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Mapping of WH-questions with question-pairs onto the Szilassi polyhedron


Now as the Ultimate Cognitive Strange Attractor (Part #12)


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Mirror pairing: As a more appropriate mapping surface, the Szilassi polyhedron (presented earlier as an animation) is indicative of its challenge to comprehension. However the symmetry of that polyhedron, facilitating its comprehension, is evident from the image below through the manner in which the shapes of 6 faces are paired as mirror images. Only one of the 7 faces is unpaired. As a preliminary exercise, each of the WH-questions is arbitrarily mapped onto a face. The following images and animations have been prepared with the aid of the remarkable Stella: Polyhedron Navigator software, produced by Robert Webb (Stella: Polyhedron Navigator, Culture and Science, 2000).

Face shapes of Szilassi polyhedron presented to show pattern of mirroring
with tentative attribution of questions to faces
Faces of Szilassi polyhedron show ing mirroring

Question-pairs: The association of question-pairs with the edges of the configuration is somewhat clearer from the following image. Again the space-time dimension of physics can be considered as associated with the where-when question-pair. The other question-pairs can be explored as associated with a variety of intangibles fundamental to decision-making in the moment.

Selected views of an indicative mapping of question-pairs onto edges of Szilassi polyhedron
(variously rotated; pairs of edge types are identically coloured)
"How" and "Which" faces (in red above) rendered transparent Only central face rendered visible
Mapping of question-pairs onto edges of Szilassi polyhedron Mapping of question-pairs onto edges of Szilassi polyhedron
Mapping of question-pairs onto edges of Szilassi polyhedron Mapping of question-pairs onto edges of Szilassi polyhedron
Mapping of question-pairs onto edges of Szilassi polyhedron Mapping of question-pairs onto edges of Szilassi polyhedron

Nets and folding: The Szilassi polyhedron, when unfolded into 2-dimensions, can be understood as being composed of two distinct nets, as presented below. These offer a sense of how the 6-sides faces bind together when folded -- each face being in contact with each of the 6 other faces. Each edge is then a question-pair (as discussed above). These correspond to the contiguous faces. This is clearer in the case of each net separately. It is however vital to emphasize that, like the space-time of fundamental physics, each edge is a continuum. This may be more meaningful in experience in the moment than it is through any formal description. Assciation of the 3 umbilic catastrophes to the smaller net merits future reflection in the light of semiophysics.

Face shapes of Szilassi polyhedron (from above) presented in their 2 distinct nets
with the tentative attribution of questions to faces -- and consequent indication of some edge pairing when folded
(first and second net on different scales; paired shapes coloured distinctively)
Faces of Szilassi polyhedron with WH-questions

A sense of how the two nets are folded separately into three dimensions is given in the following two sets of images. It is appropriate to note that the software by which these forms have been explored permits the 2-dimensional nets to be output on paper with tabs, permitting them to be folded and glued into 3-dimensional models. Paired shapes are coloured the same (below), in contrast to the distinct colouring of all shapes in the net presentation (above).

Stages in folding first net of the Szilassi polyhedron
(images at various scales to facilitate representation; ; paired shapes coloured identically)
Flat (as above) Slightly folded 1 More folded Completely folded
Folding net of Szilassi polyhedron Folding net of Szilassi polyhedron Folding net of Szilassi polyhedron Folding net of Szilassi polyhedron

With the completed folding of the second net (final image below), it is then possible to get a sense of how the folded first net "fits" within it. This is of course much clearer in the animation which follows below.

Stages in folding second net of the Szilassi polyhedron
(images at various scales; ; paired shapes coloured identically)
Flat (as above) Slightly folded More folded Completely folded
Folding net of Szilassi polyhedron Folding net of Szilassi polyhedron Folding net of Szilassi polyhedron Folding net of Szilassi polyhedron

It is potentially significant that comprehension is facilitated through manipulation of the 3-dimensional form -- in contrast to the expectations and assumptions of what might be termed a "flat Earth" mentality dependent on a 2-dimensional representation. The point has been specifically argued with respect to a more complex polyhedron (Carlo H. Séquin and Jaron Lanier, Hyperseeing the Regular Hendecachoron). The need for "hypercomprehension" can be explored more generally (Hyperaction through Hypercomprehension and Hyperdrive, 2006).

