Circular configuration of cognitive phases framing toroidal experience?

Zen of Facticity: Bull, Ox or Otherwise? (Part #9)

[Parts: First | Prev | Next | Last | All | PDF] [Links: To-K | From-K | From-Kx | Refs ]

Configuration as a circle (of eternal return): As a further exemplification of the above argument, the traditional linear sequence of those "ox-herding" images can however be configured in a circle as follows -- as previously explored in the Experimental configuration of nothingness as an "eightfold way" (2012). The magic square might be understood as an approximation to this -- exemplifying other aspects of the connectivity vital to systemic comprehension.

Circular representation of the classic Ten Ox-Herding Pictures of Zen Buddhism

Adaptation into circular form of the traditional linear version in Wikipedia, as derived from Nyogen Senzaki and Paul Reps, Zen Flesh, Zen Bones: a collection of Zen and pre-Zen writings (Charles E. Tuttle, 1957)

With respect to the above argument for an "eightfold way", it is appropriate to note that the last two images of the sequence relate to cognitive modalities transcending the 8 dynamics. These may be otherwise be held by a configuration of Möbius strips (Experimental configuration of nothingness as an "eightfold way", 2012).These are effectively subsequent to the cognitive modality of the eighth. They might be understood as relating to the two distinct axes of the toroidal form of the "necklace" -- that threading the individual strips together and that associated with its wearer.

The circular configuration of 10 images could then be reconfigured (as below) into an 8-fold pattern with the two additional modalities (or dimensions) represented as overlapping, encompassing or englobing the first in the original sequence. The possible correspondence between the 10-fold pattern and an 8-fold pattern is further highlighted by the existence of references such as the following (The Eight Ox Herding Pictures, 2012; The Eight Ox Herding Pictures: a Chan/Zen allegory).

It has been argued that the original set of the ox-herding pictures ended with the eighth "drawing" -- a blank space, indicating that anything that was depicted above the eighth and final stage would be misleading (How to Practice Zen, 2014). This could be considered to be consistent with arguments for "unsaying", notably as characteristic of apophatic discourse (Michael A. Sells, Mystical Languages of Unsaying, 1994).

Silence on certain matters has traditionally been highly recommended -- even framed as basic to civilized discourse. The core study by Ludwig Wittgenstein (Tractatus Logico-Philosophicus, 1921) concludes with the much-cited phrase: Whereof one cannot speak, thereof one must pass over in silence. The most fundamentally significant experiences of life may indeed be those of which nothing can be said. Nothing that can be said is worth saying?

Suggestive reconfiguration of the above 10-fold circular ox-herding representation into the form of an "8-fold way"

Configuration as a torus: In terms of comprehensibility and memorabilty, of interest is the possibility of presenting the above pattern within a torus (as shown below, left). The same approach may be used for any potential correspondence with the BaGua trigram pattern (below right). Such a depiction, especially when animated, evokes other ways of thinking about both in dynamic rather than static terms, as phases in a cycle rather than as stages in a linear progression. The representation is also helpful in suggesting the manner in which the phases may coexist, if only potentially, rather than each being superceded by the next. Any assumption of non-reversibility of time may also be called into question.

Variants of the animation may then enhance imaginative consideration of this possibility, as implied by other variants which could be produced (as noted below).

Animation of simplest torus as container for dynamic cognitive relations
10-fold pattern of Zen bulls (also .mp4, .wrl and .x3d versions)	8-fold BaGua pattern of trigrams (also .mp4, .wrl and .x3d versions)

Use of the torus usefully frames the question as to what might be understood as circulating through the "tunnel", as can be variously discussed (Circulation of the Light: essential metaphor of global sustainability? 2010; Enactivating a Cognitive Fusion Reactor: Imaginal Transformation of Energy Resourcing (ITER-8), 2006). The forces channelled through the major axis are well-illustrated metaphorically in the case of electromagnetism with the operation of a solenoid.

