 2012 ).
With respect to the aesthetic dimension of this argument, of interest is the envisaged use of a new material to define a hole otherwise (Mary Nichols, Researchers Create 'Super Black' Material So Dark The Human Eye Cannot Process It, Design and Trend, 14 July 2014). Known as Vantablack, it has become the focus of aesthetic and copyright controversy (Artists at war after top sculptor is given exclusive rights to the purest black paint ever which is used on stealth jets, Daily Mail Online, 15 March 2016; Jonathan Jones, Can an artist ever really own a colour? The Guardian, 29 February 2016).
Holes and "The-O"? A more provocative approach challenges the connotations of "theo" (The-O ring: Theory, Theorem, Theology, Theosophy? a playful intercultural quest for fruitful complementarity, 2014). It explores the patterns of cognitive functions associated with variants of "theo". In a second part, the concluding argument is usefully clarified using a variety of illustrations relating the theme to current global preoccupations (The-O Ring and The Bull Ring as Spectacular Archetypes: dramatic correlation of theatre, theory, theorem, theology, and theosophy, 2014).
The emphasis there is on how dramatic incorporation of the feminine can enhance "interestingness, suggestiveness and memorability" of the "theo" modalities -- otherwise to be recognized as increasingly sterile, infertile and "unfit for purpose". Could it be argued that a degree of recognition of their dimensions has encouraged Freemasonry to transform the dnamics of society into an instructive drama -- whether tragedy, comedy or farce?
The mysterious nature of holes and their implication can be given another focus through comprehension of the paradoxical hole in the Klein bottle as studied by mathematics. The psychological implications of this form have been extensively argued from a phenomenological perspective by Steven M. Rosen (Wholeness as the Body of Paradox, 1996; Pauli's Dream: Jung, Modern Physics, and Alchemy, Quadrant, 44, 2014; A Plea for the Possibility of Visualizing Existence, Scientia, 108, 1973).
Given the mysterious nature of that hole as a focus for comprehension, it could be argued that its conventional representation in terms of a "bottle" is fundamentally misleading, given its hyperdimensional nature. Provocatively it could be further argued both that that rendering is curiously reminiscent of a uterus and that understandings of "holiness" merit reflection in such terms (Is the World View of a Holy Father Necessarily Full of Holes? 2014). What are the implications of being (w)hole? Does engagement with a hole have implications for engagement with globality, as separately explored (Intercourse with Globality through Enacting a Klein bottle: cognitive implication in a polysensorial "lens", 2009).
With respect to aspirations to the initiation of humanity into galactic civilization, further speculation might suggest that the recognition of the cognitive skills to navigate paradoxical wormholes might be an essential prequisite (Self-reflective Embodiment of Transdisciplinary Integration (SETI): the universal criterion of species maturity? 2008). A related point has been made through the theme Down the Rabbit Hole otherwise controversially presented as What the Bleep Do We Know!? (2004).
"Con" as a prefix: As noted above the inspiration of this argument derived from earlier recognition of the remarkable function of "con", most notably in English (Considerable Conglomeration of "Cons" of Global Concern, 2012). This resulted in the preceding concern with the Concordian Mandala (Con-quest Aesthetically Reframed via the Concordian Mandala: inspired by implications of the systemantics of the Discordian Mandala, 2016). The concern in the latter was the possibility of designing a suitable "container" for the set of essentially elusive human values. The container was understood as a mapping device for the set of 5x9 clusters of such values resulting from the Human Values Project -- especially given the 5x9 configuration of the Concordian Mandala.
Of particular relevance to the times is how confidence is to be engendered and sustained in a context in which it is in process of being eroded in every domain. The argument has been developed from an economic perspective (Trust Busting: the dangerous contradiction between economic reality and political rhetoric, The Economist, 17 September 2016). The latter notes:
People no longer trust mainstream politicians, nor indeed the media that report on them. This of course has huge economic as well as political consequences.... Trust is built into the heart of of almost all economic activity... Any hint of a general erosion of trust... would be a very worrying sign.... Yet complacency would be unwarranted. The 2007-08 financial crisis sjowed wat can happen when belief in the financial system breaks down.... Today's economy and financial system depend on global co-operation; today's political system is one where such co-operation is increasingly seen by voters as intrinsically suspicious. That is a dangerous disconnect.
