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

13th December 2009 | Draft

Geometry, Topology and Dynamics of Identity

Cognitive implication in fundamental strategic questions and dilemmas

- / -

Simpler forms offering a support for identity
Eliciting cognitive implications of formal relationships
Appropriation of geometry as a support for development of identity
Viability of cognitive engagement with geometrical objects
Re-cognizing freedom of personal identification
Tentative design of a cognitive array of geometric elements (Annex A)
Framing the identity of an other (and an other)
Complexification and simplification of identity
-- Simplification of identity
-- Complexification of identity
-- Transformations of identity
-- Communication with "extraterrestrials"
-- Communication with the alienated
Array of geometric forms as a musical instrument


The concern here is with the interplay between a sense of identity and the forms through which identity is expressed and patterned by psychological processes of identification. The focus is on how the range of simpler forms identified by geometry and topology function in support of articulation of individual or group identity -- in the moment and dynamically over time. In particular the concern is with implicit forms serving this function and the degree to which they are rendered conscious and explicit, notably through their use in guiding, key and generative metaphors (Guiding Metaphors and Configuring Choices, 1991).

The collective emphasis follows from arguments in a set of papers (Metaphorical Geometry: in quest of globality in response to global governance challenges, 2009; Geometry of Thinking for Sustainable Global Governance, 2009). The individual emphasis follows from exploration of the challenges of embodiment, especially their dynamic implication (Existential Embodiment of Externalities: radical cognitive engagement with environmental categories and disciplines, 2009; Emergence of Cyclical Psycho-social Identity: sustainability as "psyclically" defined, 2007).

A range of authors and disciplines have explored aspects of the possibilities highlighted here. A primary concern is however to show the intuited cognitive importance of geometrical forms as accessible to all -- independently of the sophisticated descriptions offered by such studies. These arguments follow from those of George Lakoff and Mark Johnson (Metaphors We Live By, 1980) and of George Lakoff and Rafael Núñez (Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being, 2000).

Of particular interest here is how people (or groups of any size) may variously comprehend such forms and the support they offer for identity and cognition -- whether or not any more sophisticated explanations are experienced as being of assistance in this process (or a source of confusion). Identity is understood as highly dependent on the construct with which, or through which, that identity is patterned by a process of identification and embodiment

The approach has notably been inspired by the arguments of Ron Atkin with respect to comprehension of the geometry through which communication and comprehension take place (Multidimensional Man: can man live in three dimensions? 1981; Combinatorial Connectivities in Social Systems; an application of simplicial complex structures to the study of large organizations, 1977). Their implications have been summarized with respect to Social organization determined by incommunicability of insight (1995).

Simpler forms offering a support for identity

Extensive use is made of geometrical metaphor in the articulation of identity and strategy. With respect to the latter, for example, a forthcoming book by Robert W. Keidel (The Geometry of Strategy: concepts for strategic management, 2010) uses the following chapter structure:

  1. Decoding complexity by isolating form
  2. Point thinking and organizational persona
  3. Linear thinking and organizational performance
  4. Angular thinking and organizational puzzle
  5. Triangular thinking and organizational pattern

As noted previously (Metaphorical Geometry: in quest of globality in response to global governance challenges, 2009; Geometry of Thinking for Sustainable Global Governance, 2009), such simple geometry is most evident in the "points" made in any "line" of argument. This is notable in strategic presentations made with "bullet points", and in the emphasis on value "pillars", "axes" and "poles" in terms of which strategies and belief systems are structured. The bullet points may be associated with budget lines appropriately configured together into a coherent structure -- or not. Comprehending that coherence may constitute a real challenge for policy-makers and their constituencies -- being able to "see the wood for the trees" and the "pattern that connects". Similarly, in any creative process, emergence of an idea or insight may be understood as a point -- with the challenge of eliciting related points and developing each, effectively as a line of argument in a configuration. (Polly Toynbee, We needed revolution from Gordon Brown but we got triangulation, The Guardian, 29 September 2009).

Discourse within the European Union often focuses on the appropriate "geometry" and the necessity for "variable geometry" (cf Alternation between Variable Geometries: a brokership style for the United Nations as a guarantee of its requisite variety, 1985).

Whilst most evident in mainstream initiatives, "geometry" has also been significant to the coherence of alternative initiatives. Pierre Rousset (The World Social Forum: a sustainable model?, Global Research, 2009) writes:

Some new features have been tried out in recent years to ensure a more efficient process: meetings of thematic assemblies in the forums, the definition of 'axes' around which the initiatives are grouped, proposals for the 'clustering' of workshops to increase exchanges between constituents and improve the visibility of the programme, the call for 'strategic' reflection, etc... But interesting as these experiments are, a politics which has become out of date cannot be addressed solely by dealing with the operating procedures of the WSF.