Animations of Szilassi polyhedron
(paired shapes coloured identically)
Folding together of both nets indicated above
(some phases illustrated in screen shots above)
Rotation of polyhedron
(variation on version presented above)
Animation of foldng of Szilassi polyhedron Animation of foldng of Szilassi polyhedron

Toroidal hole: As a toroidal structure, further insight may be obtained from consideration of the central "hole". Remembering that the attribution of questions to the surface of the structure has been basically arbitrary, associating "why?" with the central platform as the unpaired shape has a degree of justification.

Detailed views through the toroidal "hole" of the Szilassi polyhedron
View from one side View from second side
Detailed views through the toroidal 'hole' of the Szilassi polyhedron Detailed views through the toroidal'hole' of the Szilassi polyhedron

As being potentially indicative of the dynamic experience of "now":

  • the hole is framed by 3 question surfaces (Who? What? Why?) , with 4 remaining WH-questions being "invisible" from within it -- namely the pairs Where? / When? and How? / Which? (in the current mapping). This could be consistent with a sense in which the 3 framing the hole are those more complex, having been provisionally associated (above) with the umbilic catastrophes.
  • the hole is bounded by 11 question-pairs as edges:
    • the 6 necessarily framing Why? (given the properties of the polyhedron): Where-Why, When-Why, How-Why, Which-Why, Who-Why, What-Why
    • the 5 not directly relating to Why?: Which-Who, What-Where, Who-When, How-What, Who-What
  • the hole has 4 distant edges which meet it only in a vertex point: Which-Where, How-When, How-Who, What-Which
  • the 6 edges completely dissociated from the hole are then: Where-When, Where-How, Where-Who, When-Which, When-What, Which-How

Of potential significance is the meaning of (in)visibility in this context, especially if it is suggestive of question-pairs whose nature is ignored or characterized by a unquestioning assumption. Being especially characteristic of (executive) decision, Where-When and Which-How (although notably orthogonal) may be deliberately or inadvertently dissociated from those potentially subtler questioning processes directly associated with the central hole.

The pattern of question-pairs is fruitfully suggestive of a system of cognitive decision processes whose integrative nature is lost when mapped into the 2 dimensions of any systems diagram or semantic map. It can be understood as indicative of the flow of attention associated with the sense of "now" -- and sustaining it.

Bounding spheres: The following images display the Szilassi polyhedron with circumsphere and midsphere.

Szilassi polyhedron with circumsphere
Spheres with faces visible Spheres with faces transparent
Szilassi polyhedron with circumsphere Szilassi polyhedron with circumsphere

Of potential relevance is the sense in which the outer sphere relates to externalities (objective knowns) typical of conventional executive decision-making, whereas the inner sphere relates to questions otherwise held to be implicit or abstract (mysteriously subjective and paradoxically self-reflexive) -- a contrast between explication and implication.

Types of edges as question-pairs: The Szilassi polyhedron is also of interest as a mapping surface because it offers insights into the relationship between the 12 types of its 21 edges. This can be more readily understood in two dimensions through the following schematic. This treats the 12 types as axes through the centre of a circle -- a focus for the sense of "now". The edges common to the two distinct nets are used as the primary orthogonal axes -- marked with an asterisk. The face shapes are then positioned at the 24 positions on the circumference -- with the contiguous edges on that axis. There are two separate variants of each in the polyhedron -- implying an alternation between the two in the image (with the exception of those markeed with an asterisk). The 24 circumferential positions are therefore reduced by 3 due to those marked with the asterisk.

