Sunday, March 3, 2013

Fu Xi and King Wen

Fu Xi 's complementary arrangement and King Wen's pairs coincide on the 8 x 8 square

Wen's pairs can be said to fall into two categories: 8 pairs that are complementary, and 24 pairs that are figurate inverses.

The pairs of the first kind are found along both of the major diagonals which reflect across each other.  The pairs of the second kind are found reflected across the vertical axis of the diagram.

The colors indicate the line-composition of the figures: 
Black: 6 YANG 0 YIN
Indigo: 5 YANG 1 YIN
Blue: 4 YANG 2 YIN
Green: 3 YANG 3 YIN
Yellow: 2 YANG 4 YIN
Orange: 1 YANG 5 YIN
Red: 0 YANG 6 YIN

Fu Xi's arrangement, Wen's pairs, and the 8 x 8 square

This diagram shows the affinity between Wen's pairing and Fu Xi's arrangement of complementary opposites on the 8 x 8 square.

Wednesday, January 16, 2013

"Heaven and Earth, that is Tao"

In the first chapter of The Great Treatise we find:
The changes and transformations of Heaven are in the symbols of Change
The changes and transformations of Earth are in the forms of Change

What moves and completes the symbols is called Ch'ien.
What unfolds them into patterns of living is called K'un.

Thus, Ch'ien represents enfolded or implicate order, creative, symbolic reality; while K'un is explicated/unfolded, structive, formative reality.

Earth (K'un) is represented by the form of the simple square. 
Heaven (Ch'ien) is immaterial or intangible and has no form as such, but we may represent it through an overlaid series of lines that partition the square into four discrete regions:

The resulting figure can be resolved into fractions of sixteen.  The respective areas of the four discrete regions are 1/16, 3/16, 5/16, and 7/16, comprising a unity.  These four fractions are identical to the probabilities, given by the yarrow oracle, of measuring the four xiang symbols during divination.  

Having received the four symbols by combining Heaven and Earth, we project them onto our figure, yielding a full field to support the existence of the 64 Images:

From this we can infer that the  Eight Diagrams emerge from the 64 Images, consistent with Dr. Stephen Karcher's argument that the 64 Images "preceded" the Eight Diagrams (Ba Gua), which are used for identification and interpretation of the Images.

The ancients are said to have devised many implements by imitating Change.  Here, and elsewhere, I suggest that that the yarrow oracle was devised to imitate the intermingling of Heaven and Earth that reveals the xiang.  The figures presented here are purely geometric, hence they are antecedent of human artifice.  The xiang are eternal; coeval with, and fairly indistinct or inseparable from Change.  The ancient sages' far-reaching powers of sentience and cognition brought forth the xiang into consensus reality for the use of mankind.

The point that I hope to have made is that the xiang are entangled with Change, embedded or encoded in Change by way of this intermingling of the square (representing Earth) and the enfoldment that Heaven represents.  Without the enfoldment, we cannot discern the symbols in the square; without the square, the symbols have no basis for existence.  Indeed, "Heaven and Earth, that is Tao."

If we accept this line of reasoning, we can posit a natural, mathematical interpretation for Change that can be extended with further investigation.

Tuesday, January 1, 2013

Quantum Measurement

Systems exhibiting tangled hierarchy are said to have some degree of self-awareness or 'soul', per the writings of Douglas Hofstadter.  The following illustrations are provided to demonstrate the tangled hierarchy of the system called Change.  

Physicist Amit Goswami argues that systems demonstrating tangled hierarchy (plus non-locality and discontinuity) are related to quantum systems.  A sub-theme of these illustrations concerns observers and measurement, salient themes of quantum mechanics

In the Field diagram above, the four concentric regions indicate four "densities" and represent the four Symbols. 
The numbers in the diagram indicate the areas of the bounded regions enclosing those numbers.  
Each quadrant of the diagram encloses an area of 16 units; thus the two cardinal axes and four densities enfold 6 dimensions and 64 Elements.

We never experience any of the 64 elements directly; we perceive them only indirectly by way of the four Symbols, which we can experience probabilistically, through random selection detailed by the measurement Ritual.

The four Symbols, represented on the Field with colors, give us probability densities represented by their respective bounded areas.  The densities reflect the chance of sampling the given Symbol during the measurement Ritual.

The Field comprises six dimensions.  An object represented on the Field, or by the Field, or which existence depends on the Field, should reasonably comprise 6 dimensions.  Likewise, any such object should be expressible in terms of 6 dimensions.  
The figure above right is a stack made of six Fields; thus, these six Places are means of expressing the 6 dimensions. 

The Ritual is conducted upon the Field (above left) by randomly sampling a single Symbol from each of its six dimensions, represented by the stack (above center), then interpreting the ordered collection of samples (above right) as one of the 64 Elements.

What is the subject of measurement?
What is the object of measurement?