Individual and collective identity may be variously given sustainable form through identification with such geometric elements. Individuals en masse are readily reduced to "points" -- variously "aligned" in the pursuit of a collective agenda. Dialogue and debate are characterized by "polarization", identifying with one "pole" or the other. There is a quest for "angles" to make a case convincingly or to exploit an opportunity.

From a psychological perspective, a key question is the geometric metaphor with which one might associate one's own identity -- consciously or unconsciously. To what extent does one frame one's own identity as a point (as with one's other contacts), a collection of points, a line (perhaps as being on a career path or journey), caught in an "eternal triangle" of relationships, as a node in a network of relationships, or as a focal point of a circle of friends? More conventionally importance may be attached to a kinship network or to one's place in a social hierarchy of relationships.

Eliciting cognitive implications of formal relationships

Central to this argument is the degree to which people have ready experiential access to geometric forms that are fundamental to their thinking and sense of identity. The following indication of disciplines, which have explored this connection, is inserted here primarily to reinforce the legitimacy of this perspective for those uncertain of the validity of the geometric argument. These indications are not however central to this argument. This section may be skipped as the kind of unnecessary reference to external authority which is the subject of the next section.

The following examples derive from earlier work on Patterns of N-foldness: comparison of integrated multi-set concept schemes as forms of presentation (1980) as related to Representation, comprehension, and communication of sets: the role of number (1978). The issue here however is the extent to which such sets and patterns are represented geometrically, whether explicitly or implicitly.

Mathematics and physics: The range of geometric elements and their relationships is the subject of an extensive literature in mathematics. Mathematics also has an interest in the mathematical identity of mathematical objects, with the latter abstractions also of interest to philosophy. Geometrical form is also of concern in

Philosophy and religion: Principles fundamental to belief systems may well be represented in geometric form, most notably through the triangle, the square and the circle. A striking historical example is provided by early Christian understanding of the Trinity as challenged by the heretical doctrines of Arianism, subsequently used to refer by contrast to other nontrinitarian theological systems (Wade Cox, The Unitarian/Trinitarian Wars, Christian Churches of God, No. 268). Trinitarian warfare is a topic of continuing religious preoccupation (Gregory A. Boyd, Satan and the Problem of Evil: Constructing a Trinitarian Warfare Theodicy, 2001). Polygons may be valued by some belief systems as with the use of octagonal forms in Sufism and the Ba Gua of Taoism or some yantras of Hinduism..

These forms may all be used in spiritual practices as being conducive to fruitful meditation. A notable advocate of geometrical representation was Giordano Bruno using a qualitative approach to mathematics, applying spatial paradigms of geometry to language (Arielle Saiber, Giordano Bruno and the Geometry of Language, 2005; Alessandro G. Farinella and Carole Preston, Giordano Bruno: Neoplatonism and the Wheel of Memory in the 'De Umbris Idearum', 2002).

Gaetano Rametta (Fichte: El dominio de la razón: quintuplictà e individualità -- la costruziones dell'Io in WL 1807, Daimon, 9, 1994) indicates that Johann Fichte sets the problem of individuality within a twofold concept of quintuplicity. On the one hand, quintuplicity is seen as the fundamental structure of intuition. On the other hand, quintuplicity is opened by the free acts through which the thinking I project itself on that appearance. Individuality is thereby set within five possible kinds of vision (Einsicht).

A valuable elaboration of the cognitive implications of geometrical representation, termed systematics, is offered by J. G Bennett (The Dramatic Universe, 1956-1966) with regard to sets involving from 1 to 12 elements. It notably focuses on the enneagram. (A. G. E. Blake, The Intelligent Enneagram, 1996).

The system of anthroposophy developed by Rudolf Steiner has given much attention to projective geometry as a means of working in a precise manner with aspects of reality which cannot be described in terms of ordinary physical measurements. George Adams took his descriptions of how this space is experienced and found a special geometric representation (George Adams, The Lemniscatory Ruled Surface in Space and Counterspace, 1979; Lawrence Edwards, Projective Geometry, 1985). 

Design and symbolism: The possibilities of a "geometry of thinking" were central to the two-volume magnum opus of R. Buckminster Fuller (Synergetics: Explorations in the Geometry of Thinking, 1975; Synergetics 2: Further Explorations in the Geometry of Thinking, 1979). However, as noted previously, the experiential nature of his insights is perhaps necessarily implicit rather than explicit given his design and architectural preoccupations (Geometry of Thinking for Sustainable Global Governance, 2009).

