Projective geometry of discourse: points, lines, frames and hidden perspectives

Time for Provocative Mnemonic Aids to Systemic Connectivity? (Part #5)

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Commonalities to discourse terminology: Typically missing from use of polyhedral mapping possibilities is the manner in which the contents of discourse get associated with the chosen geometry. It could be considered remarkable that the process of discourse uses terminology which is typical of descriptions of the geometry. Most obvious is reference to "making a point", a "line of argument", or the "frame of a discussion".

Any effort to transcend the polarization typical of discourse on refugees (the trigger for the current argument) merits reflection in that light. These issues can be explored more extensively (Engaging with Globality -- through cognitive lines, circlets, crowns or holes, 2009).

Locus of "values" in projective geometry? A primary characteristic of divisive and polarized discourse is associated with reference to values and contrasting perspectives with regard to them. It could be argued that the elusively implicit nature of values is usefully located by being "framed".

In the case of a regular polygon the configuration of positions of edges or lines frames a centre which is an implicit feature of the polygon. The point with which the centre is associated is not an explicit feature of the geometry. The same applies to a (semi)regular polyhedron, although in this case it is the "framing" provided by the configuration of the sides which locates the implicit centre.

Associating a primary value with such am elusive centre is consistent with reference to the "central" nature of values, and to any reference to "axis" -- implicit in its own way. Especially intriguing with respect to implicit values are those axes of symmetry which pass through features on opposite sides of the polyhedron and are typically associated with an implicit great circle.

With respect to a polyhedron in 3D, any regular polygons forming the sides can then each be understood as defining (or defined by) "secondary" values -- which together imply the primary value at the centre of the polyhedron. potentially far more intriguing is the even "subtler" values which are framed by polyhedra in 4D, and higher (4-polytopes and n-polytopes), also known as polychora. Their (virtual) existence, extensively studied, offers scope for further investigation in terms of their strategic implications, as separately discussed (Four-dimensional requisite for a time-bound global civilization?; Comprehending the shapes of time through four-dimensional uniform polychora; Five-fold ordering of strategic engagement with time, 2015).

Interplay of mapping possibilities: In the quest for a 12-fold pattern of significance, it is appropriate to note that the focus on the dodecahedron emphasizes a mapping on 12 faces, in contrast with the cube where such a mapping could be envisaged with respect to the 12 edges. Another approach, of considerable relevance to this argument, was taken from a cybernetic perspective by Stafford Beer (Beyond Dispute: the invention of team syntegrity, 1994). That focused on the icosahedron of 12 vertices, as indicated with respect to videos of the resulting syntegration process (Team Syntegrity International, Syntegration -- for achieving solutions, YouTube, 3 November 2013; Olaf Brugman, Syntegration accelerates problem solving in complex systems: the case of Responsible Soy, YouTube, 27 May 2016 and text).

The argument here relates to the challenge of interrelating -- through the geometry -- the distinctive points and lines of discourse in order to honour the manner in which the discourse can be variously framed. Of particular importance, as in the case discussed above, is how different constituencies identify with different parts of the geometry in opposition to other portions (Oppositional Logic as Comprehensible Key to Sustainable Democracy: configuring patterns of anti-otherness, 2018). The latter notes the relevance of geometry in 3D and 4D.

Dynamic mapping possibilities as "dancing patterns": It is useful to recall the properties of some of the polyhedra discussed here in order to recognize how the number of elements (with which mapping may be required) may be shifted by using one polyhedron rather than another. This is notably the case through transformation into the dual of the polyhedron -- as between dodecahedron and icosahedron.

Of particular relevance to this argument is that endeavouring to view any of these forms limits recognition to a limited number of elements -- typically half in each case -- the others being "hidden", as with whatever is mapped onto them (unless the faces are rendered transparent, or the form is rotated). The sociopolitical and cognitive implications then merit consideration.

