Challenges to Comprehension Implied by the Logo
of Laetus in Praesens
Laetus in Praesens Alternative view of segmented documents via Kairos

5 June 2017 | Draft

Envisaging NATO Otherwise -- in 3D and 4D?

Potentially hidden faces of global strategy highlighted through polyhedra

-- / --

Explanation as interplay of projection and "conjection"?
Surrogates of "conjection" as an unrecognized cognitive process?
Reframing NATO and The Pentagon?
Simple possibilities of exploring the complexity of NATO otherwise
Polyhedral duality as a means of engaging otherwise with dualism: NATO's shadow?
Comprehension of NATO as a four-dimensional dynamic entity?
Implications of stellation of the NATO star symbol?
Comprehending NATO through biomimicry?
NATO as a dynamic configuration of coats of arms and feedback cycles?
Classification of "projections" versus Classification of "conjections"?


This exercise is a development of the possibility noted in an earlier document with regard to rendering the 2D NATO logo into 3D (Framing Global Transformation through the Polyhedral Merkabah: neglected implicit cognitive cycles in viable complex systems, 2017). This was briefly illustrated with images (reproduced for convenience below). A similar argument was made with respect to envisaging The Pentagon otherwise.

That argument followed from several earlier exercises of a more general nature (Towards Polyhedral Global Governance: complexifying oversimplistic strategic metaphors, 2008; Polyhedral Pattern Language: software facilitation of emergence, representation and transformation of psycho-social organization, 2008; Configuring Global Governance Groups: experimental visualization of possible integrative relationships (2008) and Polyhedral Empowerment of Networks through Symmetry: psycho-social implications for organization and global governance, 2008; Polyhedral Empowerment of Networks through Symmetry: psycho-social implications for organization and global governance, 2008; Dynamic Exploration of Value Configurations: polyhedral animation of conventional value frameworks, 2008).

The question is whether new insights into the nature and potential of NATO (or The Pentagon) could be derived from depiction in 3D -- or 4D, if dynamics can be reflected in suitable animations. Whether these aspects are already implicit, or could be usefully rendered explicit, remains to be investigated. The previous arguments notably explored the use of such polyhedral projections as mapping surfaces through which otherwise disparate insights could be configured in an integrative manner in order to facilitate comprehension of complex systems of organization -- beyond the possibilities in 2D.

Use of NATO as an example is relevant at this time when its future has been called into question -- despite dependence on it for security. There is of course the further issue of the controversy and criticism surrounding its functions. Depiction in 3D and 4D offers a means of reframing such issues. This may be especially valuable in a period in which ever greater use will be made of virtual reality and augmented reality technologies to comprehend complexity in ways which are challenging in 2D representations.

The focus of the argument here is on overly simplistic conventional use of a logo or symbol to provide a succinct insight into the nature of a complex entity -- whether an organization or a system of concepts or values. As illustrated by any map of the planet, if a logo or symbol is a form of map, of what subtlety is it a map -- a cognitive map? A conventional map of the globe is a "projection" of a sphere onto the flatness of a two-dimensional surface. Rather than projection, the question then might be how the "conjection" of the multidimensional form (from which the logo derives) is to be determined and understood -- rather than through the commonly problematic process of "interpretation", as in the case of any symbol.

Curiously the question then also suggests the value of reflecting on the strange relationship between that geographical sense of projection and the understanding of projection extensively explored by psychology and psychoanalysis -- with respect to engagement with an "other", most notably one framed as an opponent. What is the process of "conjection" in that respect -- through which a coherent, integrative "global" understanding is achieved?

Arguably, richer visualization in support of "conjection" enables more fruitful imagination.

Explanation as interplay of projection and "conjection"?

Projection as explanation? The following argument is best clarified by the geographical example with which most are familiar to varying degrees. The question of how a 3D globe (the Earth) can be represented in 2D (a map) is the focus of a wide variety of map projections (see List of map projections). These are the subject of extensive commentary and innovation, notably by John P. Snyder (Flattening the Earth: two thousand years of map projections, 1993; L. M. Bugayevskiy and John Snyder, Map Projections: A Reference Manual, 1995). It is the primary concern of mathematical cartography (Qihe Yang, John Snyder and Waldo Tobler, Map Projection Transformation: principles and applications, 1999).

As noted by the GIS Encyclopedia, there are an infinite number of possible map projections of which many have been developed for use by cartographers. By nature, all map projections distort different properties of the spheroid-based features. Different projections exist in order to preserve certain features but at the expense of others. Depending on the purpose of the map, some distortions are acceptable while others are not (Robert Lloyd and Theodore Steinke, Recognition of Disoriented Maps: the Cognitive Process. The Cartographic Journal: The World of Mapping, 1984).In what sense could the 2D NATO symbol be considered a distortion of a more complex reality.

Comprehension of maps and cognition: Aside from the extensive literature on map projection, there is a related concern with the comprehension of maps, namely on cartography and cognition (J. S. Keates, Understanding Maps, 2014; Melita Kennedy and Steve Kopp, Understanding Map Projections, 2001).

These place emphasis to different degrees on the cognitive process from a psychological perspective (Chen Yufen, Map Spatial Cognition Theory: the interface of cartography and cognitive science, In: Proceedings of the International Cartographic Association, 2001; M. Blades and C. Spencer, The Implications of Psychological Theory and Methodology for Cognitive Cartography, Cartographica, 1986; P. P. Gilmartin, The Interface of Cognitive and Psychophysical Research in Cartography, Cartographica: the international journal for geographic information and geovisualization, 1981). The International Cartographic Association has a Commission on Cognitive Issues in Geographic Information Visualization which offers a limited list of resources. A wider range has been included in the references below to provide a context for this argument.