Whether in the form of architecture or illustration, design is necessarily based on the configuration and manipulation of geometric elements -- especially as inspired by nature (Christopher Alexander, The Nature of Order, 2003-2004). This is most evident in the design options specifically offered by the vector graphic software packages used (Adobe Illustrator, CorelDRAW, etc). The interface between art and mathematics is notably explored in the annual international events of The Bridges Organization (Reza Sarhangi, Bridges: Mathematical Connections in Art, Music, and Science, 1998). Importance may be attached to particular geometric forms in sacred geometry and its embodiment in sacred architecture. Importance is also attached to polygonal forms in various traditions of esotericism and magic as cultivated by secret societies whose insights and ritual practices are associated symbols of special significance.

Especially challenging, in terms of their potential significance for comprehension and communication, are the complex geometric forms regularly observed in crop circles worldwide. The controversy over the origins of crop circles also serves the purpose of highlighting the extent to which purported significance associated with geometric forms is subject to abuse and fraud, as with the use of such symbols in greenwashing the image of multinational corporations. This pattern follows the appropriation of the Greek pantheon by the fashion industry (Religious "Plastic Turkeys" -- Hermes vs. the Hijab, 2003) .

Psychology: As noted by Paul Talor (Geometry of Thinking, 2001), in addition to the approach of Fuller:

Psychoanalysis and psychotherapy: Geometric forms have been of interest to the therapeutic professions as exemplified by:

Political and military strategy: As noted above, much reference is made to the "pillars", "axes" and "poles" in terms of which strategies and belief systems are structured (Coherent Value Frameworks: pillar-ization, polarization and polyhedral frames of reference, 2008). Political systems polarized into political parties ("government" and "opposition") may on occasion be reconfigured for purposes of "bipartisan" initiatives. Relatively little is heard of "tripartisan", "quadripartisan", or more complex configurations, especially on a global scale where it might be assumed that they were essential. One exception is the tripartite International Labour Organization.

A contrast with geometric connotations (strangely echoing the early Christian preoccupation) is currently made between trinitarian and nontrinitarian concepts of warfare (notably in the light of the continuing debate regarding just war and the response to "insurgency"). As formulated by Carl von Clausewitz (On War), the former consisted of three elements: the government, the army and the people, with the government having the ultimate authority over the army, which functions as its instrument with the support of the population (Edward J. Villacres and Christopher Bassford, Reclaiming the Clauswitizian Trinity, Parameters, Autumn, 1995).

Trinitarian warfare is therefore understood to be between states, prosecuted by professional militaries, with the people off-limits. In non-trinitarian warfare the people are the military and the state. War is total and absolute. Nontrinitarian wars therefore tend to involve non-state entities, notably warlords (T. S. Westhusing, American Pre-emption, Trinitarian and Nontrinitarian War, and Justice, 2005). As articulated by Martin Levi van Creveld (The Transformation of War, 1991) the non-trinitarian theory of warfare identifies five key issues of war:

  1. By whom war is fought -- whether by states or by non-state actors
  2. What is war all about -- the relationships between the actors, and between them and the non-combatants
  3. How war is fought -- issues of strategy and tactics
  4. What war is fought for -- whether to enhance national power, or as an end to itself
  5. Why war is fought -- the motivations of the individual soldier.

In Sun Tzu's classic The Art of War, the five traditional elements of Chinese philosophy are replaced by the five strategic elements that define the competitive world: mission (path), ground, climate (timing), command (leadership), and methods. As a martial art Ba Gua Zhang is based on the theory of continuously changing in response to the situation at hand in order to overcome an opponent with skill rather than brute force.

Management cybernetics: At its simplest, the importance of configuration is evident in issues of "table shape" in diplomatic negotiations (Pattern of Meeting Participant Roles: shadowy 'roundtable' hidden within every meeting, 1993; Spherical Configuration of Interlocking Roundtables: internet enhancement of global self-organization through patterns of dialogue, 1998). A degree of concern with comprehension is evident in the adaptation of Fuller's work on tensegrity to (psycho)social organization by management cybernetician Stafford Beer (Beyond Dispute: the invention of team syntegrity, 1994) -- and the subsequent development of syntegrity (Allenna D. Leonard, Team Syntegrity Background. 2002; J. Truss, et al. The Coherent Architecture of Team Syntegrity: from small to mega forms, 2003).

It is Stafford Beer and his collaborators who endeavoured to give functional significance to polyhedral representation of psychosocial systems -- with his focus on the icosahedron through syntegrity. Associated with this work has been the further development of his management cybernetics into the elaboration of a viable system model (VSM). However the psychosocial implications of this are much diluted, except in the extension into knowledge cybernetics by Maurice Yolles (Knowledge Cybernetics: a new metaphor for social collectives, Organisational Transformation and Social Change, 2006; Exploring Cultures Through Knowledge Cybernetics, Journal of Cross-Cultural Competence and Management, 2007).

With the development of internet-enabled networks, further possibilities become apparent (Polyhedral Empowerment of Networks through Symmetry: psycho-social implications for organization and global governance, 2008; Sacralization of Hyperlink Geometry, 1997).