Presentation of the 12 types of the 21 edges of the Szilassi polyhedron
(each of the 12 axes is an edge type; two edge variants for those without asterisk, according to label;
hexagonal face shapes are to scale; paired shapes coloured identically)
12 types of the 21 edges of the Szilassi polyhedron

The distribution of shapes/axes could no doubt be improved to render the pattern more memorable -- especially through an animated variant. The criteria for an improved distribution include: significant positioning of the 3 axes marked with an asterisk; special (related treatment) of unpaired (green) element; handling of colouring of paired face types and their distinction as individual types; achieving memorable patterning of colours/shapes within quadrants (or octants), maximizing variety and minimizing replication of colour or shape within them; exploitation of animation to hold more complex significance (and render the display visually attractive). Given the seeming simplicity (only 4 face types), the challenge is somewhat reminiscent of the arrangement of codons in the genetic code, or of the 8-fold arrangement of trigrams in the Chinese BaGua system -- suggestive of the possibility of a memetic code (M. Pitkänen, Could one *find a geometric realization for genetic and memetic codes? 2013).

Experimental alternation to clarify relation between question-pairs (preliminary exercise)
(faces in pairs are coloured distinctively in contrast to the single colour per face type above)
Configuration A Configuration B
12 types of the 21 edges of the Szilassi polyhedron 12 types of the 21 edges of the Szilassi polyhedron
Alternation between Configuration A and B
Animation of 12 types of the 21 edges of the Szilassi polyhedron

The focus on 12 can be associated with the mapping above onto the truncated icosahedron with its 12 pentagonal faces -- providing centers for a configuration totalling 20 hexagonal faces. However the focus on 12 is especially important given the strategic preference for 12-fold articulations and the systemic possibility of integrative approaches (Checklist of 12-fold Principles, Plans, Symbols and Concepts: web resources, 2011; Eliciting a 12-fold Pattern of Generic Operational Insights: recognition of memory constraints on collective strategic comprehension, 2011; Implication of the 12 Knights in any Strategic Round Table, 2014). Potentially valuable to this possibility is the clarification of the relation of WH-question pairs to preferences for a 12-fold pattern.

Further mapping possibilities: Various indications can be taken into account in elaborating a more memorable pattern -- potentially with mathematical assistance. Of interest is whether each quadrant offers a distinctive "story" by suitably sequencing the items there according to possibilities such as: edge length on axis, area of face, variation of faces between first and second net, positioning of unpaired face. There are various design possibilities for the animation: timing, changing direction, of forms, and colouring. As noted above, although matching edges to shapes derives directly from the properties of the polyhedron, attribution of "questions" to faces is somewhat arbitrary (despite the suggestions from catastrophe theory).

Use of the cuboctahedron as a mapping surface would seem to offer a number of complementary possibilities. As with the Császár polyhedron (the dual of the Szilassi polyhedron), it has 14 faces corresponding to the 14 vertex types of the Szilassi polyhedron. It is however a quasiregular polyhedron, with more symmetry properties, potentially an aid to memory with appropriate colouring. The 24 edges correspond to the mapping of Szilassi edges to the circle (prior to the special the reduction by 3). The cuboctahedron is central to the reflections of R. Buckminster Fuller (Synergetics: explorations in the geometry of thinking, 1975/1979), as discussed separately (Geometry of Thinking for Sustainable Global Governance: cognitive implication of synergetics, 2009).

Related insights could be derived from the work on management cybernetics of Stafford Beer (Beyond Dispute: the invention of team syntegrity, 1994; Platform for Change, 1978). This combined the mapping applications of the icosahedron with the development of a viable system model (Gunter Nittbaur, Stafford Beer's Syntegration as a Renascence of the Ancient Greek Agora in Present-day Organizations, Journal of Universal Knowledge Management, 2005). This raises the question as to whether a fruitful experience of "now" could be understood as enabled by a "viable cognitive system" of which the dynamic configuration of questions offers an indication.