Now known as cognitive geography, the cognitive challenges of cartography have been pointedly highlighted in an early paper of Barbara Petchenik (Cognition in Cartography. Cartographica, 19, 1977):

A considerable amount of perceptual research within the general framework of behavioral psychology has been conducted by cartographers during the last ten or fifteen years. However, as one reviews the findings of this research in connection with problems encountered during the normal process of making maps, it doesn't seem to add up to much. No whole theory or set of principles, greater than the sum of the small component parts, has emerged.... The important shift that has occurred in psychology is away from an emphasis on strict behaviorism, and toward an emphasis on thought processes, toward what has come to be called cognitive psychology....The terminology employed attracts the cartographer's attention, for the notion of "cognitive mapping" appears frequently in the psychological literature. The term suggests that there might be something in it for us, and indeed, I believe there is.... The real problem is this: How does
a map user develop internal, personal knowledge of relations among things in space on the basis of viewing a sheet of paper covered with ink marks? How, in common language, does one read a map?

If the function of a map is to trigger meaning, then meaning becomes all-important. We must determine what the meaning of a map is, and how research could take account of such meaning.... Meaning seems to come from an all-at-once grasp of the relation of the stimulus to the reader's previous knowledge structures, rather than from a bit-by-bit build up.... One interesting point has been made recently in cognitive psychology, having to do with the question of the form in which knowledge exists in the mind. In the past, some have argued for verbal encoding, others for imagery, still others for a combination of the two.

So far cartographic research in map reading has not penetrated at all deeply into matters of this sort. In fact, comparing what has been done in map reading with what has been done in text reading, it seems fair to say that what has been done with maps in the way of symbol perception, is to total map reading, as typeface perceptibility studies are to reading comprehension research. We really lack a word that describes the apprehension of spatial knowledge from maps, a word that would compare to "comprehension" for text....The difficulty.. is that much of what goes on in cartography is not explicitly understood, particularly at the level of the nature of the human knowledge transfer that is involved. [emphasis added]

Writing much later, Wieslawa Zyszkowska (Map Perception: theories and research in the second half of the twentieth century) notes:

Until the 1990s map perception research was one of the main parts of cartography as a scientific discipline. In the last years of the century map perception research fell out of favor as cartographers turned their attention to the new computer technology. In the first decade of the 21th century the problems of map perception became more frequent in cartographic journals. (Polish Cartographical Review, 47, 2015)

Word-play indicative of cognitive dysfunction? Whether "conjection" or "conjunction", naming the process offers a provocative contrast to "projection" -- in a context in which much is made of the need for "joined-up thinking" through which seemingly disparate elements are integratively "conjoined". The "word-play" may be explored in the light of previous efforts to clarify the question (New Paradigms via a Renewed Set of Prefixes? Dependence of international policy-making on an array of operational terms, 2003; Exploration of Prefixes of Global Discourse: implications for sustainable confidelity, 2011; Checklist of words prefixed by "con" with frequency of usage, 2016 ). The latter distinguished the following, with other terms associated with prefixes in relation to the truncation "-ject":

The processes of interest may be clarified by the following, with frequencies derived Google search results (with all the reservations this may imply).

Frequency of selected uses of pro- and con- prefixes
  -jection(s) -ject(s) -jecture(s) -junct(s) -join(s)
pro- projection(s): 337 m. project(s): 4,300 m. projecture(s): 35,300 projunction(s): 2,100 projoin(s): 2,600
con- conjection(s): 31,400 conject(s): 152,000 conjecture(s): 19,550 m. conjunction(s): 157 m, conjoin(s): 915,000

This strange mix might be interpreted as follow:

There is a sense in which project as a verb implies rendering explicit a disparate array diverging from a common perspective -- echoing the use of "projectile". In contrast with this sense of "throwing apart", a complementary process merits recognition as "drawing together". Hence the case for "conjection", especially given the current value attached to the prefix (Considerable Conglomeration of "Cons" of Global Concern: eightfold constraint on constructive conflict control? 2012). However there is the strange contradictory sense in which "pro" is considered "positive" and supportive, whereas "con" is considered "negative" and opposed.

Also known as the North Atlantic Alliance, in terms of "projection", NATO can be understood as a "project" of the North Atlantic Parliamentary Assembly which sets its broad strategic goals. NATO adopted the Alliance Strategic Concept during its Washington Summit (1999) -- emphasizing conflict prevention and crisis management, despite its very extensive investment in "projectiles". The inadequacy of that investment, despite past commitments, was a widely publicized feature of the Brussels Summit of 2017 (Trump lectures NATO allies on defense spending, USA Today, 25 May 2017; Trump insists NATO members boost defense spending, Politico, 25 May 2017; President Trump Pushes NATO Members to Pay 'Their Fair Share', Time, 25 May 2017).

Given the extensive investment in the NATO "project" and its instruments -- the "projectiles" -- is it appropriate to ask of what it is a "conjection"? How is the underlying "conject" to be understood -- and what might be the associated "conjectiles" -- in cognitive terms?

Surrogates of "conjection" as an unrecognized cognitive process?

Rather than focus on any complementary term to projection. it can be readily assumed that some such process is recognized through other terms, techniques and metaphors. These could include:

In a post-truth era, conventional modes of "explanation" are clearly no longer "fit for purpose" -- as remarkably demonstrated by explanations of climate change and other emerging crises. This calls into question the adequacy of the responses to the Edge Foundation Question of 2012: What is yours favorite deep, elegant, or beautiful exaplantion? In a compilation of 150 selected responses, the editor notes that:

The common thread is that a simple and nonobvious idea is proposed as the explanation for a diverse and complicated set of phenomena. (John Brockman, This Explains Everything, 2013).

It is curiously assumed that any "simple and nonobvious idea" is readily and widely comprehensible. There is however no reference to "comprehension" in the extensive index or to how it is enabled -- other than through the significance attributed to "deep, elegant, or beautiful".

"Conjection" as an unrecognized cognitive process? With the emphasis on mapping, much of the literature conflates projection with explanation. This reinforces any assumption that if it can be mapped then this constitutes an explanation. Arguably this conflation can be recognized in the recently emergent possibilities of visualization. Visualization as explanation?