Prediction, divination and gambling: In many of the traditional forms of prediction, great significance is associated with geometric configuration. This is most obvious in astrology (triplicities and quadruplicities) but also to some degree in those processes involving casting a set of objects and the psychological engagement with the result for purposes of interpretation. Traces of the associated psychological engagement are to be found in many forms of gambling.

Music and dance: Beyond the relationship traditionally recognized between mathematics and music, a new way of analyzing and categorizing music has recently beern developed to take advantage of the deep, complex mathematics seen to be enmeshed in its very fabric (Music Has Its Own Geometry, Researchers Find, ScienceDaily, 18 April 2008; Marc West, Geometrical music theory, Plus, 19 May 2008; Clifton Callender, Ian Quinn and Dmitri Tymoczko, Geometrical Music Theory, Science, 18 April 2008, 320, 5874, pp. 328 - 329).

The language of musical theory has been translated into that of contemporary geometry. A sequences of notes, like chords, rhythms and scales, are categorized so they can be grouped into "families." to which a mathematical structure can be assigned. They can then be represented by points in complex geometrical spaces. The basis of geometrical music theory is that it provides a unified mathematical framework for musical events that are described differently depending on the scenario, but are fundamentally the same. This work is indicative of the complex ways in which music is understood and is supportive of identity, notably in traditional sacred music..

The many patterns of dance (including breakdancing) may also be understood as expressions of geometrical relationships, again especially in the case of the patterns of sacred dance.

Games and puzzles: Geometrical forms are fundamental to engagement in positional games. This is notably evident in board games such as chess and go. In ball games, such as football, passing patterns are of great significance (Mark Weston, Passing Patterns, 2006; Athalie Redwood-Brown, Passing patterns before and after goal scoring in FA Premier League Soccer, International Journal of Performance Analysis in Sport, 2008; Association for Soccer Education and Teaching, Passing Patterns and Small Sided Games, 2008; Alan Reifman, Network Analysis of Basketball Passing Patterns II, 2006).

Pattern completion is fundamental to the engagement with such as sudoku, crossword puzzles, and Rubik's cube, as previously discussed (Rethinking Rubik's Cube: a mnemonic device for ways of knowing and engagement? 2009; Augmenting the psychoactive function of a mnemotechnical device, 2009).

Indicative clustering of domains implying
various degrees of identification through geometric forms
Indicative clustering of domains implying various degrees of identification through geometric forms .

Appropriation of geometry as a support for development of identity

The argument here might be framed through an adaptation of the classic phrase of Georges Clemenceau: War is too important to be left to the generals (Issues too Important to be Left to Specialists: selected web resources, 2004). The adaptation might take the form of: Geometry is too important to be left to mathematicians. A commentary regarding the Bourbaki Archives notes: I've sometimes had the feeling that differential geometry is too important to be left to people with background too much in analysis…(Blog of Peter Woit, Not Even Wrong, 2009).

The point might be made more strongly in the light of the huge development of the field of mathematics and a degree of pride in the irrelevance of many of its implications for a struggling world. It is indeed difficult to detect any application of mathematics, and especially geometry, to enable new forms of psychosocial organization -- in contrast to its applications for exploitation, surveillance and destructive purposes (as primarily funded for purposes of "defence").

Most pathetic is the failure to apply mathematical insight to simplistic territorial claims -- as in the challenge of the Middle East, as previously argued (And When the Bombing Stops? Territorial conflict as a challenge to mathematicians, 2000). Equally pathetic is the inability to propose alternatives to the simplistic mathematics on which "democracy" is so problematically based (at the expense of lives and cultures), or to elaborate new understandings of collective "harmony" and "unity" for which appeals are so plaintively made.

Curiously the understandings of the complexity sciences have been most assiduously applied to develop the toxic financial derivatives market and not to the possibility of more fruitful forms of organization (Consciously Self-reflexive Global Initiatives: Renaissance zones, complex adaptive systems, and third order organizations, 2007).

Viability of cognitive engagement with geometrical objects

Various constraints on access to knowledge have been previously discussed (Emergent characteristics of knowledge-based society, 2009; Emerging Memetic Singularity in the Global Knowledge Society, 2009; Memetic and Information Diseases in a Knowledge Society: speculations towards the development of cures and preventive measures, 2008). Constraints are also associated with the effective use of knowledge (Coherent Policy-making Beyond the Information Barrier: circumventing dependence on access, classification, penetration, dissemination, property, surveillance, interpretation, disinformation, and credibility, 1999; Recognizing the Psychosocial Boundaries of Remedial Action: constraints on ensuring a safe operating space for humanity, 2009).

