Five-fold ordering of strategic engagement with time

Enhancing Strategic Discourse Systematically using Climate Metaphors (Part #7)

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Clearly the many polychora depicted above are a major challenge to comprehension -- as is very probably appropriate to the subtleties of the spectrum of human collective engagement with time. One possibility for addressing this complexity as a cognitive challenge is -- somewhat ironically -- to view the set of polychora from a "five-dimensional" perspective -- rather than through their distracting variety in 4D.

This arguably follows from the degree of clarity and memorability achieved more generally through mapping of higher dimensionality onto a pattern of lower dimension:

• line to point -- as in "what is the point" of a "line of argument"
• matrix to line -- as in generation of check lists (of points) from information presented in tabular or 2D matrix form (the list of 64 uniform convex polychora provides an example, in contrast with their tabular presentation giving characteristics)
• sphere to flat map -- as with the generation of 2D maps of parts of the 3D globe (illustrated by the Wikipedia List of map projections, notably using a Table of projections)

It follows, in a quest for memorability, that there is a case for representing the set of 4D polychora in 3D. More intriguing is the use of one or more polyhedra to that end, or even projections into 3D of particular polychora. This gives focus to the possibilities and desirability of "self-mapping" (as discussed below).

Assuming 30 categories of polychora: In the spirit of "following the numbers", the above-mentioned set of 29 uniform polychora (including the nonconvex) is assumed here to be a set of 30 -- especially given a lack (as yet) of any final conclusion as to the number of categories in the set, of the ongoing processes of reallocation of polychora between them, and of the lack of proof regarding the completeness of the set. Given the geometrical constraints, there is therefore greater probability that the number will be 30 rather than 29.

With this provisional assumption, the 30 categories can then be usefully -- if not significantly -- mapped onto a single polyhedral form (Spherical Configuration of Categories -- to reflect systemic patterns of environmental checks and balances, 1994; Triangulation of Incommensurable Concepts for Global Configuration, 2011). The form selected here for that purpose is the rhombic triacontahedron , given its unique properties in reconciling the forms of the 5 Platonic solids (K. J. M. MacLean, The Rhombic Triacontahedron, 2007; Robert W. Gray, Rhombic Triacontahedron, Encyclopedia Polyhedra, 2007).

With respect to the argument here regarding the climate of strategic discourse, a set of 30 categories in polyhedral configuration is a primary feature of the analysis by management cybernetician Stafford Beer (Beyond Dispute: the invention of team syntegrity, 1994). A more concrete appreciation of the significance of this form is offered by (X. W. Fang, et al. Spatially Resolved Distribution Function and the Medium-Range Order in Metallic Liquid and Glass, Scientific Reports, 2011). Given Beer's approach, there is some probability that a proof of a 30-fold set might be enabled by his insight. Together with that of Fuller (1975), Beer's insight was one basis for a polyhedral tensegrity mapping of the strategic issues of the 1992 Earth Summit (Configuring Globally and Contending Locally: shaping the global network of local bargains by decoding and mapping Earth Summit inter-sectoral issues, 1992).

Animation of rhombic triacontahedron (associating polychora categories with vertices)

*** force directed version

Nesting polyhedra: The rhombic triacontahedron then suggests the value of nesting polyhedra of greater spherical symmetry within it. This possibility of Nesting polyhedra to enable comparison of patterns of discourse was previously considered and extensively illustrated in a section of a more general argument (Embodying Global Hegemony through a Sustaining Pattern of Discourse: cognitive challenge of dominion over all one surveys, 2015). That gave rise to the following images (with links to interactive 3D variants in virtual reality).

Dodecahedron (blue) and Icosahedron (red) nested within Rhombic Triacontahedron (green) (accessible via virtual reality viewers/browser plugins -- VRML97 version or X3D version)	Tetrahedron (cyan) and Tetrahedron (magenta), with Octahedron (yellow) nested within Cube (grey) (accessible in virtual reality viewers/browser plugins -- VRML97 version or X3D version)

The complex on the right (above) can then be nested within the complex on the left (above) to give that on the left (below) -- recalling Johannes Kepler's memorable model of the solar system (on the right).

Nested Platonic polyhedra
Rhombic Triacontahedron (green) as a nesting framework (virtual reality variants static: vrml or x3d; mutual rotation: vrml or x3d; "pumping": vrml or x3d; videos: "pumping" mp4; "rotation" mp4)	Polyhedral model of solar system of Johannes Kepler Reproduced from Wikipedia entry in Mysterium Cosmographicum (1596)

Dynamic interpretation through weather/whether metaphors: At the centre of the complex on the right is the most fundamental Platonic form, namely the tetrahedron -- in two complementary forms. The tetrahedron is typically understood in static geometrical terms -- forgetting the argument of R. Buckminster Fuller that all polyhedra are more appropriately understood as systems (Synergetics: Explorations in the Geometry of Thinking, 1975). The question to be clarified is the cognitive dynamics associated with the tetrahedral pattern -- notably in relation to discourse, especially in the light of the complementarity between the two forms (Geometry of Thinking for Sustainable Global Governance: cognitive implication of synergetics, 2009).