The implications of 9/11 destruction of the Twin Towers, together with the attack at the time of writing on London Bridge, suggest that it is symbols that empower, not explanation. The issue has become especially evident in disregarded explanations regarding climate change, or the psychic numbing evoked by starvation imagery (Starvation Imagery as Humanitarian Trump Card? Counterproductive emotional blackmail engendering worldwide indifference, 2016). Clearly there are concerns with whether, and how, any map can be understood.

Given the extensive mathematical literature on "projection" of a global reality onto a flat surface, it is appropriate to ask whether there is an inverse or reverse process (not to be confused with "back projection"). This might be described as taking one map projection (or more) and deducing from the configuration in 2D the form in 3D from which the map was derived -- most obviously in the case of a globe. This process may be variously implied in the preoccupation with projection in mathematical cartography. This is not immediately relevant from the literature, and may well be considered of little interest -- except in the sense of stereographic projection. Metaphorically it could also be explored in terms of decryption, with the map understood as a form of encryption -- two or more maps being of value to the decoding process.

The challenge to comprehension of "conjection" has been remarkably clarified using the mathematics of q-analysis as developed by Ron Atkin (Multidimensional Man; can man live in 3-dimensional space?, 1981), as separately summarized (Comprehension: Social organization determined by incommunicability of insights). Atkin illustrates the challenge of comprehension in relation to experience "within" the geometry of a triangle.

Comprehension through transcending the plane
(as indicated by the q-analysis of Ron Atkin)
Conjection as recognizing "white" Relative orders of comprehension
Atkin triangle

Red, Green or Blue

Yellow (=Red/Green)
Purple (=Red/Blue); or
Turquoise (=Blue/Green)

White (=Red/Green/Blue)

Beyond the "plane" of explanation? The geographical challenge of reducing the sphere to a plane as a means of explanation highlights the continuing challenge of a "flat earth" understanding of global reality, however absurd (The Flat Earth Theory has seen a resurgence, with people trying to prove our planet is not a sphere,, 1 June 2017). This has been famously emphasized by Thomas Friedman (The World is Flat, 2005) and the focus for criticism (Irresponsible Dependence on a Flat Earth Mentality -- in response to global governance challenges, 2008).

The inadequacy of the quest to reduce complexity through projection onto a plane is only too evident in the extension of the metaphor into "planning", namely the conception of strategic "plans" in response to "global" challenges. This tendency is curiously exemplified in the interpretation of the religious agendas by which governance may be especially informed (Adhering to God's Plan in a Global Society, 2014).

Conventional explanation may therefore be characterized as the projection onto a plane -- supposedly offering a perspective from "elsewhere", as indicated by "ex". Some form of "extraction" from the plane is then implied as a key to comprehension. Greater coherence regarding the disparate elements forming the explanation -- in the plane -- might then be more significantly related to a process of "conjection" from the plane. This is variously confused and conflated with surrogates, most notably those associated with interpretation and inference.

With respect to maps, it has been famously stressed by Alfred Korzybski that "the map is not the territory", The available maps are clearly inadequate for a civilization characterized by a combination of high degrees of uncertainty and high degrees of certainty -- itself questionable. Many offer guidance (E. F. Schumacher, A Guide for the Perplexed, 1977; Maimonides, The Guide for the Perplexed). Clarifying "conjection" in relation to mapping may then offer a more fruitful means of engaging with the "territory" (The Territory Construed as the Map: in search of radical design innovations in the representation of human activities and their relationships, 1979). Missing is a process which is the reverse of "zooming down" and "drilling down" -- as originally argued by Joel de Rosnay (The Macroscope, 1979). Geometry suggests the possibility of "zooming up" to the high orders of symmetry which render symbols so meaningful -- to whatever degree they are comprehensible.

How indeed can invisible coherence be rendered comprehensible? How might the process of projection be reversed in the light of mathematical insights -- starting with a 2D representation and extending it into 3D, with the possibility of 4D animation? Is "conjection" the key to empowerment in a period of crisis, as can be otherwise argued (Polyhedral Empowerment of Networks through Symmetry: psycho-social implications for organization and global governance, 2008).

Rather than offering commentary and explanation, the focus here is on the use of such imagery to evoke interest in the possibility and its potential. The exercise relies on use of the possibilities of morphing the basic 3D analogue to the 2D NATO symbol through the geometrical transformations enabled by Stella Polyhedron Navigator.

Reframing NATO and The Pentagon?

NATO: At the time of writing, with new strategic directions of NATO as a principal theme of the presence of Donald Trump at a NATO Summit, the approach explored previously with respect to the Star of David and the Merkabah may also be explored with respect to the symbol with which NATO is identified (Framing Global Transformation through the Polyhedral Merkabah: neglected implicit cognitive cycles in viable complex systems, 2017). This can be illustrated by the images below.

The NATO symbol or logo is necessarily in 2D (A Star is Born: choice of design, 2016). The question is whether the geometric extension of the NATO logo into 3D would enable recognition of "missing strategic dimensions" -- whatever this might imply in strategic terms. Just as the 2D depiction may be assumed to reinforce certain cognitive processes, would the reinforcement of a more complex set of processes be reinforced -- of greater relevance to global governance?

The previous discussion argued for the value of basing this design on a four-fold augmentation of a fundamental central polyhedron, namely the tetrahedron -- informally named the augmented tetrahedron, composed of 5 tetrahedra. Each tetrahedron, as the most fundamental polyhedron, could be understood as embodying one of the 5 fundamental values of NATO, and of the Europe it is designed to defend: Democracy, Freedom of speech, Freedom of the media, Independence of the judiciary, Protection of minorities. These were reaffirmed by NATO Secretary General Jens Stoltenberg (NATO's Enduring Mission -- Defending Values, Together, NATO, 8 September 2016).

The augmented tetrahedron depicted below is composed of 12 faces (of 1 type), 18 edges (of 2 types) and 8 vertices (of 2 types). It can be understood as composed of 5 tetrahedra.

