9th December 2009
| Uncompleted
Tentative Design of a Cognitive Array of Geometric Elements
***
-- / --
Introduction
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
References
Annex A of
Geometry,
Topology and Dynamics of Identity (2009)
Introduction
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):
- geometric forms: point, line, plane, polygon, circle, polyhedron sphere,
torus
- characteristic metaphors of cognitive complexification, notably as depicted
in Zen Buddhism by a traditional sequence of 10 ox-herding pictures, each
with a brief commentary (cf D T Suzuki; Kubota Ji'un, Ten
Ox-herding Pictures with the Verses Composed by Kakuan Zenji,
1996). These are of special interest because of their indication of a person's
progressive discovery and interplay with a shadowy element denoted by an
ox. In a Commentary
on the Integration of perceived Problems in the Human Development
section of the Encyclopedia
of World Problems and Human Potential, an attempt was
made to suggest how that classical sequence might be interpreted for clues
to an unfolding relationship between humanity and its shadow (in the shape
of the problematique as the complex of world problems). The interpretation
was developed under the following succession of headings (Progressive
integration of the shadow of non-self-reflexivity, 2007):
- Undisciplined exploration of the problematique
- Recognizing traces of the problematique as an integrated
system
- Focusing on the problematique as a whole
- Encompassing the problematique
- Orienting the problematique
- Using the problematique as a vehicle for sustainable development
- Transcending the realm of the problematique
- Disappearance of both humanity and the problematique
- Expression of essential humanity
- Human intervention in the world
- catastrophes identified in catastrophe
theory as articulated by Rene Thom (Structural
Stability and Morphogenesis: an outline of a general theory of models,
1975), expressed in a form of value to this argument by Denis Postle
(Catastrophe Theory: predict and avoid personal
disasters, 1980) as commented in Catastrophe
Theory: Examples of Integrated, Multi-set Concept Schemes (1984):
- WH-questions: (Conformality
of 7 WH-questions to 7 Elementary Catastrophes: an exploration of potential
psychosocial implications, 2006; Interrelating
Cognitive Catastrophes in a Grail-chalice Proto-model: implications
of WH-questions for self-reflexivity and dialogue, 2006)
- characteristic cognitive processes
- as suggested by the set of prefixes: projection, objection, subjection,
interjection, dejection, abjection, ejection, rejection, injection
(New Paradigms
via a Renewed Set of Prefixes? Dependence of international policy-making
on an array of operational terms, 2003)
- potentially associated with "complex" archetypal interaction
morphologies, identified in Thom's study
of catastrophes, whose
graphs are labelled as follows: capturing, sending, crossing, "almost",
fastening, giving, rejecting, failing, taking, stirring, emitting,
cutting
- as Roget **** : elongation, collapse, explosion/expansion/growth,
implosion/contraction/recession, curving, curling, encounter, disassociation,
solidifying, configuration, emptying, completion, initiation, clustering
- spectrum of white light ***
-- 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
.......... |
design
operational transformations |
scale
uniform/ nonuniform |
cluster |
radial |
reflect |
mirror |
. |
. |
. |
. |
questions |
unchanged. |
enlarged. |
multiplied
(scattered). |
multiplied
(related). |
extended
(axially). |
extended
(non-axially). |
divided. |
hollowed. |
rotated. |
what is the point
bullets
. |
point |
solid circle |
several
unrelated
points |
configuration
of points |
straight line |
curve |
segmented circle |
empty circle |
. |
which is the line
target of bullets
alignment
thematic pillars
axes, menu
Drupal
taxa
. |
line |
solid cylinder |
several
unrelated
lines |
intersecting
(network,
triangle,
polygon) |
flat plane |
curved surface |
polarized |
tube |
. |
where in the matrix (set of pillars)
cloth of many colours
field of decision
tangential
where to look
. |
plane |
beam |
several planes |
configuration
(polygon) . |
. |
. |
. |
. |
. |
when
timing in
learning/action
cycle
12 tribes
roundtable
implied centre ;
to intervene |
circle |
. |
. |
. |
. |
. |
. |
. |
. |
how to
construct
polyhedron
viable construction |
polyhedron |
. |
. |
. |
. |
. |
. |
. |
. |
who
articulation of identity in sphere |
sphere |
. |
. |
. |
. |
. |
. |
. |
. |
why
self-reflexivity
torus / Klein |
torus |
. |
. |
. |
. |
. |
. |
. |
. |
umbilics |
. |
. |
. |
. |
. |
. |
. |
. |
. |
- columns (to the "base" of how many before completion and starting new
row?):
- more Thom-like (see Roget) -- underlying cognitive geometry:
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:
- point: most obviously in the case of bullet points and making
a point (with further indications below)
- line: as with a line of thought or argument, or an
ideological line, and exemplifed in use of pillars, poles and axes. This
frame may be evident in reference to line of responsibility or command,
thematic thread, career pathway, line of credit, or alignment within any
group. Lines of communication may be established
- matrix or table: as the mode most fundamental to cognitive organization
of projects (through spreadsheet
software) and of conceptual models
- cylinder:
emphasizing the cognitive constraint of a line and its channelling capacity
for attention. Exemplified pejoratively by the sense of silo thinking and
tunnel vision.