With respect to the dubious connotations of "con", other uses merit concern, as in the case of neocon -- with its implication of a new form of confidence trickery. As noted above, if neoliberal agendas could be explored as a "makeover" of those of Freemasonry, is there a case for "reconnaissance" with regard to the emergence of equally suspicious "neoegalitarianism" and "neofraternalism"? In that sense, is reconnaissance an appropriate prelude to renaissance?
Conflation and disconnectivity: The dynamics of the mandala are also suggestive of conditions under which the 5 nonagonal sets may be confused and conflated, as with the 9 pentagonal sets. In this sense any pentagonal configuration may be inadequately informed by the 9 forms of 5-foldness, and any nonagonal configuration by the 5 forms of 9-foldness.
The mandala configuration of 5 sets of 9-fold rings is potentially valuable in ensuring forms of connectivity and disconnectivity which may be systemically appropriate. The dynamics of the mandala are also valuable in indicating conditions under which connectivity and direct linkage between values may be inappropriate and dysfunctional. The Borromean ring configuration frames a higher order of connectivity -- especially given the possibility of variants of even higher order (Nils A. Baas, Higher order architecture of collections of objects, International Journal of General Systems, 44, 2015, 1).
Semiotic nonagon: Of potential relevance to this argument is the criticism by Claudio F. Guerri (Philosophy and Symmetry in the Semiotic Nonagon. International Interdisciplinary Conference "Symmetry of Forms and Structures", 2009)
There was a significant gap between the complex philosophical proposals of Charles S. Peirce, as such, and the potential to extract out of it useful applied semiotics without descending into a reductivism or an injurious distortions of the original purpose. The aim of this paper is to show that the well-known triadic sign classification of Peirce can be designed to perform as a diagrammatic icon in order to work as an operative model. In the design format of the nonagon, symmetry becomes a helpful tool of an operational and cognitive sort.
Following Peirce, Guerri stresses that All valid necessary reasoning is in fact diagrammatic. He introduces the Semiotic Nonagon by taking account of Peirce's concept of the sign
According to Peirce, a sign is not determined by its proximity to other signs as in Saussure'?s theory, but is chained, in a semiotic process, to all other signs through a triadic relationship. The Semiotic Nonagon acting in different ways, prioritizes the possibility to comprehend, to analyze, and to produce signs: that is, the matrix renders the possibility to organize within a logical framework, the nine fields, and allows a phenomenological description, and the relationships of the nine fields of the matrix sustain the cognitive processes. Peirce maintains that diagrammatic icons always contribute to knowledge. The most important aspect of the Semiotic Nonagon, the value of the Nonagon, is to establish the internal dynamics of the nine parts of the sign, which are strongly interdependent, and the relation of the sign to the Dynamic Object.
Guerri argues that there is no symmetry in Peirce?'s proposal of sign classification, at least not in the limits of human knowledge, only the real incommensurable universe of knowledge may possibly be completely symmetric. In the semiotic nonagon, as presented below, there is an apparent diagonal mirror from FF through EE to VV, even the letters FE and EF are mirror images; but it is a mirror with a sort of a palindromic twist, since FE and EF are not interchangeable in conceptual content.
|Semiotic nonagon of Claudio Guerri
(reproduced from Philosophy and Symmetry in the Semiotic Nonagon. 2009)
|Indication hree superposed mirror symmetries
along this diagonal: F of F, E of E and V of V.
|Indication of complementarity and specificity
of the horizontal and vertical axes.
Identity containment and embodiment: Engagement with a mandala, especially as a portal, is intimately associated with identity, its comprehension, its dynamic and its transformation.
This can be variously recognized and enabled through use of other symbolic devices most notably a circlet of beads, a sash, a ring, a necklace, a torc or a crown. These can be endowed with unusual properties and practices (Designing Cultural Rosaries and Meaning Malas to Sustain Associations within the Pattern that Connects, 2000; Engaging with Globality through Cognitive Crowns, 2009).
The multiple nonagons of the mandala presented offer the sense of a multiplicity of such devices -- interweaving dynamically to frame what is otherwise understood statically and therefore only to a limited degree, as argued otherwise (From Statics to Dynamics in Sustainable Community, 1998; Dynamic Transformation of Static Reporting of Global Processes, 2013). As noted, there is necessarily a degree of paradox and a challenge to comprehension of what is thereby contained, embodied or implied. Rather than the self-reflexive emphasis on the singular by Douglas Hofstadter (I Am a Strange Loop, 2007), there is necessarily a multiplicity -- to be understood as a requisite variety (Sustaining a Community of Strange Loops: comprehension and engagement through aesthetic ring transformation, 2010).