Juxtaposition of NATO 2D symbol with views of augmented tetrahedron in 3D
Reproduced from Framing Global Transformation through the Polyhedral Merkabah: neglected implicit cognitive cycles in viable complex systems (2017)
NATO symbol Screen shots of 3D augmented tetrahedron
( variously rotated)
design in 2D schematic in 3D?
NATO symbol in 2D Schematic basis for NATO symbol in 3D on augmented tetrahedron Schematic basis for NATO symbol in 3D on augmented tetrahedron Schematic basis for NATO symbol in 3D on augmented tetrahedron
  Images prepared using Stella Polyhedron Navigator

Curiously, but perhaps appropriately, the 3D form recalls that of the morning star spiked club of medieval times -- of which a variant, as a mace, is now a central feature of some parliamentary ceremonies. The structure also resembles that of certain types of road traffic spike deployed by security services.

The Pentagon: Given the fundamental strategic importance of The Pentagon (as headquarters of the US Department of Defense) with respect to global security -- understood in terms of full-spectrum dominance -- there is a case for extending the above exploration in terms of the 5-fold symmetry of that body as a symbol. As noted, the augmented tetrahedron, indicated above as a means of depicting NATO otherwise, is composed of 5 tetrahedra, one of which is central to the configuration.

This suggests the possibility combining 5 such tetrahedra otherwise in what is known as a tetrahedron 5-compound (with 20 faces, 30 edges and 20 vertices). The vertices of the compound then form the 20 vertices of the dodecahedron. It is one stellation of the regular icosahedron. The compound is unusual in that the dual figure is the enantiomorph of the original. If the faces are twisted to the right, then the vertices are twisted to the left. The compound has been extensively studied by geometers, notably as a geometric illustration of the notion of orbits and stabilizers; it is one of five regular polyhedral compounds. Two such tetrahedron 5-compounds of opposite chirality combine to make a tetrahedron 10-compound (with 40 faces, 60 edges, and 20 vertices).

Symbolic reframing of The Pentagon?
Alternative views of the compound of 5 tetrahedra
Possible symbolic reframing of The Pentagon in 3D based on compound of 5 tetrahedra Possible symbolic reframing of The Pentagon in 3D based on compound of 5 tetrahedra
Images reproduced from Wikipedia

Such structures in 3D (with geometry reflective of the number of members of NATO, for example) are an indication that the conventional simplicity of the Pentagon as a symbol may be inherently misleading.

Simple possibilities of exploring the complexity of NATO otherwise

The following images of a 3D form are essentially self-explanatory. The possibility of switching between alternative variants (transparent or non-transparent) is also of interest. The suggestion above that there was additional value in a "4D" representation is partially indicated here in the rotations of the structure. It is also the case that with the use of virtual reality software the structure can be explored interactively -- variously rotated and zoomed. Additional colouring of faces was not applied in these examples.

Augmented tetrahedron as a symbol of NATO in 3D?
Faces non-transparent Faces transparent (wireframe rendering)
Static Rotation Static Rotation
Augmented tetrahedron as a symbol of NATO in 3D? Augmented tetrahedron as a symbol of NATO in 3D Augmented tetrahedron as a symbol of NATO in 3D Augmented tetrahedron as a symbol of NATO in 3D
Images prepared using Stella Polyhedron Navigator

Even in this simple form, the symbol may be used as a mapping surface to which images, text and hyperlinks can be attached -- as one means of articulating the significance of the form. One example is illustrated in the animation below. This derives from an articulation by Christopher E. Van Avery (12 New Principles of Warfare, Armed Forces Journal, 1 July 2007).

Mapping selected sets of strategic principles onto polyhedra
Principles of War?
Universal Declaration
of Human Rights
Arab Charter
on Human Rights
NATO Principles of War? Universal Declaration of Human Rights
Images prepared using Stella Polyhedron Navigator

As shown above (centre and right), a wider variety of possibilities of mapping text onto polyhedra is demonstrated in a previous exercise focusing on declarations of human rights (Dynamic Exploration of Value Configurations: polyhedral animation of conventional value frameworks, 2008). Although now inactive, the issues recognized by the NATO Committee on the Challenges of Modern Society could have been coherently portrayed in that way, as with those of its successor: NATO Science for Peace and Security.

A more complex form was, for example, used to give insight into the implicit coherence of a conference of the International Peace Research Association, as illustrated separately (Polyhedral Conference Representation as a Catalyst for Innovation: polyhedral animation of IPRA 2008, 2008).

Assuming an equivalence to the augmented tetrahedron, the following visualizations offer insights into NATO from an analytical perspective. These are variously reminiscent of systems and other diagrams favoured by management. Could a polyhedral form be elicited from such conventional depictions -- notably network diagrams, as argued separately (Polyhedral Empowerment of Networks through Symmetry: psycho-social implications for organization and global governance, 2008).

Analytical depictions of augmented tetrahedron geometry
Stellation diagram Cell diagram Vertex figure Faceting diagram
Stellation diagram of augmented tetrahedron Cell diagram of augmented tetrahedron Vertex figure of augmented tetrahedron Faceting diagram of augmented tetrahedron
Images prepared using Stella Polyhedron Navigator

Polyhedral duality as a means of engaging otherwise with dualism: NATO's shadow?

In addition to the representations above, the basic form may be "unfolded" or "exploded" into its component tetrahedra, as illustrated in the animation on the left below.

Duality and double standards: More intriguing, as with any polyhedron, is the nature and significance of its polyhedral dual. This contrasting structure is indicated below, where the vertices of one correspond to the faces of the other and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. The dual is rotated on its own below -- and separately with the augmented tetrahedron embedded within it. Again, text and imagery could be added to the dual, in contrast to what might be associated with faces, vertices or lines of the base polyhedron.

Augmented tetrahedron and its dual
Tetra unfolding Dual Dual in rotation Tetra with dual rotating
Augmented tetrahedron unfolding Dual of Augmented tetrahedron Rotation of dual of Augmented tetrahedron Rotation of Augmented tetrahedron with its dual
Images prepared using Stella Polyhedron Navigator

In a psychosocial system much challenged by duality and "otherness", what might be the significance of the interplay between the base polyhedron and its dual? What might any symbolic representation of this complex dynamic encompass? Arguably this offers a means of encompassing the tendency to double standards, as variously described (Enabling Suffering through Doublespeak and Doublethink, 2013; David Rohde, Ending NATO's Double Standard on International Justice, The Atlantic, 18 May 2012; Taras Kuzio, NATO's Double Standards: Why Montenegro but Not Ukraine? Atlantic Council, 31 May 2017; Double Standards? Nato's Afghan Errors, HonestReporting, 17 February 2010).