- polygon: as exemplified by the triangle (Christian
Trinity) and the square (Masonic "on the square") -- possibly implying
a 3D polyhedron
- circle (or hole): implying a form of holistic containment
(exemplified by the sense of security of a stockade, or a circle of trust),
or an opening into another domain of thought. A hole may of course imply
a weakness in any system of containment, as with a "hole" in
an argument.
- polyhedron: as in tightening up a proof or an argument,
but physically evident through the necessary tightening of guy lines
and rigging. Notions associated with this are necessary balance, complementarity,
symmetry and proportion.
- sphere: a more developed sense of completion and comprehensiveness
than the circle, exemplified by "sphere of influence"
- dodecameral packing disparate domains ***
Cognitive dynamics of identity associated with elements
(of an array)
The following are indicative of the dynamics associated with each of the
above elements:
- point: the typical process is "making a point" (a
set of points), or
"pointing out"
- line: the associated process is constituting
a line by aligning points (of an argument, or "articles" in an
agreement). On the other hand a "line may be drawn" to constitute
a boundary ("thus
far and no further") or to make a distinction. A line, in the form
of a polarity, may relate (and separate) two distinct conditions. Such
a line, appropriately tensed, may
"vibrate" through "resonance" with other dynamics,
or may even be "plucked" to achieve vibration (Polarities
as Pluckable Tensed Strings, 2006)
- string theory ***
- walking / alternation ***
- matrix:
the dynamics may be recognized in:
- engagement with the process of balancing input/output in a spreadsheet,
namely with the interweaving pattern of flows of funds or resources
- weaving together threads of thematic information as symbolized
by the art of of textile or carpet weaving
- any exercise in urban grid planning, whether balancing claimed
attributions or traffic flows
- cross-fertilization of ideas between sectors or disciplines, metaphorically
represented by a genetic approach to cross-breeding or inter-marrying
to favour emergence of fruitful traits from disparate "bloodlines"
- ensuring fruitful co-existence of disparate perspectives and methodologies
metaphorically represented by inter-planting (as in permaculture),
multicultural communities, or interfaith gatherings.
- lines of poetry ***
- polygon: passing patterns, roundtable, dynamics --
network? sacred dance ***
- circle: cyclic
dynamics may be recognized
in the learning/action cycles explored by Arthur Young (***), through the
feedback loops of knowledge cybernetics (****), and metaphorically through:
- the process of circling the wagons and the role of the circus ringmaster
- the operation of a crime rings, of call girl rings, and of carousel
fraud
- the process of crop rotation (***)
- the process of recycling
- ****
- making a hole (in the integrity of another)
- dynamics of a hole are on off -- bytes
- presages
- ring, smoke ring, etc
- orifice -- consumption, excretion
- smoke ring, etc
- orifice -- consumption, excretion
- dyson ring (sphere?) ** fan
- digderdoo
- cylinder: wormhole / rabbit hole
- activity of the day is like moving down a cognitive tunnel -- spaceship
hyperdrive
- implicit line of travel
- visible objects passing by
- polyhedral dynamics -- tensegrity
- sphere: given the coherence of a globe and sphere, and
their role as integrative symbols -- the "stellar" celebrities and "balls
of energy" in any gathering or community --
the associated cognitive dynamics (potentially internalized) might be explored
metaphorically in terms of characteristic stellar dynamics:
- coriolis
- convection
- ejection (planets, etc) -- solar wind, etc
- Bucky ***
- accounting
- "sphere of" -- google
- attention
- influence
- defined by lines
- torus: the cognitive implication of toroidal dynamics
may be readily comprehensible through the metaphor of smoke rings (tori
***)
- dynamics of a dodekatheon of deities understood as the archetypal metaphor
of the ubiquitous roundtable (of a jury or any council of the wise), or
a system of "12 tribes" (**tribes, internet). The issue is how such cognitively
disparate functions interact to sustain the stability of the whoile. What
might be the interaction between the appropriately organized set of perspectives
of a dodecameral mind? (***)
- dodekatheon of identity >>> 36 plots
- metaphors of:
***
- translate
- shear
- reflect (glide)
- rotation (different orientation)
- intelligences / Ba Gua ?
- lyre
Transformational dynamics of identity across an array
- sphere to torus transition
- sphere/torus to Klein
- transitions:
- polygon to circle (incircle, midcircle, circumcircle)
- polyhedron to sphere (insphere, midsphere, circumsphere)
- sphere (event), turn in on itself
- torus, make a hole
- enantiodromia
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.
alternation
Ralph Abraham. Dynamic Geometry
- dynamical geometry / topology
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? |
point |
|
|
|
|
line |
|
|
|
|
|
|
|
|
|
sphere |
|
|
|
|
torus |
|
|
|
|
|
|
|
|
|
12 seats at roundtable (Tribes):
- EAFM
- CFM: point (space?), line, curve?
- opposites
- triplicity
- quadruplicity
- dodecameral
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] . |
. |