Dancing cognitively inside the box -- and beyond

Framing Cognitive Space for Higher Order Coherence (Part #12)

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Within the cubic framework: Given the cubic framework of this exploration, the point was made above that the six sides of the cube offered six windows through which the wider world may be seen. Using the Platonic polyhedra, the relationship between that 6-foldness, as with the 6-foldness of the I Ching hexagrams, can be explored with respect to the 8-foldness of the cube vertices highlighted above. Each of the polyhedra can be undestood as a "cognitive box" by which one is "boxed in".

Platonic polyhedra as "cognitive boxes"
	tetrahedron	octahedron	cube	dodecahedron	icosahedron
edges	6	12	12	30	30
sides/faces	4	8	6	12	20
vertices	4	6	8	20	12
axes		6 edged-centered	6 edged-centered	6 face-centered	6 vertex
great circles

The cube is the easiest to comprehend in this respect. As mapping surfaces any of the 6 windows could be transparent or opaque -- or possibly distinctively tinted. In the hexagram encoding, transparency could (for example) be denoted by a broken line and opacity by an unbroken line. The four vertices of any window could then be understood as distinctive perspectives. Opacity could then signify a lack of connectivity between those 4 perspectives -- whether as beliefs, faiths, disciplines, models, ideologies, or the like. Transparency of the window could then imply fruitful 4-fold connectivity between them -- enabling a view through the window.

Clearly any combination of transparent and opaque windows is then possible for the 6 windows -- possibly to be understood as "glass ceilings" or "glass doorways" of some kind. The glass cube then offers conditions ranging from complete non-transparency to total transparency. Which windows are imagined to be up or down, front or back, or right or left, is necessarily another matter -- depending on one's orientation. The transparency can be imagined as changing over a period of time in response to various conditions -- although the condtions could be usefully understood as cognitive rather than externl.

The octahedron (as dual of the cube) can be used otherwise -- with the vertices holding the 6-fold distinction between openness and closedness, and the sides a 3-fold pattern of connectivity between perspectives. In the case of the tetrahedron, it is the edges which then hold the conditions of openness and closedness. In the case of the dodecahedron, it is 6 face-centered axes which could hold those distinctions -- also held by the 6-vertex axes of the icosahedron.

The following are exercises in imagining the cognitive "dance". Rather than a tetrahedron, the equivalent number of edges of the Star of David can be used for a 2-dimensional animation. Of interest is the convention regarding allocation of trigram lines to triangle positions and whether alternative allocations are anyway of significance in their own right. The experimental animation on the right has the signs of the zodiac "dancing" within the constraints of a 12-sided "cage" in 2D, suggestive of the connectivity of such signs if represented in 3D within a dodecahedron (or in relation to the edges of a cube). The exercise derives from arguments of Arthur Young (Geometry of Meaning, 1976) regarding an experiential system of interwoven creative processes -- embodied in alchemical processes encoded by the forms of zodiacal signs, as discussed separately (Geometry of meaning: an alchemical Rosetta Stone? 2013; Representation of Creative Processes through Dynamics in Three Dimensions, 2014).

Representation of I Ching hexagram line codes		Use of 2D 12-fold pattern
Mapped onto edges of Star of David Reproduced from Mapping of I Ching hexagram coding onto Star of David (2008)		Use of signs of zodiac to suggest patterns of creative conectivity through "dancing" within a dodecahedron
64 I Ching hexagrams configured as double triangles (as in animation on the right)	Animation of cycle of 64 hexagrams suggestive of dynamics of triadic bonding

The following animations are exercises in representing a hexagram by the sides of a cube, one line per side. The design metaphor chosen was to use transparency and opacity to indicate unbroken or broken lines. This offers two alternatives represented below. An unbroken line could indicate opacity with a broken line indicating transparency (as on the right, as suggested by the openning). This would not however emphasize the "openness" associated with the first hexagram in red. In the reverse case an unbroken line could indicate transparency with a broken line indicating opacity (as on the left).

Experimental animations of one group of 8 hexagrams (a "house") using sides of a cube (alternative conventions with lower trigram unchanging -- 3 sides at bottom left of cube)
Convention: hexagram unbroken lines "transparent" (therefore lower left sides always transparent)	Convention: hexagram broken lines "transparent" (therefore lower left sides always opaque)

The experiment is indicative but relatively unsuccessful. Unlike the Star of David animations in 2D of all 64 hexagrams (above), it is not practical to cycle through more than a small group. Hence the use of the 8 in a "house" with one invariant trigram. More problematic is the default shading in the 3D rendering which effectively undermines the distinction between opaque and transparent (unless using a 3D viewer), since the opaque is rendered dark grey according to the orientation of the cube to the light.

It has been noted that in the much-cited poem Wallace Stevens (Thirteen Ways of Looking at a Blackbird, 1917), the description of a Turdus in a snowy autumn landscape alludes to the Cubist painting tradition of observing subjects simultaneously from numerous viewpoints to present a novel perspective. Of some relevance is the sense of a thirteenth perspective beyond the 12-fold pattern of the sube or the dodecahedron, as discussed separately (Ways of looking at ways of looking -- in a period of invasive surveillance, 2014; Post-modern challenge to simplistic binary framing of the other, 2014)

Beyond the cubic framework:

Maps indicating relationships between Platonic and Archimedean polyhedra
Polyhedra flow chart	Distinctive relationships pathways between spherically symmetrical polyhedra
Reproduced from Reddit (2018)	From Pathway "route maps" of potential psychosocial transformation? (2015)

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