Many of the regular and semi-regular polyhedra can be transformed into one another through geometrical operations, most notably through duality. The following simplified map offers a sense of particular transformational pathways between patterns of order -- in which prime numbers appear to play a determining role as argued separately (Memetic Analogue to the 20 Amino Acids as vital to Psychosocial Life? 2015, and annex of Changing Patterns using Transformation Pathways, 2015).

The colouring of the "routes" in the map serves to highlight pathways of contrasting significance. Arguably some of the features derive simply from design choices, although the degree of symmetry calls for future comment.

Map highlighting distinctive relationships pathways between spherically symmetrical polyhedra (regular and semi-regular) F=faces, E=edges, V=vertices (Total of these in parenthesis) [Total reduced to prime number, other than 2, in square brackets]

The dancing metaphor with respect to such patterns is explored separately with respect to classic Chinese encoding systems (Sustainability through Magically Dancing Patterns: 8x8, 9x9, 19x19 -- I Ching, Tao Te Ching / T'ai HsÜan Ching, Wéiqí (Go), 2008). Those classics have the considerable merit in that they indicate both how content can be distinctively associated with the elements distinguished and how the relationships between the elements (with their content) can be understood systemically.

Is the dancing metaphor relevant to the switch from the 8-fold Millennium Development Goals of the UN to its current 16-fold Sustainable Developmet Goals (the 17th being the coordination of the 16)? In relation to this argument regarding the 12-fold, it could be considered curious that the former was 4 less than the 12-fold, and the latter is now 4 greater than the 12-fold, with the latter being twice the former.

Points, ball-games and "passing patterns"? Not only is "point" of value in discourse, it is also extremely evident in ball-games. Use of "ball" could even be considered as competitive play with a "point", or vice versa. "Line" is similarly important in relation to ball "passing patterns" (many videos), recognition of openings, and framing the boundary of a game. Technology is increasingly used in games, most obviously in football, and in relation to scoring. Perhaps the most extreme irony is that the basic design of an association football is that of a truncated icosahedron -- ironic in that the familiarity with the ball kicked around in games worldwide has in no way contributed to reframing global discourse more fruitfully.

It is indeed the case that divisive discourse is highly reminiscent of a competitive ball game with each endeavouring to score points and to downgrade the other within any ranking with other teams (Nature of the "ball" in game-playing and governance, 2016). The pattern is echoed in competitive point-scoring in parliamentary discourse. Curiously little effort is made in the latter case, and in the case of discourse more generally, to track points and passing patterns. Technology is limited to recording votes and voting patterns.

More fundamentally, there is a degree of irony to the manner in which the geometry of point and ball echo that of "global" -- especially in the case of discourse of global issues. To the extent that (semi)regular polyhedra are approximations to a sphere, the relevance to tracking global discourse (and its integrity) clearly merits exploration.

The argument can be further developed with respect to the contrast between bipolar discourse (as a fallback modality) with the multipolar discourse characteristic of a more complex global society. In particular it can be fruitfully asked why investigation of multi-sided games is so rare, as discussed separately (Destabilizing Multipolar Society through Binary Decision-making: alternatives to "2-stroke democracy" suggested by 4-sided ball games, 2016). It is appropriate to suggest that an unexplored reason for non-convergence of global discourse on the purported ideal of a unified perspective is the inherent interest in the dynamics of games -- whether as spectator, participant or gambler -- and familiarity with the default format of 2-sided games. People "like to watch" conflictual interaction -- to be understood in several connotations.

Given the familiarity with football, no effort has seemingly been made to experiment with four-team football, however three-sided football (also known as 3SF or d3fc) has indeed been developed. The three teams play over a hexagonal pitch (Geoff Andrews, The Three Sided Football Revolution: football's new idea, Philosophy Football, 9 June 2013; Sachin Nakrani, Three-sided football gives players something to think about. The Guardian, 7 May 2013; A game of three halves, Philosophy Football; see video and video and d3fc blog).

The point to be emphasized is that in each case the other side has a different "view" of the game. None see it whole in the absence of other forms of visualization and mapping.

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