Further challenges and possibilities are indicated by the subsequent work of Douglas Hofstadter and Emmanuel Sander (Surfaces and Essences: analogy as the fuel and fire of thinking, 2013). Any sense of a central hole can be cultivated as a portal -- potentially reinforced by experiential sexual connotations and associated metaphors. The singular strange loop recalls the widespread depiction of a halo in relation to a person esteemeed holy -- possibly now presented as an infinity sign (or a Mobius strip) to imply more fundamental complexity.
The argument here is that the singular form could be presented as multiple rings -- perhaps recalling electron orbitals and their wave-like implications. Whether together or separately the sense they offer of a portal could be playfully framed as a porthole -- or even a "portwhole" -- as a means of transporting a way of seeing or a perspective. In that sense the 5 rings are also usefully reminiscent as framing the 5 senses enabling engagement with physical reality.
As argued with respect to correspondences, metaphor would appear to offer a key to more coherent engagement with globality (Theories of Correspondences -- and potential equivalences between them in correlative thinking, 2007; Metaphorical Geometry in Quest of Globality -- in response to global governance challenges, 2009).
Missing however is any geometric or topological indication of degrees of self-reflexivity, however these might be implied by degrees of insight, of rebirth and of freemasonry. The need for such in relation to coherent identity can be presented in terms of the viable systems model, and more specifically in relation to knowledge cybernetics (Maurice Yolles, Knowledge Cybernetics: a metaphor for post-normal science, 2012). Beyond the requirements for double loop learning, as variously advocated, the arguments of Maurice Yolles and Gerhard Fink point to the potential role of system comprehension in terms of third-, fourth- and fifth-order cybernetics (Generic Agency Theory, Cybernetic Orders and New Paradigms, 2014).
Clearly there is however a collective challenge of comprehensibility and incomprehension in a learning-constrained society. Neglect of these dimensions of identity has every probability of contributing to system failure, as discussed separately (Variety of System Failures Engendered by Negligent Distinctions, 2016).
In contrast there is the sense in which people are already obliged to live "between worlds", irrespective of whether they are imaginatively attracted to some such modality (Living as an Imaginal Bridge between Worlds: global implications of "betwixt and between" and liminality, 2012). With reality in process of being reframed in terms of quantum mechanics, and by identification with music, indvidual identity may already be better understood through wave metaphors (Being a Waveform of Potential as an Experiential Choice: emergent dynamic qualities of identity and integrity, 2013; Being Neither a-Waving Nor a-Parting: cognitive implications of wave-particle duality in the light of science and spirituality, 2013).
Reframing the I-conic: Given the degree to which authority makes explicit use of a symbolic sash -- especially the highest authority with which importance is implied and expressed -- the implications of donning such a device merit the most careful attention with regard to the containment and embodiment of identity. This can be fruitfully explored (as with other symbols worn in this manner) in terms of traditional use of wampum. This could be reframed with the newly recognized implications of quantum reality (Quantum Wampum Essential to Navigating Ragnarok: thrival in crisis through embodying turbulent flow, 2014). Would multiple sashes be more consistent with the implications for identity-- or the desperate quest to embody their meaning and cognitive implication?
The set of patterns, especially those which are centro-symmetric, merits similar attention -- especially through the manner in which they carry cognitive significance. It is argued above that the simplest in 2D, which are characteristic of many symbols, are variously associated with psychosocial processes -- whether familiar or unrecognized (even held to be "secret").
Current preoccupation with pattern recognition calls for its consideration in cognitive terms. As argued, this is equally true in 3D, and potentially more pregnant with significance. The cognitive implications of 4D remain to be explored through polychora, given widespread preoccupation with time and cycles (Four-dimensional requisite for a time-bound global civilization? 2015; Comprehending the shapes of time through four-dimensional uniform polychora, 2015; Marco Möller, Four-dimensional Archimedean Polytopes, 2004).
