Comprehensive mapping of distinction dynamics in governance

Unquestioned Bias in Governance from Direction of Reading? (Part #5)

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Limits of tabular representation in 2D: As noted, the table above indicates 16 "reading modalities". Understood as pairs --as evident from the table -- this offers 8 dimensions or axes.

Of interest here is whether these distinctive "dimensions" in two dimensions can be configured in other ways such as to offer further insight into the challenge of governance through this 8-fold set of 16 extremes. Polyhedra in three dimensions offer one approach, suggestive of how arenas of distinction could be configured together -- appropriately separated from each other as a reflection of the challenge of "reconciling" contrasting "reading styles" and the tensions between extremes. These tensions are significantly and fruitfully explored in music.

Given the preoccupations of governance, the merit of moving beyond a two-dimensional representation can be stressed otherwise. Much is made of the negotiation "table" with respect to resolving the challenges of governance and the manner by which they can be "contained". There is however good reason to consider that their complexities cannot be contained by a table -- from which they may "roll off" or be "designed off", especially when much is handled "under the table". In that context, metaphorically speaking, any 2D map bears an interesting resemblance to a "table cloth" of a particular design.

In considering a 3D container, reference can be usefully made to Pandora's box -- the mythical container of all the evils of the world, according to the Greek culture from which democracy derived. Purportedly this was a sealed "jar", rather than a closed "box", the latter resulting from a mistranslation. If the challenge of governance is to gather the problems of the world -- wicked and otherwise -- into a container. The cognitive and other implications of the design of that container metaphor merit careful attention in the light of the influential work of George Lakoff and Mark Johnson (Metaphors We Live By, 1980). What indeed is the design appropriate to the challenges of the times?

Simpler 3D mapping surfaces: The dimensions could be mapped as followed, using 3 of the 5 Platonic solids:

• using the cube with its 8 vertices offers one memorable configuration
• using the octahedron with its 8 faces
• using the 4 faces and 4 edges of the tetrahedron

These are especially interesting if the purpose is to map only a sub-set of the 16 modalities, notably excluding the 8 characterized as "both", due to their use of the bidirectional pattern (the boustrophedon). The cube and octahedron are then suitable. Potentially more interesting, using the 4 faces of the tetrahedron, is the mapping of the three predominant modalities -- left-to-right, right-to-left and top-down -- respectively characteristic of "Western", Arabic and "Eastern" languages. This allows one face to hold the bottom-up language currently missing from conventional reading directions. The first image highlights the conventional assumption that only the conventional directions are recognized as necessary to global governance -- obscuring the necessity for the missing language, made evident in the other images to varying degrees. In strategic terms, "ISIS" might be questionably recognized as representing one such bottom-up "language".

Screen shots of a tetrahedron mapping conventional directions of reading -- and the missing bottom-up language (animation of rotation)

However the argument can be taken further by suggesting that mapping "dimensions", such as left-right, does not hold in a sufficiently memorable manner the problematic dynamics of the necessity of choosing one or the other in practice. Such mappings are in a sense too abstract for the reality of governance.

The question is then whether the 16 reading modalities could be distinctively mapped onto some other polyhedron. Curiously this is not the case. The possibility is excluded by Euler's polyhedron formula governing the relatiion between the number of faces, edges and vertices.

Reduction from 16 to 14 reading modalities? It is intriguing to consider that -- reading in the light of a chosen reading modality -- a mapping implies both a "reader" of the map and the consequent exclusion of a contrary reading modality, obscured by the preference (or rendered "unconscious" thereby). These two modalities are necessarily not represented within the mapping. The second is effectively hidden "beneath" the map -- "on its other side".

Although a seemingly questionable device, it is part of the reality of creation and use of maps. In this sense 2 modalities are necessarily excluded from any map of reading modalities (however that choice is made). This leaves 14 modalities which might then be mapped onto a suitable polyhedron.

The question is then how the process of choice of the reading modality is made -- "external" to the map -- in relation to the 16 in the table above. It could be suggested that any observation of a map (and its "repressed" complementary modality) become possible when:

• any pair of two reading modalities are excluded -- namely by excluding a dimension
• or any two (complementary) pairs of reading modalities (in the table) are each "conflated", so that the distinction between their extremes is "lost" in each case. Alternatively this could be better understood as the "collapsing" of the two dimensions into one, namely an inability to distinguish between them.

Cuboctahedral mapping: One polyhedron that could be used for such a mapping is the cuboctahedron -- one of the 13 semi-regular Archimedean polyhedra. Reduced from 16 to 14 -- two being implied -- the 14 directional modalities of reading can then be associated with its 14 faces (of 2 types), linked in pairs by 7 axes. That polyhedron has 12 faces and 24 edges.