Challenging images of otherness: Such questions are especially pertinent in the case of NATO which is fundamentally structured and configured to respond to the threat of particular forms of otherness -- namely potential opponents and enemies. In any conventional depiction of NATO these threats are effectively extra-systemic, although essentially messy representations of them may be made as in the case of Afghanistan, as discussed separately (Cultivating Global Strategic Fantasies of Choice: learnings from Islamic Al-Qaida and the Republican Tea Party movement, 2010). This included the following image adapted from that by the PA Consulting Group and presented to the Office of the Joint Chiefs of Staff of the US of counterinsurgency (COIN). This takes the form of a map, notably publicized on behalf of McClatchy Newspapers by Dion Nissenbaum (The great Afghan spaghetti monster, Checkpoint Kabul, 20 December 2009; Graphic Shows Complexity of US Counterinsurgency in Afghanistan, The Huffington Post, 22 December 2009). Coincidentally this map was publicized over the web at the end of the 2009 Copenhagen climate change event.

To illustrate the possibilities (as in the above exercise), the "spaghetti map" was adapted (as reproduced below) to suggest the relevance of such a perspective to the climate change debate (Insights for the Future from the Change of Climate in Copenhagen, 2010). As with other enduring "wars", the need for another strategic approach is remarkably demonstrated by the decades-long conflict in Afghanistan (Transforming the Unsustainable Cost of General Education: strategic insights from Afghanistan, 2009; Review of the Range of Virtual Wars: strategic comparison with the global war against terrorism, 2005).).

Adaptation to climate change
of a representation of counterinsurgency operations in Afghanistan

(click on image for larger version)
Adaptation to climate change of a representation of counterinsurgency operations in Afghanistan
Afghanistan COIN dynamic
(clusters in original map)
Climate change COIN dynamic
(clusters in adapted map)
Population/Popular support
Infrastructure, Economy and Services
Afghanistan Security Forces (ANSF)
Crime and Narcotics
Coalition Forces and Actions
Physical Environment
Population/Popular support
Infrastructure, Economy and Services
Activist NGO Strategic Forces (ANSF)
Dissenters ("Them")
Crime and Distractions
Initiatives of Coalition of the Willful ("US")
Physical Environment

NATO's shadow? Any polyhedral dual, generated from a more complex polyhedral representation of NATO, then offers at least three distinct possibilities for exploration:

Clearly there is an interplay between such understandings to be explored in the light of the insights of Carl Jung who argued: None of us stands outside humanity's black collective shadow (On the collective shadow and evil). Others write of its current implications (Paul Levy, Shadow Projection: the fuel of war, 2010; Mats Winther, Thanatos: a contribution to the understanding of the collective shadow, 2012; Nadine Kreisberger, Trump's election and what it reveals about our collective Shadow, 2016; Rosko Green, Trumpism: the collective shadow rises again, 24 December 2016).

Dynamics of engagement with the other: To give a degree of form to this shadow, through a process of "conjection" the possibility of geometric morphology may be used to transform the dual of the proposed NATO symbol through various procedures consistent with mathematical cartography. These are illustrated in the animations below. There is clearly some advantage to the colour contrast between the blue of the NATO symbol and the "threatening" red of the dual as it variously emerges. Are such emergent forms in red helpful in comprehension of what terrorism (ISIS, Al Qaida, etc) can be framed and understood in strategic terms from a NATO perspective? What are the possible cognitive analogues to these geometric transformations?

Morphing duals of augmented tetrahedron - A
By sizing By truncation By augmentation By expansion
Morphing duals of augmented tetrahedron (sizing) Morphing duals of augmented tetrahedron (truncation) Morphing duals of augmented tetrahedron (augmentation) Morphing duals of augmented tetrahedron (expansion)
Images prepared using Stella Polyhedron Navigator
Morphing duals of augmented tetrahedron- B
By tilting quads By tilting triangles By tilting to compound By tilting to rectify
Morphing duals of augmented tetrahedron (tilting quads) Morphing duals of augmented tetrahedron (tilting triangles) Morphing duals of augmented tetrahedron (tilting to compound) Morphing duals of augmented tetrahedron (tlting to rectify)
Images prepared using Stella Polyhedron Navigator

The animations offer a sense of the struggle of NATO with its own shadow -- whether the shadow "within" or that projected without, as with the struggle with Islamic terrorism. Curiously this is consistent with the Islamic understanding of jihad as a continuing struggle, especially one with a praiseworthy aim. As noted by Wikipedia: Jihad can have many shades of meaning in an Islamic context, such as struggle against one's evil inclinations, or efforts toward the moral betterment of society. In classical Islamic law, the term refers to armed struggle against unbelievers, while modernist Islamic scholars generally equate military jihad with defensive warfare In Sufi and pious circles, spiritual and moral jihad has been traditionally distinguished under the term "greater jihad"

Such interpretations naturally find their analogues in the Christian understanding of crusade -- by which NATO is inspired to a greater degree -- and in the light of the traditional Christian response to Islam (Richard C. Csaplar, Jr, 1,400 Years of Christian/Islamic Struggle, CBN). However the sense of internal struggle has also figured in Christian mysticism (Neil Holman, The Struggles of a Christian, Bible Study, 21 December 2009; John Edmiston, The Christian's Struggle with Sin, Global Christians, 2010).