As a feature of this emerging context, various interrelated factors inhibit or constrain cognitive engagement with geometrical objects. These include:

In practice these dysfunctional constraints contribute significantly:

Mathematicians have offered a variety of delightful fictional descriptions of aspects of this cognitive condition: Edwin A. Abbott's Flatland: a romance of many dimensions (1884), Charles Howard Hinton's An Episode on Flatland: or how a plain folk discovered the third dimension (1907), A. K. Dewdney's The Planiverse (1984), Ian Stewart's Flatterland (2001), and Rudy Rucker's Spaceland (2002). The 1884 novel has recently taken the form of an animated version (Flatland, 2007) to highlight the challenges otherwise.

Re-cognizing freedom of identification

The focus in what follows is to identify how individuals remain free to explore and benefit from geometrical objects in support of the development of their own identity -- despite the above constraints then understood as framing windows of cognitive opportunity. The exploration is consistent with the principles of such as Paul Feyerabend (Against Method: outline of an anarchistic theory of knowledge, 1975; Conquest of Abundance: a tale of abstraction versus the richness of being, 1999) -- as previously discussed (Value Embodiment: participatory engagement with environmental reality, 2008; Declaration of Universal Independence: delinking from detachment through radical questioning, 2009).

The windows of opportunity are conveniently identified with the aid of the following figure.

Tentative map relating "closed" and "open" arenas
Map relating  closed  and  open  arenas .

This cognitive freedom is a feature of:

Examples of maps with related concerns

AQAL Map: "All Quadrants All Levels"
Open Source Integral
[click for larger version]

Interrelating problematique, resolutique,
"imaginatique" and "irresolutique"

[click for larger version]
AQAL Map: "All Quadrants All Levels". Interrelating problematique, resolutique, imaginatique and irresolutique.

Tentative design of a cognitive array of geometric elements

Presented separately as Annex A
-- Sense of static identity through cognitive elements (in an array)
-- Cognitive dynamics of identity associated with elements of an array
-- Transformational dynamics of identity across an array

Framing the identity of an other (and an other)

Irrespective of the framing of one's own identity as a point, a line, or a circle, etc, a corresponding issue is how the identity of any other person (or group) is then to be framed. Examples possibly include framing the other as:

Each such framing of identity raises the question of how "agreement" or "disagreement" with the other is experienced and what is invariant in the sensed identity. Clearly there is also the question of how one's own "geometry" connects and engages with that of any other.

Complexification and simplification of identity

Simplification of identity: This may be understood in two senses:

Where identity is associated with a variable geometry, simplification may be a temporary reconfiguration for a particular purpose. The geometry with which a richer understanding of identity is associated is then implicit in the simpler form. Expression of identity may then be effectively unfolded and enfolded (as illustrated by some forms of origami).

Complexification of identity: This may also be understood in two senses through:

With respect to complexification, of great interest are the very complex geometric objects discovered by mathematicians. These may be understood to some degree through their symmetry. The key question is how such symmetry may be used as a support for more complexified identity. How vital is greater complexification of individual or group identity as a means of sustaining higher degrees of order, especially in situations which are otherwise completely problematic? (Engaging with Questions of Higher Order: cognitive vigilance required for higher degrees of twistedness, 2004).

Transformations of identity: However, with respect to both simplification and complexification, the argument here is for an ability analogous to that now well-recognized with respect to any maps on the web, namely the ability to "zoom" into greater detail, or out of it, as required. How is identity to be understood in such terms?

What might be the cognitive implications of these transformations and the attraction of their ultimate forms, as noted above with respect to the Mandelbrot fractal, the E8 group and the Monster group (Potential Psychosocial Significance of Monstrous Moonshine: an exceptional form of symmetry as a Rosetta stone for cognitive frameworks, 2007; Psycho-social Significance of the Mandelbrot Set: a sustainable boundary between chaos and order, 2005; Cardioid Attractor Fundamental to Sustainability: 8 transactional games forming the heart of sustainable relationship, 2005; Hyperaction through Hypercomprehension and Hyperdrive, 2006; Comprehension of Requisite Variety for Sustainable Psychosocial Dynamics, 2006).

Of particular interest is the relevance of more complex geometry to adaptive resilience under turbulent conditions as the possible requirement for:

Communication with "extraterrestrials": The challenges of communication with hypothetical extraterrestrials of the future have long constituted an opportunity to reflect on the forms appropriate to such contact (Communicating with Aliens: the psychological dimension of dialogue, 2000). The matter was given due consideration in the design of the famous plaque affixed to the Pioneer 10 (1972) and Pioneer 11 (1973) spacecraft. The plaques show the nude figures of a human male and female along with several symbols that are designed to provide information about the origin of the spacecraft. The controversial nude figures were removed from the cover of the Voyager Golden Record included in the two Voyager spacecraft launched in 1977.

In terms of the argument here, these plaques are unique in their effort to represent human identity to a potential other -- hence the irony of the deliberate removal of the contrasting human figures.