Mapping of 14 reading modalities onto cuboctahedron faces
Showing bidirectional reading faces (animation of rotation)	Hiding bidirectional reading faces (animation of rotation)

The following images show the cuboctahedron unfolding into a flat 2D representation. That on the right is potentially indicative of the challenge of global governance, namely to configure the reading directions together globally.

Folding-Unfolding of cuboctahedron mapping
Completed unfolding into 2D	Partial unfolding (animation of folding from 2D to 3D)

The cuboctahedron is especially interesting given the arguments made in that regard by Buckminster Fuller in relation to world resouce management. It served as the basis for an intial version of his Dymaxion map (1946). Of particular interest is the manner in which any such unfolded 2D "map" is folded into a 3D global form. This suggests, as noted, that the challenge of global governance may include "folding" the pattern of potential reading modalities into a cognitively significant global form.

Mapping complexity consistent with the complex challenge of governance? However that choice is made to enable map reading, a quite different approach is possible. This would then be suggestive of the challenging problem of comprehending and governing the relationships between the set of dimensions -- appropriately honouring that complexity as a whole, to which reference is frequently made.

One possibility involves the use of the highly unusual Szilassi polyhedron. It is unique in that each face of that polyhedron shares an edge with each other face. As a result, it requires seven colours to colour distinctively each adjacent face, providing the lower bound for the seven colour theorem. The question is whether the complexity of that polyhedron is commensurate with that of global governance -- as is arguably required at this time.

Use was made of that polyhedron in a previous exploration of the challenge of Ways of Looking at Ways of Looking (2014). This took as its point of departure the much-cited poem by Wallace Stevens -- Thirteen Ways of Looking at a Blackbird -- with its allusions to the Cubist painting tradition of observing subjects simultaneously from numerous viewpoints to present a novel perspective. In the current period, this reframes concern with direction of reading in terms of how might a variety of ways of looking be elicited and juxtaposed -- perhaps such that together their strange integrity rendered them meaningless to conventional observation.

"Blackbird" is then best understood generically as an alternative voice, transcending any conventional framework and therefore implying a degree of dissent through its challenge to conventional modes of comprehension. Umberto Eco might be said to offer an example (Eternal Fascism: Fourteen Ways of Looking at a Blackshirt, New York Review of Books, 22 June 1995, pp.12-15).

In this case it is the 14 vertices which can be used to hold the contrasting modes of reading -- linked by the 21 edges (of 7 contrasting types). The faces are of 4 distinct types -- possibly to be associated with particular modalities distinguished in the tabular representation.

Mapping reading directions using Szilassi polyhedron

Animations of Szilassi polyhedron of 7 faces (4 types), 21 edges (12 types), 14 vertices (7 types) [totalling 42=2x3x7; product=2x3273] (paired shapes coloured identically; prepared using Stella Polyhedron Navigator)
Folding together of the two complement nets	Rotation of polyhedron

Further discussion of the Szilassi polyhedron is presented below. The previous exploration of its value to consideration of the variety of ways of looking formed part of an argument with respect to Anticipating When Blackbirds Sing Chinese (2014) -- in sections on:

• Post-modern challenge to simplistic binary framing of the other
• Imaginative composition of ways of looking or listening
• Embodying a multiverse of uncertainly ordered incongruity
• Thirteen ways of apprehending blackbird song
• Imagining future communication integrity enabled by aesthetics

The Szilassie polyhedron is complemented by its dual, the Császár polyhedron, necessarily of equivalent complexity and therefore lending itself to a complementary mapping approach. This polyhedron has no diagonals; every pair of the 7 vertices (topologically equivalent) is connected by an edge, together numbering 21. The tetrahedron and the Császár polyhedron are the only two known polyhedra (having a manifold boundary) without any diagonals.

Unfortunately, compared to the Szilassi polyhedron, Császár polyhedron is less visually interesting in its folded form and therefore less memorable as an orgranisation of the biases potentially significicant to globel governance.

Császár polyhedron
	Mapping reading directions using Császár polyhedron

Both the Császár polyhedron and the Szilassi polyhedron have the topology of a torus -- allowing for further interpretation (Spreadsheet to Torus, 2004; Comprehension of Requisite Variety for Sustainable Psychosocial Dynamics: transforming a matrix classification onto intertwined tori, 2006; Comprehension framed by "Torus", 2012; Complexification of Globalization and Toroidal Transformation, 2010).

In contrast to right-left, up-down thinking, are such depicitions adequately representative of the non-trivial complexity of global governance?

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