Exercise in imagining hypercomputing via hexagram patterning

Imagining Order as Hypercomputing (Part #9)

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The potential relevance of hypercomputing can be usefully explored in relation to religions. Many might argue these constitute a particular understanding of hypercomputing in their own right, especially when understood as the creative imagining of order through analogy. Of special interest is the sense in which the imagined order then engenders credibility and fundamental belief. Also of relevance, however, is the degree to which the order imagined by different religions is seemingly incompatible -- despite claims to the contrary -- with all the consequences for the conflict these differences then engender (Stephen Prothero, God Is Not One: the eight rival religions that run the world, 2011).

Given the special understanding of number variously cultivated by religions, of particular interest is the possibility that development of mathematical theology could prove to be intimately related to the comprehension and credibility of hypercomputing (Mathematical Theology: future science of confidence in belief, 2011). Especially relevant with respect to religion is the binary focus on "belief" or "unbelief" -- variously played out so problematically. The possibility is increased by the arguments of cognitive psychology of George Lakoff and Rafael Nuñez (Where Mathematics Comes From: how the embodied mind brings mathematics into being, 2001). Hence the examples explored in the following exercise.

Six lines are configured in a number of cultures to form a double triangular "hexagram". This is most commonly associated in the West with the Hebrew Star of David symbol. It is however also used by Christians (notably the Mormons) and in Islam. Six pointed stars are also to be found in the cosmological diagrams of Hinduism, Buddhism, and Jainism. The non-Jewish Kabbalah (also called Christian or Hermetic Kabbalah) interprets the hexagram to mean the divine union of male and female energy. In traditional alchemy, the two triangles represent the reconciliation of the opposites of fire and water.

Triangular configuration of I Ching hexagrams: It is therefore useful to assume that the pattern is indicative of a form of hypercomputing through the manner in which it is possible to engage with the configuration -- or perhaps some intuitive sense of that possibility. Given that implication, this suggests the possibilty of configuring the traditional I Ching hexagrams in a triangular form as previously presented (Triangular representation of 64 I Ching hexagrams, 2008; Sustainability through Magically Dancing Patterns 8x8, 9x9, 19x19 -- I Ching, Tao Te Ching / T'ai Hsüan Ching, Wéiqí, 2008).

As presented there, the following table constitutes an alternative to the columnar representations of hexagrams. It is an effort to explore other mnemonic possibilities. The order follows that used in in the Richard Wilhelm translation of the I Ching. Although this order is used in the Wikipedia entry on the I Ching, it is not the order of the classical King Wen sequence. The lower, internal triangle, corresponds here to the lower trigram (in which the horizontal line corresponds to the lowest in the columnar hexagram representation). The upper lines correspond to the upper trigram (in which the central vertical line is the uppermost in the columnar hexagram representation). The convention of complete and broken line is maintained.

Triangular representation of 64 hexagrams

Mapping of I Ching hexagrams onto Star of David: It is therefore also of interest to explore such representations together with a possible configuration of the 6 lines of the I Ching hexagram into a double triangle consistent with such traditional symbolic use. The following table was previously presented (Mapping of I Ching hexagram coding onto Star of David; Double triangular representation of hexagrams: Star of David)

64 I Ching hexagrams configured as double triangles [tentative: 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 above set can now be presented as a dynamic pattern (as had been suggested in that earlier exercise) in what follows.

Dynamic representation of hexagram codes mapped onto the Star of David (note if the SWF format animation does not display automatically, it may do so more readily in Internet Explorer)

It is interesting that Barbara G. Walker, in a discussion of the earlier Fu Hsi (Earlier Heaven) arrangement of the hexagrams, in what is described as the only feminist interpretation of the I Ching (I Ching of the Goddess, 2002), explains a still earlier representation based on two interlinked triangles: one pointing down and the other up pointing.

Imagining static line depictions in 2D as circular dynamics in 3D: By "entangling" Eastern and Western symbol systems in this way, a degree of "stereoscopic" understanding of hypercomputing may become apparent, as previously argued (Enhancing the Quality of Knowing through Integration of East-West metaphors, 2000). Also of interest is the degree to which a triangle-based configuration is suggestive of the highly complex relationship between the three Abrahamic religions so fundamental to the current challenges and conflicts of global civilization (Root Irresponsibility for Major World Problems: the unexamined role of Abrahamic faiths in sustaining unrestrained population growth, 2007; Triangulation of Incommensurable Concepts for Global Configuration, 2011).

The argument for "re-cognizing" the lines in any symbolic depiction as implying a circle can be partially understood through the following.

Association of circles with edges of configurations of triangles

The argument for understanding the circles in 3D rather than in 2D can be partially clarified from the following, remembering that interlocking of circles have been a representation of the Christian Trinity. However, further to the suggestion above, Borromean rings are also a potentially valuable indication of the complex relationships between the three Abrahamic religions. The choice of the Borromean depiction in 3D is also significant as the logo of the International Mathematical Union -- notably given the argument of George Lakoff and Rafael Nuñez (Where Mathematics Comes From: how the embodied mind brings mathematics into being, 2001).