Representations of human identity by NASA for extraterrestrials
Plaque affixed to Pioneer spacecraft Cover of the Voyager Golden Record
Plaque affixed to Pioneer spacecraft Cover of the Voyager Golden Record

In the light of the argument previously presented, the interesting feature of these plaques is whether they say more about human communication preferences (and inhibitions) than about how it may be fruitful to engage with others (Self-reflective Embodiment of Transdisciplinary Integration (SETI) the universal criterion of species maturity? 2008). The emphasis is placed in the above images on the fundamentals of number theory with only a minimalistic use of geometry. It may however be the case that extraterrestrials attach greater significance to geometric forms, as suggested in the controversy regarding the origins of crop circles (Jenny Haworth, Is crop circle pi from the sky or just another con?, 18 June 2008; Graham Tucker, What's behind the symbolism found in formations? Medway Crop Circle, 2007).

Ironically again, rather than abstract geometry, it also the case that humans attach greater significance to the geometries of the human form -- fundamental as (strange) attractors to the process of reproduction, and currently the source of the major problem of humanity and the planet. Also of relevance is the Protagorean dictum that "man is the measure of all things" -- a focus for the Renaissance and famously depicted by Leonardo da Vinci as Vitruvian Man (naked). Curiously this image was used as the basis for the astronaut patch of the Earth-orbiting Skylab Expedition 2 in 1973 (with a variant patch for the wives of astronauts) -- presumably not for the attention of extraterrestrials.

Vitruvian Man by Leonardo da Vinci
[click for larger version from Wikipedia]
Vitruvian Man by Leonardo da Vinci

The cognitive role of aesthetics in relation to an "extraterrestrial" challenge was provocatively framed at that time by Marsilio Ficino (Thomas Moore, The Planets Within: the astrological psychology of Marsilio Ficino, 1990). The cognitive implications of reproductive geometry have been separately explored (Intercourse with Globality through Enacting a Klein bottle: cognitive implication in a polysensorial "lens", 2009).

In the light of the argument above, how might a degree of communication be ensured through geometric pattern? The question is especially pertinent if it is assumed that communication is driven by sets of values and understandings of order and harmony -- configured beyond the simplistic conventional use of "pillars", "poles" and "axes".

If such cognitive order, and the psychosocial order with which it is associated, was more fully articulated through such unique mathematical objects as the the Mandelbrot fractal, the E8 group and the Monster group, how would humanity seek to engage with such expressions of identity? Is the "key" to ("unlocking") such communication a question of matching geometries rather than one derived from decryption based on number theory?

Some possibilities for communication reflecting a degree of transcendence of duality can be highlighted through the manner in which precious stones -- traditionally emblematic of human values -- are cut into jewels (Patterning Archetypal Templates of Emergent Order: implications of diamond faceting for enlightening dialogue, 2002). This may correspond to Celtic and other traditions regarding the elder races, "from the time before", who thereafter "withdrew into the stones".

To the extent that such integrative communication relates to spherical geometry, and "approximations" to a sphere, any notional human representation of extraterrestrials as "angels" might also be fruitfully considered in terms of the "angles" at which such "stellated" entities variously engage with an all-encompassing sphere of global insight. By comparison humans may simply be "geometrically dyslexic" -- despite the role of geometry in courtship and reproduction !

Communication with the alienated: Rather than any focus on extraterrestrials, of dramatic current significance are the "terrestrial extras" -- the alienated with whom communication is a major challenge of increasing political relevance. The term might be interpreted as geometrically delinked or "out of line" (with respect to social geometry).

Both "angels" and "angles" are widely used metaphorically. Of relevance to the argument here is the sense in which "angle" is then associated with insight as in an "angle of research" or "angle of negotiation". "Angel" is widely used as an exemplification of value. "Dyslexic" or not, curiously any web search for items containing both "angels" and "angles" reveals an unexpectedly rich assortment of documents in which an association between them has been variously recognized. This implies a degree of recognition of a bridge between two seemingly unrelated cognitive functions -- the logical and the intuitive. The references include:

Array of geometric forms as a musical instrument

Perhaps appropriately, the explicit use of geometry in the Pioneer and Voyager imagery was replaced in the latter case by music and song as an expression of human identity. As noted above, there is a geometry to music and its cognition that is potentially richer than that which can be visually represented -- and clearly attractive to "terrestrial extras" and potentially to "extraterrestrials".

There is the suggestive possibility that any array of geometric forms, with their cognitive implications, might be fruitfully considered as a kind of musical instrument -- with the strings of the traditional lyre bearing a resemblance to such an array, or the guitar with its fret. "Playing" on such an instrument then elicits cognitive associations patterned by the many possible geometric transformations between those forms. Of particular interest is the implication in such stringed instruments that the "columns" of an array are explicit as "strings", but the connections between the "rows" (and across the array as a whole) is supplied by the cognitive engagement of the "player". It is through playing that the Gregory Batespon's "pattern that connects" is rendered consciously explicit.