One approach is through the unit cube within which the two tetrahedra are nested. As noted by Martin Gardner (The combinatorial basis of the "I Ching", the Chinese book of divination and wisdom, Scientific American, January 1974), a natural way of generating the eight Ba Gua trigrams in terms of the unit cube was indicated by Z. D. Sung (Symbols of Yi King, 1934). Noting that argument, that illustration was more recently reproduced by Pieng-Lam Kho (YiJing (I-Ching) Matrices, 2004). As the latter indicates, the convention used is 0 for broken line and 1 for solid line. Note the possibility of alternative readings of the encoding, whether from left to right (of the numbers) or from bottom to top of the lines -- significant in the dynamics governed by chirality. The eight sets of coordinates correspond to the eight trigrams, are complementary to the opposite cornered trigram of the cube.

Association of Ba Gua trigrams with unit cube (reproduced from Z. D. Sung, Symbols of Yi King, 1934)	Animation of adaptation of Ba Gua trigrams (based on image on left)

The tetrahedron can be used as a valuable mapping device for the quadrilemma articulated by Kinhide Mushakoji (Global Issues and Interparadigmatic Dialogue; essays on multipolar politics, 1988). This addresses the issue of transcending any dysfunctional oversimplification in discourse into binary form -- the "sidedness" of argument as challenged above. The four vertices of the tetrahedron can then be used to hold and distinguish:

Using the polyhedral framework (above), the dimensions in relation to knowing (comprehension, understanding. certainty) and to not-knowing (incomprehension, misunderstanding, uncertainty, ignorance) can be suggestively interrelated as follows. This is consistent with the notorious poem of Donald Rumsfeld regarding the known unknowns, as discussed separately (Unknown Undoing: challenge of incomprehensibility of systemic neglect, 2008).

The great value of the complementarity between the tetrahedra is that together they can then hold the dynamics of discourse. Use of the trigrams, understood through their weather-related metaphors, then offers a degree of coherence in systemic terms.

Animation of adaptation of cube to quadrilemma	Dominion dynamically challenged by incomprehension and uncertainty

The challenge is then to acquire greater insight into the dynamics between the tetrahedra, as is explored in the animations below.

Exploratory animations of tetrahedral morphing -- whether to be understood as distinctively 2-fold, 4-fold or 8-fold
Animations prepared using Stella Polyhedron Navigator

Cognitive "pumping" cycle with implications for collective discourse? The argument with respect to the dynamics of the complementary polyhedra suggests further reflection on the manner in which the polyhedral patterns (within which they are nested) should also be considered as necessarily dynamic rather than static.

The question is how the tetrahedra, octahedron, cube, icosahedron and dodecahedron "move" in relation to one another -- notably along axes of symmetry, as indicated in separate animations (in preparation). What are then the implications for discourse relating to governance? In the light of the animations above, such cyclic movement could be readily explored through animation. There is a sense in which such dynamics perform one or more "pumping" functions in cognitive terms. Those above are already reminiscent of the operation of chambers of the heart. Is the collective "heart of humanity" to be better articulated through such frameworks?

In relation to discourse and strategic formulation, one valuable indication is provided by the manner in which polyhedra can be formed by operations on each other -- most notably that of truncation. Stella4D provides a range of facilities through which a given polyhedra can be variously morphed into other forms through standard geometrical operations.

Much work in this respect has been done by Pieter Huybers (Nested Polyhedra, Newsletter of the Structural Morphology Group, 2007). He notes there that the envelope of a polyhedron can be formed by operations on polygons, notably rotation. This rotation takes place along circular routes around the X- and Y-axis of the co-ordinate system, but not before the polygonal face has been translated over a certain distance along the Z-axis. In some cases an initial rotation around the Z-axis is also necessary. He has developed the CORDIN computer programme to offer ways for the visual presentation of spatial forms -- without the need to build them in reality.

This has been usefully contextualized by Peter Forbes (Spatial Relations: the science of morphing has created a resurgence of geometry-led architecture. The Guardian, 27 March 2003)

D'Arcy Thompson was primarily concerned with the shapes of living things and how they got that way, but he often referred to parallels with engineering. One of his most perceptive observations showed how the form of one creature could be derived from another by means of a systematic grid deformation. By this system, two apparently different fishes can be seen to be derived from each other; ditto the human skull from chimpanzees.

Although a purely formal process, an exercise in the science of morphology, biology provides plausible mechanisms. Nature as a designer can only work through evolution; one structure by definition has to be derivable from another....Many of the new shapes are tried out on computer first and a program called CORDIN... invented by Pieter Huybers [who] applies Thompsonesque transformations to architectural designs and shows how what appear to be unrelated structures can be derived from each other. Usually, the overall shape is divided into triangles so that once the final form of a structure is found by means of the computer transformation, the space frame can be made by turning the triangles into struts.

Forbes notes that the driving force of modern architecture is geometry, and the computer has given a great fillip to the investigation of novel shapes -- as exemplified by the work of the artist Tony Robbin (Engineering the New Architecture, 1996), as a manifesto for the geometry-led architecture which is a particular focus of the Space Structures Research Centre (University of Surrey) using the Formian computer language to generate the triangulated truss structures of large complex domes and polyhedral configurations -- as well as four-dimensional structures (Tony Robbin, Formian for Art and Mathematics, Space Structures 5, 2002; Shadows of Reality: the fourth dimension in cubism, relativity, and modern thought, 2006).

Consistent with this approach, following his study of the Nature of Order (2003-4), the recent work of Christopher Alexander has focused on geometric analysis Christopher Alexander (Harmony-Seeking Computations: a science of non-classical dynamics based on the progressive evolution of the larger whole, 2009).

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