This cybernetic approach has been notably developed by Maurice Yolles (Knowledge Cybernetics: a new metaphor for social collectives 2005; Exploring Mindset Agency Theory, 2013), taking account of both Chinese frameworks and the mindscape perspective of Magoroh Maruyama. There is a case for associating the distinctive "cognitive feel" of WH-questions -- framing "now" -- with the distinctive sense of different mindscapes (Mindscapes, social patterns and future development of scientific theory types, Cybernetica, 1980, 23, 1, pp. 5-25; Context and Complexity: cultivating contextual understanding, 1992). It is in this light that Maruyama could be understood as arguing for "polyocular vision" (Polyocular vision or subunderstanding? Organization Studies, 2004). Such arguments are indicative of the possibility of a "cybernetics of now", potentially consistent with various philosophies.

Indication of the association of one WH-question with another
(based on data on the Szilassi polyhedron from the Stella Polyhedron Navigator)
Indication of the association of one WH-question with another Indicates the association between a given WH-question and other WH-questions, as indicated by edge-length, as a percentage of total edge-length for the shape to which the WH-question is mapped. (NB: Self-contiguity is treated as 0%)

Clearly evident is the relative lack of "involvement" of the cluster What, Who and Why in When and Where, or How and Which. This is seemingly indicative of the relative lack of signifiance of the latter questions to the earlier cluster. The following schematic assumes that the set of WH-questions constitutes a viable system, meaning that the contribution of each is vital to the cognitive operations of the whole. In such terms, apparent over-representation may well be an indication of relative lack of significance, whereas relative under-representation may be indicative of greater proportionate "weight".

Indication of the explicit and implicit "weight" of WH-questions in discourse
Indication of the explicit and implicit 'weight' of WH-questions in discourse

Edge-lengths and Areas for a given face, to which a WH-question has been mapped, are presented as a percentage of the total edge-lengths or areas for the polyhedron.

On the assumption that the polyhedral configuration is an indication of a viable cognitive system, Edge-lengths and Areas are also presented in terms of the percentage by which they are greater or smaller than an equal contribution (namely one seventh of the total edge-length or total surface area of the polyhedron.

Current research on word association makes use of MRI brain scanning (Morgan Kelly, Word association: Princeton study matches brain scans with complex thought, News at Princeton, 31 August 2011). There is a case for applying such methods to clarify the nature of the "cognitive feel" associated with use of WH-questions, and issues relating to under-use or over-use of particular sets of of such questions. Of particular interest is any relation to the sense of "now", especially in comparison with uses of corresponding words in other languages (maintenant, ahora, jetzt, nunc, etc). Current work on the conversion of brain waves to music is expected to give rise to an application that can be run on a mobile device for personal experimentation within a year, as has already been done with biofeedback (Kat Austen, We turn brainwaves into sound for music and medicine, New Scientist, 12 April 2014).

Potential insights for operation of the "winged self": As developer of the Bell helicopter, Arthur M. Young was concerned with the decision-making and information processes in piloting one -- as they might apply to the development of a "psychopter". This was envisaged as the "winged self", requiring an analogue to "seat of the pants" skills in the moment (Geometry of Meaning, 1976), as discussed separately (Engendering a Psychopter through Biomimicry and Technomimicry: insights from the process of helicopter development, 2011). His analysis is potentially of relevance to reconciling the cognitive implications of catastrophe theory (as semiophysics) to any discussion of a 12-fold pattern of question-pair types. Its particular importance lies in the "cognitive feel" the pilot of the helicopter must necessarily have for the knowledge management in the moment.

The relevance can be explored through an argument presented separately in a section on Geometry of meaning: an alchemical Rosetta Stone? in the conclusion to a more general discussion (Eliciting a Universe of Meaning -- within a global information society of fragmenting knowledge and relationships, 2013). This included the following table.

12 "measure formulae" distinguished and clustered by Arthur Young
(reproduced from The Geometry of Meaning, p. 102)
Actions States Relationships
Position -- L Moment -- ML Power -- ML2/T3
Velocity -- L/T Momentum -- ML/T Inertia -- ML2
Acceleration -- L/T 2 Force -- ML/T 2 Action -- ML2/T
Control -- L/T3 Mass control -- ML/T3 Work -- ML2/T2
NB: Young indicates with respect to this table: The last column is displaced one place.., in order to have the three members on each line 120 degrees apart (p. 102) ... in his circular configuration presented below

As indicated there, Young comments extensively on the significance of each of these 12 in terms of the cognitive processes of learning/action cycles. Of relevance to the above argument is the co-presence of "states" and "actions" -- the latter implying the process dimension otherwise lacking in a "state-focus". Recognition of a "relationship" dimension suggests a valuable means of transcending that duality.