As explored elsewhere, identity is then associated dynamically with what is played, and the ability to play, rather than with any particular form (Polarities as Pluckable Tensed Strings: hypercomprehension through harmonics of value-based choice-making, 2006). On the other hand such "strings" might be compared to the lines of a poem -- between the elements of which cognitive associations are evoked. The challenge is of course how to "tune" such an array, as argued with respect to the array of belief systems (Tuning a Periodic Table of Religions, Epistemologies and Spirituality -- including the sciences and other belief systems, 2007).

Such an approach has possible implications for elaboration of attractive, comprehensible strategies and organizations, as variously argued in the following:

The approach also has implications for the elaboration of any set of ideas, whether:

Curiously, as with the above-mentioned tradition of the elder races "from the time before", humanity is already "withdrawing" its identity into such structures -- whether they are to be understood as jewel-like "precious stones" or, more simply, as legally fossilized (if not "petrified").

Within the context of a musical metaphor, it is useful to reflect on the insight of Jacques Attali (Noise: The Political Economy of Music, 1985) that each period of history cultivates the musical organization appropriate to its current operational and strategic challenges -- but unfortunately makes actual use to that end of the patterns from the previous era of music. Typical of this is the use of Beethoven's Ode to Joy as the Anthem of Europe (prior to the Lisbon Reform Treaty which suppressed use of an anthem). There is then a case for reflecting on the musical implications of "organ":

The metaphor also lends itself to insight into identity through the notion of "composing a life" as a work of harmony with an implicit geometry -- a dynamic geometry (Mary Catherine Bateson, Composing a Life: life as a work in progress, 1989).

As previously cited with respect to this argument, the case has been even more strongly emphasized in the words of Antonio de Nicolas with regard to any language with its epistemological basis in tonal relationships (Meditations through the Rg Veda, 1978). He distinguishes four "languages" in the Rg Veda by their intentionality: images and sacrifice, existence, embodied vision, and non-existence. Such efforts to show the functional significance of sacrifice in relation to social integration need attention in a challenging strategic period when "nobody is willing to sacrifice" advantages acquired under the present systems in crisis.

For de Nicolas: "The embodiment of Rg Vedic man was understood... as an effort at integrating the languages of Asat, Sat and Yajna to reach the dhih, the effective viewpoint, which would make these worlds continue in their efficient embodiment". The unique feature of the approach is that it is grounded in tone and the shifting relationships between tone. It is through the pattern of musical tones that the significance of the Rg Veda is to be found:

Therefore, from a linguistic and cultural perspective, we have to be aware that we are dealing with a language where tonal and arithmetical relations establish the epistemological invariances... Language grounded in music is grounded thereby on context dependency; any tone can have any possible relation to other tones, and the shift from one tone to another, which alone makes melody possible, is a shift in perspective which the singer himself embodies. Any perspective (tone) must be "sacrificed" for a new one to come into being; the song is a radical activity which requires innovation while maintaining continuity, and the "world" is the creation of the singer, who shares its dimensions with the song. (p. 57)

As previously argued, the challenge of identity is that of Existential Embodiment of Externalities: radical cognitive engagement with environmental categories and disciplines (2009).

Use of geometry to articulate cognitive insight
(generated with Stella Polyhedron Navigator as described in Polyhedral Pattern Language: software facilitation of emergence, representation and transformation of psycho-social organization, 2008)
Universal Declaration of Human Rights
Universal Declaration of Human Rights
(future detailed elaboration?)
Polyhedron of Universal Declaration of Human Rights Polyhedron with Universal Declaration of Human Rights
Articulation of a pattern of strategic insight at some future time (in a semantic web context) might use such polyhedra, or transformations between them, to interrelate (in an integrative, comprehensible manner) the elements of the pattern. The sections of a document (such as this one) might be accessible through each face -- appropriately related by the geometry to other sections


Ralph Abraham and Christopher D. Shaw. Dynamics: the geometry of behavior. Addison-Wesley, 1992.

Ron Atkin:

Ron Atkin and J. L. Casti. Polyhedral Dynamics and the Geometry of Systems. IIASA Report, Laxenburg, Austria, 1977.