Of partiular interest for NATO are the institutional games in which it may engage as notably explored through a martial arts metaphor by Thierry Gaudin (Les katas institutionnels. Transnational Associations, 30, 1977) identified 21 tactical moves (katas) open to institutions, subsequently included in English translation as The Institutional Katas in an argument with regard to Game-playing in global governance? (Engaging an Opposing Ideology via Martial Arts Philosophy: reframing the challenge of Trump and Jihadism as worthy opponents, 2016). Of relevance in this respect is that the leader of the nation framed as the primary opponent of NATO is a martial arts black belt (Vladimir Putin Earns 9th Degree Black Belt In Taekwondo, Huffington Post, 16 November 2013). It is also the case that Donald Trump has been a major sponsor of mixed martial arts.

Comprehension of NATO as a four-dimensional dynamic entity?

Despite the emphasis of NATO on "unity", frequently framed in terms of "solidarity", it is necessarily misleading to frame NATO as a static entry. There is a sense, especially from a military strategic perspective of the need for strategic nimbleness more appropriately associated with "fluidity". Any "unity" is likely to be more appropriately found in 4 (or more) dimensions. As in the metaphor evoked for corporations, NATO needs to "dance" (Rosabeth Moss Kanter, When Giants Learn To Dance: mastering the chal.lenge of strategy, management, and careers in the 1990s, 1989). Some sense of the possible dynamics can be offered by various animations of the augmented tetrahedron, as presented above -- offering one sense of a 4D entity.

Another approach is to make use of facilities to present the augmented tetrahedron in geometric emulations of 4D -- a technique offered by Stella4D (as an extension of Stella Polyhedron Navigator used to prepare the figures above). This includes four-dimensional polytopes, known as polychora (projected into 3D), with the possibility of animating 3D cross-sections in real-time and 4D rotation. Their relevance is discussed separately (Four-dimensional requisite for a time-bound global civilization? 2015; Comprehending the shapes of time through four-dimensional uniform polychora, 2015; Five-fold ordering of strategic engagement with time, 2015).

Representations of NATO as a 4-dimensional entity?
4D 4D Runcinated Runcinated and expanded
Representations of NATO as a 4-dimensional entity? Representations of NATO as a 4-dimensional entity? Representations of NATO as a 4-dimensional entity? Representations of NATO as a 4-dimensional entity?
Images prepared using Stella Polyhedron Navigator

Other techniques can be used to transform the augmented tetrahedron in ways which variously invite imaginative exploration -- especially as mapping surfaces.

Transformations of the augmented tetrahedron -- NATO explored otherwise?
Waterman polyhedron 4D polytope 4D Truncated expansion Transformation into a torus Zonohedrification
NATO as a Waterman polyhedron 4D polytope NATO as a 4D Truncated expansion Transformation of NATO into a torus Zonohedrification of NATO
Images prepared using Stella Polyhedron Navigator

Implications of stellation of the NATO star symbol?

Use of "star" as a metaphor, especially in launching the NATO symbol (A Star is Born: choice of design, 2016), suggests the merit of exploring the geometric operation of stellation with respect to the proposed augmented tetrahedron. In this context, stellation is the process of extending the polyhedron in three dimensions to form a new figure. Starting with the augmented tetrahedron, the process extends specific elements such as its edges or face planes, usually in a symmetrical way, until they meet each other again to form the closed boundary of a new figure. The new figure then constitutes a stellation of the original.

Polyhedra typically have extensive sets of stellations. For example, in the case of the Platonic polyhedra:

Given the preoccupation of NATO with "order" within the global system, there is a case for exploring the insights of Christopher Alexander in a magnum opus with a related concern (The Nature of Order: an essay on the art of building and the nature of the universe, 2003-4). This gave rise to his insights of potential relevance to further understanding of NATO (New Concepts in Complexity Theory: an overview of the four books of the Nature of Order with emphasis on the scientific problems which are raised, 2003; Harmony-Seeking Computations: a science of non-classical dynamics based on the progressive evolution of the larger whole, International Journal for Unconventional Computing (IJUC), 2009).

The latter study identified 15 "transformations" whose implications for the psychosocial realm merit clarification, as discussed separately (Comprehending Alexander's transformation principles within the psychosocial realm, 2010; Tentative adaptation of Alexander's 15 transformations to the psychosocial realm, 2010). Those considerations framed the possibility of using a subset of the stellations of the icosahedron as a guide to reflection on the Geometrical configuration of Alexander's 15 transformations (2010).

The selection of stellations of the augmented tetrahedron presented below therefore invites reflection on their potential significance in a richer understanding of NATO. Are they indicative of what NATO variously imagines itself to be -- or aspires to be -- as a "star"?

Selection of stellations of the augmented tetrahedron -- as the imagined significance of NATO?
NATO as a stellation of the augmented tetrahedron NATO as a stellation of the augmented tetrahedron NATO as a stellation of the augmented tetrahedron NATO as a stellation of the augmented tetrahedron NATO as a stellation of the augmented tetrahedron NATO as a stellation of the augmented tetrahedron
NATO as a stellation of the augmented tetrahedron NATO as a stellation of the augmented tetrahedron NATO as a stellation of the augmented tetrahedron NATO as a stellation of the augmented tetrahedron NATO as a stellation of the augmented tetrahedron NATO as a stellation of the augmented tetrahedron
Images prepared using Stella Polyhedron Navigator

The stellations above can be presented together (below left) as animations suggestive of phases in NATO's imagination of itself. The exploration can be taken further by juxtaposing that animations with a second (on the right below). This could be seen as a representation of NATO's enemy -- as the primary source of threat (whether Russia, ISIS, or the like) with which NATO is confronted and against which a defensive posture is considered vital. As such it is a dynamic depiction of NATO's worst fear -- the hydra-headed monster with which it is committed to deal.

Following the earlier argument with respect to the "shadow", the animation on the right can also be explored as a depiction of how NATO is (mis)perceived by others -- its "ugly face", as it appears in the mirror into which NATO never looks. Similarly it could be understood as inherent in NATO -- the unrecognized dimensions of the entity which have yet to be integrated into a mature form of organization.