With respect to collective governance, and especially concerns with regard to corporate megastructure of the military-industrial complex, there is a case for recognizing how these interweave in 3D and 4D -- beyond the preoccupations these engender in the many conventional 2D network representations in "plane sight". As emphasized, the problematic nature of their interlinkage (however viable) merits recognition through its "fraternal informality" -- rendered evident in aesthetic terms. The elusive structure of their Borromean ring configuration offers a provocative way of considering the interwoven rings of organized crime, espionage, or the distinctive expressions of a common belief (Catholic relgious orders, Christian denominations, masonic jurisdicictions, and the like) -- as a viable, but deniable, conspiracy.
For the individual, the illusory existence of such iconic corporate configurations might indeed be compared to that of interwoven "fairy rings" -- usefully understood in terms of confidence trickery -- an exercise in "I-conning". Far more intriguing are the many ways in which any sense of "I" is reframed for the individual in terms of the multiple connotations of "con", whether constructive or otherwise. It is in this sense that the cognitive processes associated with the nonagons serve as "cognitive sashes" through which aspects of identity are "magically" interwoven.
Should the nature of requisite variety be expressed in such terms?
Enabling more fruitful comprehension: The challenges to comprehension of complexity as emphasized above also imply the challenges to embodiment of simplicity (Engaging with Insight of a Higher Order: reconciling complexity and simplexity through memorable metaphor, 2014). As indicated, the challenges can be helpfully framed by metaphor, numbers and geometry, and the depiction of the latter.
Especially intriguing is the constraint famously identified by George Miller (The Magical Number Seven, Plus or Minus Two: some limits on our capacity for processing information, Psychological Review. 1956). The above argument is effectively an exploration of 5-fold patterns (as 7 minus 2) in relation to 9-fold patterns (as 7 plus 2). The implication is that greater simplicity (including any form of unity) poses its own challenge to comprehension, rendered problematic by the ease of oversimplication and reification -- as with dualism and the pattern associated with the thinking enabled by the Pentagon. The systemic complexity of the 9-fold is elusive in another sense -- despite intuitions of the harmony associated with the 12-fold which lies "beyond" it.
As configurations of (w)holes, polyhedra offer a language within which such patterns can be explored -- recognizing the dangers of premature closure. The extent of malnutrition worldwide suggests that civilization may be faced with analogues to physical disease and the need for analogous remedies (Memetic and Information Diseases in a Knowledge Society, 2008; Memetic Analogue to the 20 Amino Acids as vital to Psychosocial Life? 2015)
These help to frame the challenge of consensus and consummation in a context of contention and conquest.
Nils A. Baas:
Barbara R. Barry. The Philosopher's Stone: essays in the transformation of musical structure. Pendragon Press, 2000
Stafford Beer. Beyond Dispute: the invention of team syntegrity. Wiley, 1994
A. G. E. Blake. The Intelligent Enneagram. Shambhala Books, 1996
Roberto Casati and Achille C. Varzi:
Marcel Danesi. The Puzzle Instinct: the meaning of puzzles in human life. Indiana University Press, 2004
Terrence W. Deacon:
J. Duffy, J. Rooney, B. Knight and C. D. Crane. A Review of a Familiy of Tensegrity Structures with Elastic Ties. The Shock and Vibration Digest, 33. 2000, 2 [text]
Alexandre Eremenko, Andrei Gabrielov and Vitaly Tarasov. Spherical polygons and differential equations July 2015 [text]
Alina Feld. Melancholy and the Otherness of God: a study in the genealogy, hermeneutics, and therapeutics of depression. Lexington Books, 2011
A. Felix, S. Christopher and A. Victor Devadoss. A Nonagonal Fuzzy Number and Its Arithmetic Operation. International Journal of Mathematics and its Applications, 3, 2015, 2 [text]
Robert Geretschläger. Folding the Regular Nonagon [text]
C. Graciano. Tensegrity Structures: a state-of-the-art review. II Simposio Latino Americano de Tenso Estructuras, 2005 [text]
Claudio F. Guerri:
