Recognition of cultural icons within a pattern of nested circles?

Eightfold Configuration of Nested Cycles of Cognitive Transformations (Part #10)

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

If, as noted above, human perception can be understood as intimately associated with hyperbolic geometry, there is a case for exploring how this is evident in symbolic and aesthetic intuitions that have proven to be enduringly attractive to cultures -- over centuries. In the quest for a "new set of wheels" it is then appropriate to note how expression may continue to emphasize any "old set of wheels" whose value continues to be recognized. This suggests the existence of many attempts to "pack" the variety of subtle distinctions of significance into comprehensible form -- perhaps to be recognized as traditional vehicles for experiential navigating of hyperspace.

Especially intriguing is the manner in which such an "old set of wheels" may be recognized -- to some degree -- within a nested pattern of circles grounded in hyperbolic geometry. The following images are tentative experiments in that exploration.

Circle pattern placed over a threefold spiral Celtic triskele from stone carving in a prehistoric monument Brú na Bóinne, Ireland -- a World Heritage Site, built around 3200 BC	Interlinking of elements of circle pattern to form the classic configuration of the Kabbalah, originating in Judaism, and adopted by other religions.
The spiral pattern was enlarged without deformation to fit within the major horizontal circles. This positions the origin of the upper spiral within the smallest circle. Variants of the pattern exist, some having been redrawn with modern techniques. Used extensively in Ireland, the image features as the logo of the futures research organization Mankind 2000.	The classic Kabbalah configuration has been slightly deformed, although retaining topological invariances. The deformation is primarily required in fitting the upper horizontal circles -- then no longer vertically aligned with those of the two lower horizontals. Other approaches to "fitting" involved greater deformation.

The attraction of some representations of the Christian cross, most notably the High Cross of Celtic Christianity, may be partly associated with the focus it gives to some dimensions of the pattern of circles.

It is noteworthy, in an explication of the nature of hyperspace, that an image of the Kabbalah is reproduced by Clifford A. Pickover (Surfing Through Hyperspace: understanding higher universes in six easy lessons, 2001). He relates the Kabbalah to current insights into superstring theory, pointing out that the significance associated with the 10 linked circles of the image are understood to be hypostatized attributes "allowing the infinite to meet the finite". "Hypostatization" is the process of making into substance, namely making material an abstraction (with due regard to the fallacy of misplaced concreteness). For Pickover:

Similarly physical reality may be the hypostatization of these mathematical constructs called "strings"... the basic building blocks of nature... strings exist in a strange 10D universe... Using hyperspace theory, "matter" is viewed as vibrations that ripple through space and time... theoretical physicists have been using strings to explain all the forces of nature -- from atomic to gravitational... String theorists claim that six of the ten dimensions are "compactified" -- tightly curled up (in structures known as Calabi-Yau spaces) so that the extra dimensions are essentially invisible. (p. 15)

Pickover then relates superstring theory to the subsequent development of membrane theory:

In this new theory, life, the universe, and everything may arise from the interplay of membranes, strings and bubbles in higher dimensions of spacetime. The membranes may take the form of bubbles, be stretched in two dimensions like a sheet of rubber, or wrapped so tightly that they resemble a string. (p. 18).

The pattern of circles explored here also lends itself to mappings of the traditional relationship between the chakras in the tantric and yogic traditions of Hinduism and Buddhism. It could also be explored in relation to the geometrical figure composed of multiple evenly-spaced, overlapping circles, now called the Flower of Life. The circles in this case are arranged to form a flower-like pattern with a sixfold symmetry (as shown below), with the centre of each on the circumference of six surrounding circles of the same diameter. Many spiritual beliefs have been associated with the pattern, considered by some to be a symbol of sacred geometry depicting the fundamental forms of space and time.

Contrasting tiling patyterns
Circle pattern combining Tao and Lauburu (as described above)	Flower of Life (reproduced from Wikipedia)

Comparison of the circular images above makes apparent that both can be considered as tiling patterns or tesselations -- a theme of extensive mathematical investigation. The comparison highlights their respective embodiment of different principles of symmetry within a circle. The question is then the nature of the organization in a hyperspace of higher dimensionality which would hold and reconcile both such patterns -- and others. The argument has been developed with respect to the discovery of mathematics of "objects" of very high orders of dimensionality -- upheld as offering a key to the universe and its comprehension (Potential Psychosocial Significance of Monstrous Moonshine: an exceptional form of symmetry as a Rosetta stone for cognitive frameworks, 2007).

One strategically relevant application of such possibilities of "reconciliation" through higher dimensionality can be explored with respect to the 5-fold and 6-fold stars which are so fundamental to cultures currently in violent opposition. Each could be considered an "old wheel" valued by that culture -- effectively a monocycle. As tilings, these cannot be reconciled in 2D. They can however be reconciled in 3D through a particular spherical polyhedron, as argued separately (Middle East Peace Potential through Dynamics in Spherical Geometry: engendering connectivity from incommensurable 5-fold and 6-fold conceptual frameworks, 2012).

Of particular relevance is the poorly understood degree to which any such patterns are as much inventions of the collective human mind as objective representations of an external reality. There is even the possibility that such patterns could be compared to a variety of interference patterns between such creations -- as with the manner whereby the pattern of circles (as discussed here) emerged from the geometrical relations between the traditional symbols of the Tao and the Lauburu. In relation to any such instituted pattern, there is a degree of mnemonic humour to traditional references such as "animals" going into Noah's Ark "two by two" to avoid the flood, where the Ark is then to be understood as a cognitive pattern. Associated number references include the "144,000" entering the Kingdom of Heaven, or the "10,000 things" of the Tao Te Ching -- as contrasted with the Tao.

Such associations suggest the value of assuming that there is a possibility of recognizing the variety of enduring, culturally-valued, integrative symbols as intuitive understandings and depictions of a pattern of higher dimensionality -- one which can only be partially understood and communicated (as with many insights of mathematics). Hence the merit of the challenge of hyperbolic geometry, comprehension of the hypersphere, and the cognitive dynamics potentially associated with it. Relationships between partial understandings can be fruitfully explored through animation, as separately described (Dynamic Exploration of Value Configurations: interrelating traditional cultural symbols through animation, 2008).

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