Mary Catherine Bateson. Composing a Life: life as a work in progress. Plume, 1989

J. G. Bennett:

J. G. Bennett and A. G. E. Blake. Systematics: A New Technique in Thinking. Kingston-upon-Thames, The Institute for the Comparative Study of History, Philosophy, and the Sciences, 1966

E. W. Beth and Jean Piaget. Mathematical Epistemology and Psychology. Dordrecht: D. Reidel, 1966.

Antonio M. Battro. Visual Riemannian space versus Cognitive Euclidean space. Synthese, 35, 4, December, 1977, pp. 423-429 [abstract]

Martin Bliemel, Ian P McCarthy, and Elicia Maine. In Search of Entrepreneurial Network Configurations: using Q-analysis to study network structures and flows. Proceedings of the 4th European Conference on Technology Management, 6-8 September, 2009, Glasgow [text]

Luciano Boi. Geometrical and Toplogical Foundations of Theoretical Physics: from gauge theories to string program. 2003 [text]

J. L. Casti:

Mark C. Chu-Carroll. From Sets to Groups: deep meaning in simple constructs. Good Math, Bad Math, 3 December 2007 [text]

Konstantin Y. Degtiarev. Systems analysis: mathematical modeling and approach to structural complexity measure using polyhedral dynamics approach. Complexity International, 7, 2000 [text]

P. Doreian. Polyhedral dynamics and conflict mobilization in social networks. Social Networks, 3, 1981, (2), pp. 107-116.

Kalvin B. Drake. A Unitarian Universalist Theometry. The New Universalists, 2007 [text]

Michele Emmer. Mathland: the role of mathematics in virtual architecture. Nexus Network Journal, 7, 2, Autumn 2005 [text]

Alessandro G. Farinella and Carole Preston. Giordano Bruno: Neoplatonism and the Wheel of Memory in the 'De Umbris Idearum'. Renaissance Quarterly, 55, 2, (Summer, 2002), pp. 596-624

Giovanni Ferrero, Celestina Cotti, Michela Rossi and Cecilia Tedeschi. Geometries of Imaginary Space: Architectural Developments of the Ideas of M. C. Escher and Buckminster Fuller. Nexus Network Journal, 11, 2, July, 2009, pp. 305-316 doi:10.1007/s00004-008-0090-1 [abstract]

Paul Feyerabend:

R. Buckminster Fuller with E. J. Applewhite:

Howard Gardner:

P. Gould. Q-analysis, or a language of structure: an introduction for social scientists, geographers and planners.
International Journal of Man-Machine Studies, 13, 1980, pp. 169-199.

Thomas Homer-Dixon. The Ingenuity Gap: how can we solve the problems of the future? Knopf, 2000

T. Jacobson. Visualizing information seeking with Q-analysis. Paper presented at a non-divisional workshop held at the meeting of the International Communication Association, San Diego, CA., 2003

Robert W. Keidel. The Geometry of Strategy: concepts for strategic management. Routledge, 2010

George Lakoff and Mark Johnson:

George Lakoff and Rafael E. Nunez. Where Mathematics Comes From: how the embodied mind brings mathematics into being. Basic Books, 2001

Hilary Lawson:

J. Mcfadzean, N. G. Cross and J. H. Johnson. An Analysis of Architectural Visual Reasoning in Conceptual Sketching via Computational Sketch Analysis (CSA). iv, pp.258, Third International Conference on Information Visualisation (IV'99), 1999 [abstract]

Donald N. Michael. Learning to Plan and Planning to Learn, 1997

Thomas Moore. The Planets Within: The Astrological Psychology of Marsilio Ficino. Lindisfarne Books 1990

Jean Piaget:

Denis Postle. Catastrophe Theory: predict and avoid personal disasters. Fontana Paperbacks, 1980

Robert Romanyshyn. On angels and other anomalies of the imaginal life. Temenos Academy Review, The Temenos Academy, 2000

Reza Sarhangi (Ed.). Bridges: Mathematical Connections in Art, Music, and Science (Conference Proceedings 1998). Winfield, Kansas: Southwestern College, 1998 [review]

Arielle Saiber. Giordano Bruno and the Geometry of Language. Ashgate, 2005

Jonathan D. Spence. The Memory Palace of Matteo Ricci. Viking Penguin, 1984

Russell K. Standish. Theory of Nothing. BookSurge Australia, 2006 [text]

Robert J. Sternberg and Talia Ben-Zeev. The Nature of Mathematical Thinking. Routledge, 1996 [extracts]

P. Suppes, M. Pavel and J. C. Falmagne. Representations and Models in Psychology. Annual Review of Psychology, Vol. 45, 1994 [contents]

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

Bo Wang, Tian Gang Zhou, Yan Zhuo and Lin Chen. Global Topological Dominance in the Left Hemisphere. Proceedings of the National Academy of Sciences of the USA, 26 December 2007; 104(52): 21014-21019. doi:10.1073/pnas.0709664104. [text]

Olive Whicher. Projective Geometry: creative polarities in space and time. Rudolf Steiner Press, 1971

Peter Woit. Not Even Wrong. Jonathan Cape, 2006

Frances A. Yates. The Art of Memory. University of Chicago Press, 1966

creative commons license
this work is licenced under a creative commons licence.