With the reservations indicated there regarding the tricky cognitive nature inherent in the creativity of the alchemical process, it was argued that there is a case for reviewing the 12 phases of the process as they have been traditionally associated  with the signs of the zodiac (although other patterns of phases are also identified). The names for these alchemical processes (as indicated by Wikipedia) have been added to the circular representation in the schematic below. This combines the triangular and square patterns of connectivity in the schematics articulated by Young in relation to the zodiacal pattern -- valuable to many for for mnemonic purposes, if not otherwise.

Zodiacal encoding of 12 "measure formulae" with associated alchemical processes
Zodiac of Geometry of Meaning (Arthur Young)

These distinctions have been interpreted separately for a variety of psychosocial contexts (Typology of 12 complementary strategies essential to sustainable development,1998; Characteristics of phases in 12-phase learning / action cycles, 1995; Typology of 12 complementary dialogue modes essential to sustainable dialogue, 1998).

As indicative of the kinds of information of relevance to the control of a helicopter, the 12 types of question-pairs could then be very tentatively explored in relation to Young's configuration -- thereby framing the nature of their relevance to the sense of "now" potentially required for the imaginative control of the "winged self". An obvious approach would be to use the distinction between the 3 "inner" faces and the 4 "outer" faces of the polyhedron (above) to frame a table corresponding to Young's 3x4 table (above), Given the subtle nature of the "cognitive feel" involved, a potentially more appropriate approach could be to use the 7 types of vertices of the Szilassi polyhedron -- of which there are 14. With the current mapping these are as follows.

7 Pairs of vertices in Szilassi polyhedron with indication of question-pairs
(coloured by vertex pair, as with 12 edge types;
edge lengths not to scale in this rotated perspective of the polyhedron, with faces transparent)
7 Pairs of vertices in Szilassi polyhedron with question-pairs
7 Pairs of vertices in image above
(distinguished by associated mapping of question-pairs onto the Szilassi polyhedron)
What-Where / What-When / Where-When Where-When / Where-Who / When-Who
Which-Where / Which-How / Where-How Which-How / When-How / Which-When
How-Who / How-What / Who-What Who-What / What-Which / Which-Who
Which-Why / Which-Where / Where-Why When-How / How-Why / Why-When
When-Who / When-Why / Who-Why What-Where / Where-Why / Why-What
How-Why / How-What / Why-What Which-Why / Which-Who / Why-Who
Where-How / Where-Who / How-Who Which-When / What-Which / What-When

It is the characteristic of the qualitative nexus, shared within a pair, which could then (in future) be used to dimension the 3x4 table of question pairs as suggested by the pattern below (and as explored in the previous typologies relating to learning/action cycles). For example, Young himself offers various indications (p. 154) including a fourfold distinction between spontaneous act, reaction, observation, and control.

Potential reconciliation of Young's pattern of cognitive functions with that based on questions (uncompleted)
    Vertex type D Vertex type E Vertex type F Vertex type G  
    T0 T1 T2 T3  
Vertex type A Actions
Who-What Which-When
Where-How
Where-Who
What-When
Which-Where
When-How
M0L
Vertex type B States
Which-How When-Who
What-Where
How-What
Which-Who
How-Who
What-Which
ML
Vertex type C Relationships
Where-When How-Why
Which-Why
Why-What
Why-Which
Why-Where
Why-When
ML2

Given Young's helicopter inspiration, there is a certain charm to the resemblance of the Szilassi polyhedron to the twisted rotor blades of a helicopter rotor through which lift and control are ensured. This reinforces implications as to the possibility of a "winged self" -- centered on "now".


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