Configuration as interlocking pathways in 3D: The following animations derive from a pervious exercise ("Magic" and "auspiciousness" framed by ambiguity of the swastika, 2015; Integrative relevance of magical globality in 3D and 4D, 2015)

Screen shots of a 3D rendering of magic cube framing of 8-fold pathways for circulating spheres (central sphere added) (video; interactive virtual reality: VRML version or X3D version)
Solid rendering	Wireframe rendering

Further implications are suggested by two exploration of mappings of 3x3 patterns in 3D:

• As developed by James Grime, inspired by the widely-famed Rubik's Magic Cube (3x3x3), the question is whether the 2D magic square can be fruitfully incorporated into such a form. As extensively described by Katie Steckles (The Maths of the Grime Cube, The Aperiodical, 15 December 2016), the resulting Grime Cube is based upon the so-called perimeter magic square. This is an arrangement of the numbers 1-8 such that the three numbers on each edge sum to the same total (in this case 12). Only six valid perimeter magic squares exist for the numbers 1-8 -- with the consequence that when any two such patterns are overlayed they then sum to 9 as indicated below
  
• Use of a simple dice to hold the pattern of a 3x3 magic square. As discussed by the mathematicians Ie-Bin Lian and Vini Wu (Paradoxes on Chinese Dice and Magic Square, 2003) the equal-sum property of the columns, rows and diagonals on magic squares are mapped to sets of equal-expectation dice, thereby highlighting the associated paradoxes. The numbers in each row, column or diagonal of the magic square are assigned as the number of dots on faces on a dice. For a 3 by 3 magic square, an easy way is to s assign each of the three numbers to two faces of a regular dice.

The configuration of 10 and 15 can be most fruitfully held by various mappings in 3-dimensions on the icosahedron (as discussed below). Further insight may be obtain from a discussion of the enneagram in relation to the magic square and the BaGua pattern (Introducing the Celtic Enneagram)

Configuration as polyhedral great circles: The cognitive coherence of any potentially dynamic coexistence of the different conditions of the Zen pattern (or of any 8-fold trigram correspondence) can be fruitfully explored through polyhedral configurations. This would of course then have implications for any sense of facticity. The icosahedron (or its dodecahedral dual) are especially useful for this purpose. Of particular interest to any such mapping are the groups of associated great circles (as separately described and illustrated) [Ref: 'Geodesic Math And How To Use It, pp. 50-51, fig 7.6-7.8]:

• Family I: 6 great circles: Opposing vertices are used for the poles of spin. This creates six great circles which subdivide each face of the spherical icosahedron. This marks on the sphere the spherical equivalent of the icosidodecahedron with 20 spherical triangles and 12 spherical pentagons.
• Family II: 15 great circles: Opposing mid-edge points are used for poles of spin -- creating 15 great circles which outline every icosahedral face and slice through each three ways.
• Family III: 10 great circles: Opposing mid-face points are used for poles of spin -- creating 10 great circles; each one leaving the midpoint of a face edge to cross the next face edge at right angles. The circles are normal to the 10 axes of the corresponding dodecahedron. This enables one mapping of the Zen images (as shown below). The circles can be understood as dividing the icosahedral circumsphere into 20 distinct hemispheres; dividing the surface of that sphere into a total of 92 regions: 60 triangles, 20 hexagons, and 12 pentagons (N Points On Sphere all in One Hemisphere, MathPages).

Icosahedron
15 Great circles (click for dynamic variant from Wolfram)	Showing single golden rectangle (made with Stella Polyhedron Navigator)	Showing all 15 golden rectangles (made with Stella Polyhedron Navigator)	Showing dodecahedral dual (made with Stella Polyhedron Navigator)

In embodying so explicitly the 10-fold and 15-fold within a coherent framework, the icosahedron (or its dual) presents the challenge of how the relationship between such mappings is to be comprehended. These raise intriguing questions as to the relationship of the 10 to the 15, as previously suggested by the 3x3 magic square -- and to the 8-fold Bagua set. The fundamental implications are suggested by the building block of life, namely the carbon atom:

• allotropes of carbon: 8 are notably distinguished: diamond, graphite, lonsdaleite, C60 (Buckminsterfullerene or buckyball), C540, C70, amorphous carbon, and single-walled carbon nanotube (or buckytube).
• isotopes of carbon: 15 are currently distinguished (with half-lives ranging from nanoseconds to thousands of years), of which only 3 are naturally occurring

Of particular interest in this respect is how the cognitive significance of the Zen 10-fold pattern might be understood as related to the set of 15 transformations articulated by Christopher Alexander as discussed separately (Harmony-Comprehension and Wholeness-Engendering eliciting psychosocial transformational principles from design, 2010).