P. Hackney. Making Connections: total body integration through Bartenieff Fundamentals. Gordon and Breach, 1998.
Edward Haskell. Generalization of the structure of Mendeleev's periodic table. In: E. Haskell (Ed.), Full Circle: The Moral Force of Unified Science. New York, Gordon and Breach, 1972 [text]
Eleanor Heartney. Kenneth Snelson: Forces Made Visible. Hudson Hills, 2009
Peter Hilton and Jean Pedersen. A Mathematical Tapestry: demonstrating the beautiful unity of mathematics.Cambridge University Press, 2010
Christopher Hodapp. Solomon's Builders: Freemasons, Founding Fathers and the Secrets of Washington DC. Ulysses Press, 2006
Jos Hoebe. Making a nonagon with ruler and circle. 2012 [text]
Robert H. Johnson:
Daniel Kwan. Nonagon Tessellation. Flickr. [text]
George Lakoff and Mark Johnson. Metaphors We Live By. University of Chicago Press, 1980
George Lakoff and Rafael Nüñez. Where Mathematics Comes From: how the embodied mind brings mathematics into being. Basic Books, 2000 [summary]
Robert W. Lam and Shinya Fukuda. Sphincters: properties, types and applications. Nova Science, 2012
A. Leonard. Team Syntegrity: a new methodology for group work. European Management Journal, 14, 1996, 4, pp. 407-413
Albert G. Mackey. Encyclopedia Of Freemasonry. Jazzybee Verlag, 2013
Sheryl Nichols Morris. Nine: Observing Nine with Young Children (SNAP: Scaffolding for Numerical Synapses: Awakening Curiosity in the Numbers One to Ten. CreateSpace Independent Publishing Platform, 2013) [text]
Pierce Morton. Geometry: plane, solid, and spherical, in six books. Baldwin and Craddock, 1835
Rene Motro. Structural Morphology of Tensegrity Systems. Meccanica, 46, 2011, 1, pp 27-40 [abstract]
Rene. Motro (Ed.). An Anthology of Structural Morphology. World Scientific, 2009 [text]
Ranjan Mukherjee and Jay T. Pukrushpan. Class of Rotations Induced by Spherical Polygons. Journal of Guidance, Control, and Dynamics, 23, 2000, 4, pp. 746-749. http://dx.doi.org/10.2514/2.4594 [abstract]
Martin Palmer and Karl Wagner:
Tomo Perisha. Sacred Geometry: the nonagon (Method "A") [text]
Samuel Pierre-Louis and Robert Henderson. The Scottish Rite Camp (Part 1): the military might of the nonagon. Phylaxis Magazine (The Phylaaxis Society: Prince Hall Masons), 2016 [text]
Dieter Plehwe, Bernhard J. A Walpen, Gisela Neunhöffer. Neoliberal Hegemony: a global critique. Routledge, 2007
Burkard Polster. The Mathematics of Juggling. Springer, 2006
Harish Chandra Rajpoot. Mathematical Analysis of Regular Spherical Polygons by applying HCR's Theory of Polygon. 2015 [text]
Marco F. Regolini. Centrosome: is it a geometric, noise resistant, 3D interface that translates morphogenetic signals into precise locations in the cell? Italian Journal of Anatomy and Embryology, 118, n . 1: 19-66, 2013 [abstract]
Carolina Rodriguez. The Swivel Diaphragm: a geometrical examination of an alternative retractable ring structure in architecture. University of Nottingham, 2006
Carolina Rodriguez, John Chilton and Robin Wilson. Flat Grids Designs Employing the Swivel Diaphragm. In: Rene. Motro (Ed.), An Anthology of Structural Morphology, World Scientific, 2009 [text]
Brigitte Servatius. A Literature Review of Tensegrity. Worcester Polytechnic Institute [text]
Robert E. Skelton and Mauricio de Oliveira. Tensegrity Systems. Springer, 2009
Steven M. Rosen. Dreams, Death, Rebirth: a multimedia topological odyssey into alchemy's hidden dimensions. Chiron Publications, 2014 [contents]
Social Issues Research Centre. The Future of Freemasonry. 2012 [contents]
A. G. Tibert and S. Pellegrino. Review of Form-Finding Methods for Tensegrity Structures [text]
Joseph Truss, C. Cullen and A. Leonard. The Coherent Architecture of Team Syntegrity: from small to mega-forms. Team Syntegrity Inc. [text]
Stephen F. Vogel. The Pentagon: A History -- The Untold Story of the Wartime Race to Build the Pentagon and to Restore It Sixty Years Later. Random House, 2007
Timothy Wilken. UnCommon Science, 2001 [text]
this work is licenced under a creative commons licence.