Animation of phases in the stellation of the augmented tetrahedron
as it imagines itself to be?
as it aspires to be?
"isotopes" of varying "half-lives" ?
perception of its enemies?
perception by opponents?
traits denied ("shadow")?
Animation of NATO? Antimation of NATO as perceived by its opponents
Images prepared using Stella Polyhedron Navigator

Curiously, but as might be expected, there are few references to the "psychology of NATO". However some NATO studies do indeed focus on the application of psychology (Mark E. Koltko-Rivera and P. A. Hancock, Psychological Strategies for the Defence Against Terrorism, NATO RTO-MP-SCI-158, 2004). Far more problematic is the degree to which these imply NATO's deniably complicity in harbouring so-called "black sites", as notoriously used by the CIA (Jane Mayer, The Black Sites: a rare look inside the C.I.A.'s secret interrogation program, The New Yorker, 13 August 2007; Trump Administration Moves To Keep Full CIA Torture Report A Secret, Reuters, 2 June 2017; Ray McGovern, Hiding the Ugly Business of U.S. Torture, Information Clearing House, 5 June 2017).

The coluring of the complementary animations offers a usefully reminder of the existential choice in society between the "blue pill" and the "red pill", as framed in the science fiction movie The Matrix (1999). NATO might itself be understood symbolically as a remedial "pill", as framed by an earlier argument (Psychosocial Transformation by "Pill Pushing"? Model-making, strategic advocacy and the myth of the "red pill", 2017). That study used other animations in 3D to envisage the possibility of transcending that choice through a third possibility (Visualization in 3D of a trinity of connotations as a cognitive pill, 2017).

Comprehending NATO through biomimicry?

The representations above, irrespective of the animations and transformations, rely to a questionable degree on geometrical constraints without considering how they might be fruitfully relaxed -- as a source of greater insight. One approach would be to foreshorten or lengthen the lines in the depictions -- selectively shearing the depiction as a whole.

A distinctive approach, consistent with this possibility, is to build dynamics into the models otherwise. This was remarkably illustrated by the award-winning Soda Constructor a java-based physics engine, central to Soda's collaborative learning environment The facility enables users to create and share "models". These models are made of "masses" and "springs" and follow strict physical laws. By turning springs into "muscles", models can be animated and made very lifelike. Although thousands of models have been created, it is uncertain whether the facility continues to be viable, given the evolution of web technologies and incompatibilities. Many legacy websites and online fora are however partially accessible.

Of relevance to the argument with respect to NATO, the potential of the Soda Constructor approach was reviewed with respect to the institutions of Europe (Animating the Representation of Europe: visualizing the coherence of international institutions using dynamic animal-like structures, 2004).

Examples of multi-legged animated Soda models
(clicking on the images originally "brought them to life"; a feature now deactivated)

Multi-legged animated Soda models Multi-legged animated Soda models Multi-legged animated Soda models Multi-legged animated Soda models
Multi-legged animated Soda models Multi-legged animated Soda models Multi-legged animated Soda models Dynamics of an emergent multi-organizational "butterfly"

The argument in terms of biomimicry and naturomimicry has achieved greater relevance given recognition of its acknowledged value in two complementary arenas:

Such arguments help to focus the questions as to whether NATO can learn to "swim" or "fly" in a global psychosocial system (Counteracting Extremes Enabling Normal Flying: insights for global governance from birds on the wing and the dodo, 2015).

NATO as a dynamic configuration of coats of arms and feedback cycles?

Heraldic coats of arms of NATO members: It is easy to forget that central to the collective identity of each member country of NATO is an emblem or heraldic symbol -- otherwise known as a "coat of arms" (see List of coats of arms). Clearly these are symbols relevant to the military preoccupations of NATO -- and effectively implicit in the significance of its own logo. Each national coat of arms has been displayed alongside the coats of arms of other member states in the assembly chamber of the United Nations. They typically feature on the cover of passports and on other official documents.

The previous argument with respect to the star tetrahedron and its association with the Star of David focused on the fundamental symbolism of the Merkabah, noting its use (in the form (Merkavah) as the name of the primary battle tank of the Israel Defence Forces (Framing Global Transformation through the Polyhedral Merkabah: neglected implicit cognitive cycles in viable complex systems, 2017). , The visualizations there can be adapted, as a preliminary proof of concept exercise, to indicate how such heraldic devices might be dyamically configured in 3D. The images below show various phases in a complex animation (see video) using just two tetrahedra (as in the Merkabah exercise). This approach could be extended to include all 5 tetrahedra of the augmented tetrahedron proposed in the above argument. Obviously the limitation in the exercise below is restriction to the use of only 8 faces as mapping surfaces for the symbols of 8 NATO members. This could be extended to 20 (5x4) faces if the augmented tetrahedron was used. Another approach would be to associate such symbols (or some of them) with other features (edges and/or vetices).

Screen shots of phases in the process of "interpenetration" and "disassociation" of configurations of heraladic symbols of NATO members
(images depicted are merely indicative of others of that type in each case; political implications have not been considered.)
Interactive animation: X3D. MP4 video
Animation of heraldic symbols of NATO members Animation of heraldic symbols of NATO members Animation of heraldic symbols of NATO members
Coats of arms from Wikipedia. Model prepared with the aid of X3D.Edit and Stella Polyhedron Navigator.

It is appropriate to note that NATO has produced its own freely accessible video regarding NATO membership.

Feedback cycles: In the light of the arguments used with respect to the tetrahedral configuration of the Merkabah, circles can be associated with the edges of the augmented tetrahedron as an indication of implicit feedback cycles. This was illustrated in the Merkabah case with a complex animation (see video) which can be adapted (as with that above) to a complex of cycles. The argument made was that the edges could be understood as implying cycles -- reduced in the symbol to the diameter of an implied circle. As argued with respect to the Merkabah, the question is how complexity in relation to NATO can be reduced to succinct representations which are far more readily comprehensible.