Potential concurrence of cognitive conditions: The animations below emphasize the manner in which the cognitive conditions distinguished by the Zen pattern may be to some degree co-existent or activated in some complex sequence -- with some only emerging very briefly (as with the half-life of some carbon isotopes). This would contrast with the traditional sense in which the images are associated with a linear cognitive development, typically as a consequence of a lifetime of meditative experience, if at all.

Icosahedron as a container for 10 Zen images mapped onto 10 great circles
animation of cycle in .mp4 format: vertical, expanding. Also x3d source
screenshot of vertical animation (10 images)	screenshot of expanding animation (5 images)
Icosahedron edges in green, dodecahedron (dual) edges in pale blue Icosahedron face centre points in red (axes of spin)
Polyhedra generated with Stella Polyhedron Navigator Configuration of great circles enabled with the aid of Sergey Bederov of Cortona3D

The visualization options engender a complex cycle of images suggesting that their greatest value for the viewer might derive from interactively switching between such options according to their appeal, most notably between rapid rates ("busy") and slower rates ("meditative"). It should be emphasized that (in their x3d or wrl versions) changes to the source files can be readily made with a simple text editor.

With regard to any such concurrency, Conrad Pritscher quotes Thomas Merton to the effect that:

Enlightenment is not a matter of trifling with the facticity of ordinary life and spiriting it all away. As the Buddhists say, nirvana is found in the midst of the world around us, and truth is not somewhere else. To be here and now where we are in our "suchness" is to be in nirvana, but unfortunately as long as we have thirst (desire or craving) we falsify our own situation and cannot realize it as nirvana. (Einstein and Zen: learning to learn, 2010, p. 138)

Mapping onto other polyhedra: Given the division of the cricumsphere of an icosahedron into 92 regions (as noted above), it is appropriate to note the possibility of mappings onto a limited group of the Johnson solids (92 strictly convex polyhedra that have regular faces but are not uniform).

Polyhedra offering a coherent reconciliation of mapping patterns of 8, 9 and 15
Tridiminished icosahedron (8 faces, 9 vertices, 15 edges)	Elongated triangular dipyramid (9 faces, 8 vertices, 15 edges)	Triangular cupola (8 faces, 9 vertices, 15 edges)

The central image is an enneahedron, one of 2606. Seemingly also of great significance to the possibility of any such mapping is the much-studied Herschel graph (with numerous social) and the enneahedron derived from it. That polyhedron has only recently been visualized following the work of Christian Lawson-Perfect (An Enneahedron for Herschel, The Aperiodical, 1 October 2013). As he describes the graph:

It's the smallest non-Hamiltonian polyhedral graph -- you can't draw a path on it that visits each vertex exactly once, but you can make a polyhedron whose vertices and edges correspond with the graph exactly. It's also bipartite -- you can colour the vertices using two colours so that edges only connect vertices of different colours. The graph's automorphism group -- its symmetries -- is D₆ -- the symmetry group of the hexagon. That means that there's threefold rotational symmetry, as well as a couple of lines of reflection.

Herschel graph (reproduced from Wikipedia)	Herschel enneahedron (animation from Wikipedia)
11 vertices, 18 edges	11 vertices, 9 faces, 18 edges

Using the forms of carbon as a potential guideline to psychosocial organization, of particular interest is the alternation of bonding most notable in the benzene molecule fundamental to the structure of organic tissue. This alternation is termed resonance. It is then relevant to think of the possibility that distinctive mappings onto polyhedra, variously highlighting 8, 9, 10 and 15 (for example), may be associated with a dynamic process of "morphing" between such forms. Rather than any singular polyhedron providing privileged comprehension of the phase transitions exemplified by the Zen images, a variety of such polyhedra may be of significance to such comprehension and its communication.

As a container for comprehension, any ultimately integrative the "grail" may be a resonant dynamic rather than a singular structure (In Quest of Sustainability as Holy Grail of Global Governance, 2011; Interrelating Cognitive Catastrophes in a Grail-chalice Proto-model: implications of WH-questions for self-reflexivity and dialogue, 2006).

[Parts: First | Prev | Next | Last | All | PDF] [Links: To-K | From-K | From-Kx | Refs ]