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Tentative Design of a Cognitive Array of Geometric Elements


-- / --

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
Annex A: Tentative design of a cognitive array of geometric elements
-- 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)
Complexification and simplification of identity
Array as a musical instrument

Annex A of Geometry, Topology and Dynamics of Identity (2009)


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, 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).

In particular of 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).

Structural outliners and conceptual scaffolding, 1995 ***

Tentative design of a cognitive array

Following from the above-mentioned use of geometric metaphors, the question here is the cognitive implication of the larger set of geometric forms and the transformations possible between them, notably as rendered explicit in design packages. 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):

As mentioned, there is an extensive literature on these distinctions and their classification. The attributions in the tabular array below are necessarily tentative rather than definitive. They are primarily indicative of how such frames might be made and with what one might then identify. Such a table is a form in its own right and may be fruitfully transcended for that reason as argued by Michael Schiltz (Form and Medium: a mathematical reconstruction, Image [&] Narrative, 6, 2003) and previously discussed (Beyond the plane: form and medium in terms of the calculus of indications, 2006).

Any subtleties that might be derived from insights of more formal classifications would then be best included subsequently, when there is a sense of the cognitive significance of the framework and its elements. They key issue is whether the process engenders patterns of coherence with which one can identify.

Characteristic metaphors are typically associated with the forms in each cell of the table below.

Transformation options (columns): It is assumed here that the extensive articulation of transformation options on geometric forms available in design software packages (Adobe Illustrator, etc) have cognitive functions implicitly associated with them, if only as suggestive metaphors. These options are therefore used as indicative column headings.

Developmental complexification (rows): It is assumed here that the complexification of common geometric forms is also associated with development of comprehension, as indicated above. The suggestion however is that various sets may be tentatively associated with the rows of the table, with precise correspondences to be discovered (Theories of Correspondences -- and potential equivalences between them in correlative thinking, 2007):

-- and possibly transformed from one to another, or alternating between one or more of them. ***

not a question reading the literature and becoming expert in it -- flocking

through their use in metaphor that the form of what is understood can be recognized

Indicative design: see metaphor geometry table / Towards a Periodic Table of Ways of Knowing -- in the light of metaphors of mathematics, 2009

operational transformations
uniform/ nonuniform
cluster radial reflect mirror . . . .
questions unchanged. enlarged. multiplied
divided. hollowed. rotated.

what is the point


point solid circle several
unrelated points
of points
straight line curve segmented circle empty circle .

which is the line
target of bullets

thematic pillars
axes, menu

line solid cylinder several
flat plane curved surface polarized tube .

where in the matrix (set of pillars)

cloth of many colours
field of decision

where to look

plane beam several planes configuration
. . . . .
timing in
12 tribes
implied centre ;
to intervene
circle . . . . . . . .
how to
viable construction
polyhedron . . . . . . . .
articulation of identity in sphere
sphere . . . . . . . .
torus / Klein
torus . . . . . . . .
umbilics . . . . . . . . .

Sense of static identity through cognitive elements (in an array)

The following are indicative of how thinking is channelled and focused by static geometric elements:

Cognitive dynamics of identity associated with elements (of an array)

The following are indicative of the dynamics associated with each of the above elements:

Transformational dynamics of identity across an array

Such a framework points to ways in which one might be cognitively or existentially "wafted" through the geometry along "lines", down "holes", etc -- and the associated processes of reification explication/ implication. There is a sense in which the cells of the framework offer alternative cognitive "realities" between which one might "navigate" (Navigating Alternative Conceptual Realities: clues to the dynamics of enacting new paradigms through movement, 2002).

Cognitively it is curious that more powerful explanation suggests recognition from a perspective out of any plane -- ex-planation. This raises questions about the significance of in-planation -- and the planes in which identity is embedded.


Ralph Abraham. Dynamic Geometry

Ralph H. Abraham and Christopher D. Shaw (1992). Dynamics -- the geometry of behavior, 2nd edition. Addison-Wesley

dancing cognitively

Dancing dynamic array: dialogue, sex, intercourse, violence
inherent fundamental dynamic: point > line, etc; return to point (redefined)
12 seats at roundtable? . point line sphere? Klein?

12 seats at roundtable (Tribes):

Young 12, but want spatial dimension

The Relationships between Network Elements, Structures and Flows

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

Relationships between Network Elements, Structures and Flows .

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