Screen shots of phases in cyclic movement in relation to selected edges of the augmented tetrahedron variously oriented
(images depicted are merely indicative of others of that type in each case; systemic implications have not been considered.
The central image is rendered with non-transparent faces; the other images have transparent faces )
Interactive animation: X3D, WRL. Single viewpoint MP4 video
Movement in relation to selected edges of the augmented tetrahedron Movement in relation to selected edges of the augmented tetrahedron Movement in relation to selected edges of the augmented tetrahedron
Model prepared with the aid of X3D.Edit and Stella Polyhedron Navigator, with valuable guidance from Sergey Bederov of Cortona3D

The above animation is again a proof of concept exercise. Many of the features involve design choices which could be made otherwise, notably with respect to rates of movement, circle properties and colouring (as is evident in the contrast with the Merkabah video). In particular the animation only shows circles in relation to sides of the central tetrahedron, not to the four tetrahedral augmentations.

In offering the possibility of associating systemic content with the circles, as a pattern of feedback loops these give a sense of the communication pathways essential to the viability and integrity of NATO. With respect to the argument above for biomimicry, such an animation is visually reminiscent of the motile cilia in some eukaryotes, or the potentially complex movement of bird wings or fish fins. As pathways, the pattern of circles also recalls the set of metabolic pathways vital to biological organization (Comprehending cognitive metabolism, 2012).

Classification of "projections" versus Classification of "conjections"?

Projections in cartography: Many sources enadeavour to clarify the types of projection in cartography. A fundamental projection classification is based on the type of projection surface onto which the globe is conceptually projected. The projections are described in terms of placing a gigantic surface in contact with the earth, followed by an implied scaling operation. These surfaces are:

Many mathematical projections, however, do not neatly fit into any of these three conceptual projection methods. Hence other categories have been described in the literature, such as:

Properties and implied distortions distinguished include:

Projections in psychology: As noted above, psychological projection is recognized as a process whereby humans defend themselves against their own unconscious impulses or qualities (both positive and negative) by denying their existence in themselves while attributing them to others. Different schools of psychology and psychoanalysis tend to define projection in distinctive ways. A limited number of examples are offered but it is seemingly not the case that there is any classification of modes of projection in any way analogous to projection in cartography (John M. Grohol, 15 Common Defense Mechanisms, PsychCentral, 17 May 2016). With respect to NATO, it is interesting to note the shared emphasis on defence.

Ironically, given the clear distinctions made in cartography, there is a case for exploring the distortions in cartographical projections as a means of distinguishing and "mapping" the distortions in psychological projection. Any such exploration could necessarily be informed by the more systematic approach to cognitive biases and their classification (see List of cognitive biases). The Wikipedia description includes a remarkable Cognitive Bias Codex as reproduced below. This clusters some 180 biases into 20 subcategories clustered into four main categories:

Cognitive Bias Codex
(link to larger original version)
Cognitive Bias Codex
Categorized by Buster Benson using an algorithmic design by John Manogian III derived from Wikipedia data

Such a document is necessarily of value with respect to exploring collective cognitive bias, as might be evident in the case of NATO and its opponents (Robert Cordray III and Marc J. Romanych, Mapping the Information Environment, Joint Information Operations Center, 2005; Eric R. Mandel, Cognitive bias and UNSC Resolution 2334, Jerusalem Post, 3 January 2017; J. T. Byford, Conspiratorial explanations of the Nato bombing of Yugoslavia: cognitive bias or an ideological tradition. In: International Society for Political Psychology, Annual Meeting, Berlin, July 2002).

However, whilst the design of the above document necessarily emphasizes the "entropic" tendency towards some form of bias (at any point on the circumference), it offers no clues as to the nature of the reverse "negentropic" process -- framed in the argument above as "conjection". Ironically the form of the diagram echoes that of many traditional mandalas -- which have that preoccupation but call for other considerations. In the light of the cartographic approaches to projection, it is intriguing that the above design uses an "algorithmic" approach.

Forms of transdisiplinarity as indicative of the "conjection" process: Reference was made above to the various flavours of interdisciplinarity as a conventional surrogate for "conjection". It might therefore be asked how integrative cognitive procsses could be fruitfully clustered and classified. One early exercise identified some 600 such "understandings" but explicitly avoided any classification (Integrative Knowledge and Transdisciplinarity Project: overview).

A recent exercise collected 43 methods from a number of transdisciplinary research projects dealing with a variety of research topics (Matthias Bergmann, T. Jahn, T. Knobloch, W. Krohn, C. Pohl, and E. Schramm, Methods for Transdisciplinary Research: a primer for practice, 2012). These were then grouped into seven classes following an epistemological hierarchy -- starting with methods in the narrower sense, progressing to integration instruments:

  1. Integration through conceptual clarification and theoretical framing
  2. Integration through research questions and hypothesis formulation
  3. Screening, using, refining, and further developing effective integrative scientific methods
  4. Integrative assessment procedures
  5. Integration through development and application of models
  6. Integration through artifacts, services and products as boundary objects
  7. Integrative procedures and instruments of research organizations

Missing from such an approach is any sense of the nature of transdisciplinary insight. What is the "transdisciplinary experience", given use of an analogous expression with respect to mathematics (Philip J. Davis and Reuben Hersh, The Mathematical Experience, 1981, with a chapter on the Varieties of Mathematical Experience; Douglas R. Hofstadter, Metamagical Themas: questing for the essence of mind and pattern, 1985; Wolff-Michael Roth, Phenomenology and Mathematical Experience, Linguistics and Education). How is this to be related to the argument of George Lakoff and Rafael Nunez (Where Mathematics Comes From: how the embodied mind brings mathematics into being, 2001).

The latter argument reinforces the implication that "conjection" may be more intimately related to a sense of integrative comprehension otherwise understood. This was the original preoccupation in collecting some 4,400 such approaches, again explicitly avoiding any classification (Human Potential and Development Project). Such classification, through the AQAL pattern, has been a notable preoccupation of the integral theory articulated by Ken Wilber (The Spectrum of Consciousness, 1977).

Despite the extensive literature cited above, missing would appear to be any meaningful classification of "conjections". Do the classifications with respect to cartography offer indications as to how this might be explored? In terms of the polyocular argument of Maruyama (2004), this would help to clarify the variety of ways in which NATO might be understood -- and might understand itelf in relation to its opponents.


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