Borromean rings
Conventional depiction (notable for the topological implications)	Presented orthogonally as logo of International Mathematical Union (see Wolfram Mathematica animation)	Employed as a symbol of the Christian Trinity (from a 13th-century manuscript)

A sense of the 3D pattern of circles implicit in conventional 2D hexagram representations (especially including the Star of David pattern) is variously offered by the following animations. Of particular interest is the difficulty (if not impossibility) of representing the interlocking of those 6 circles, as suggested by the 3-ring Borromean rings above.

Animations indicative of circular dynamics implicit in Star of David hexagram pattern
"Remote" portion of cycles indicated in dashed form	Cycle indication as a continuous line	Emphasis on the plane framed by each cycle

Of particular interest is how contrasting understandings are carried by the 6 cycles -- differently oriented to one another. The implication of the colour reversal is usefully questionable.

As a vehicle, a cycle then functions somewhat as a conduit or tube for a particular style (or quality) of attention -- understood as a process. The question necessarily raised is the directionality of that attention through the conduit. This is partially suggested (in an unsatisfactory manner) by the reversals in the animations. The left-hand image, using dashed lines for the remote portion, also highlights the sense in which there is a degree of reversal implied by the cycle -- it returns to any starting point -- implying 12 distinct directions, inviting interpretation (Generic Reframing of the 12 Tribes of "Israel": "We have met the Zionists and them is us", 2009).

Insights from resonance structure of 6-fold benzene molecule: The challenge to comprehension is usefully illustrated by the historical process of understanding the 6-fold benzene molecule which is so fundamental to organic molecules and life. This is illustrated by the following image in the Wikipedia entry.

Historical development of understanding of the benzene molecule (reproduced from Wikipedia)

Historic benzene formulae (from left to right) by Claus (1867), Dewar (1867), Ladenburg (1869), Armstrong (1887), Thiele (1899) and Kekulé (1865). Dewar benzene and prismane are different chemicals that have Dewar's and Ladenburg's structures. Thiele and Kekulé's structures are used today.

The above image is suggestive of degrees of inadequacy in any comprehension of whatever is implied by a Star of David/Trinity pattern. In the latter case, the inadequacy is compounded by the manner in which any earlier depiction is asserted to be unquestionably correct, and the possibility that the most current is itself inadequate -- and likely to be subject to subtler interpretations (as with those of quantum mechanic understanding of molecular orbitals) as illustrated in the animation below.

Indicative animation of molecular orbital of hydrogen atom (reproduced from Wikipedia entry as the simplest illustration of molecular orbitals)

Electron wavefunctions for the 1s orbital of a lone hydrogen atom (left and right) and the corresponding bonding (bottom) and antibonding (top) molecular orbitals of the H2 molecule. The real part of the wavefunction is the blue curve, and the imaginary part is the red curve. The red dots mark the locations of the nuclei. The electron wavefunction oscillates according to the Schrödinger wave equation, and orbitals are its standing waves. The standing wave frequency is proportional to the orbital's kinetic energy. (This plot is a one-dimensional slice through the three-dimensional system.)

There are many animations of the benzene molecule on the web to illustrate its resonance structure -- so fundamental to life. As is so typical, most of them are subject to copyright or marketing restrictions. The following simple presentation illustrates the principle of oscillation between single and double bonding -- between the (carbon) atoms at the vertex positions of the hexagonal structure. Although conventionally depicted in 2D, the configuration is necessarily to be understood as being in 3D (at least).

Benzene molecule animations indicative of resonance between configurations A and B
Schematic of bonding configuration (A)	Schematic of bonding configuration (B)

Insights from tetrahedral folding: The contrast between the linear representation of hexagrams in 2D, and any richer comprehension implied by cycles in 3D, is suggestive of further insight. The Star of David pattern can be transformed by reducing the size of one triangle relative to the other in the hexagram -- nesting the smaller within the larger, with all that that may imply symbolically. The corresponding parallel lines then constitute metaphors of one another. Of interest then is how the lines in 2D are to be understood as "edge-on" views of circular processes in 3D (as explored above).

The right-hand image from the earlier exploration can then be used to take the argument further as shown in the accompanying images below.

Alternative association of circles with edges of configurations of triangles

The point to be stressed is that the 2D configurations of triangles can be considered as an unfolded tetrahedron. A mapping of the I Ching hexagrams onto a tetrahedral form was presented above. However, when folded into the 3D tetrahedral form, the challenge is how the implicit circular dynamics are to be understood. Is the folding/unfolding dynamic also intrinsic to hypercomputing -- as suggested by the explication/implication argument with respect to the holomovement of David Bohm? Some understanding is offered by the animations (left and centre) below. The animation on the right is presented in lieu of ability to associate circles with edges.

Animations suggestive of transformations through 3D (produced with Stella Polyhedron Navigator)

Especially relevant is the manner in which any movement "along" a 2D line of the nested triangle is matched by analogous movement along the corresponding parallel (of the co-linear two-lines in the external triangle). However, when folded into a tetrahedron, such double lines merge into a single line -- oriented quite distinctively in 3D relative to that with which they were parallel in 2D. The movement "along" that merged line is then better understood as movement around the circle it implies. Reaching any apparent end-of line-then implies reversal of direction. The significance of this reversal may be "re-cognized" in terms of enantiodromia or through the much-quoted lines of T S Eliot (in Little Gidding, 1942):

We shall not cease from exploration
And the end of all our exploring
Will be to arrive where we started
And know the place for the first time.

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