Wednesday, April 25, 2012

Hypothetical Structure of Space-time

Our hypercube is used here to model space-time.
The core of the 4 x 4 x 4 cube is a 2 x 2 x 2  cube.  Since each of these are composed of cubes, the entire object is considered an hypercube.
The core may be regarded as the hypercube's inner dimension, while the corners of the shell may be taken as framing the corresponding outer dimension.  For the sake of clarity, the remaining 48 cubes are not depicted .

oracle inner outer
prob %   color corners corners remains dimension
 4/64    red        2       2       0       0th
12/64    blue       2       2       8       1st
20/64  green      2       2      16       2nd
28/64    yellow     2       2      24       3rd

In the table above, the yarrow-stalk divination probabilities are treated in a reductive fashion.  The inner and outer dimensions are framed at the corners, and are assumed to be analogues of each other.  We observe that the red probabilities are completely consumed by the framework, leaving no remainder for the sides of the hypercube.  This suggests that, when regarded as a dimension, the red probability is unmanifest or non-spatial, serving perhaps as the ground/field for the remaining dimensions.  Given that there are four dimensions in our model, time may be regarded as the zeroth dimension. 

The other three colors (blue green, and yellow), after contributing to the framework, leave remainders that are multiples of 8, reinforcing the notion that eight, expressed cubically, is template for dimensional space.  Continuing our analogy, these three colors provide for three manifest spatial dimensions.  These [sixteen] cubes, comprising fully one-quarter of the hypercube complete the framework, leaving 48 cubes (not shown) to fulfill their respective dimensional spaces.

Hypercubes: Strange & Loopy

Abstract: Hypercubes, or tesseracts, are proposed as instances of, or equivalent to "strange loops," self-referential, paradoxical constructs discussed by Douglas Hofstedter in Gödel, Escher, Bach (GEB).

Strange loops are often characterized as "level-crossing feedback loops" that inexplicably create cycles from hierarchies.  On traversing the hierarchy in an apparently monotonic manner, one is returned to the origin, often signaled by a change in context.

geometric hierarchy
We begin with a geometric hierarchy of points, lines, and areas; these give rise to volume. Points or vertices in space give rise to line segments or edges.  Connecting line segments end-to-end to form a closed figure yields area. Areas connected by their edges can enclose space (think "soccer ball").  

Some form of subspace, however, is presumed to permit the very existence of points.  The very location of a point begs the question, "where?"  What is the position of these points?  If points are to be assigned definite locations, axes are also presumed to exist.  In the graphic at upper right, the colored arrows denote the axes of extension/projection.

"Seed, Tree, & Fruit"
Once space, itself a "container," is enclosed, the strange loop manifests .  A tesseract may be produced by retracing our sample strange loop and substituting cubes for points.  Descriptive rhetoric lacks the force of mathematical argument, but compensates for this with  simplicity and accessibility.

The crux of GEB is Hofstedter's proposition that strange loops are the prime material of consciousness itself.

Yao-Numbers: Complementary to 65 and 360

This graph demonstrates how yao-numbers (complement 360) and xiantian (complement 65) and Wen pairing correlate. Wen pairs are consecutively-numbered (odd-even) pairs; e.g., (33,34)

Five groups, lettered A, C, F, T, and L, have subgroups 1 and 2, according to their yao-number.  Group T has no known subgroups.

Given any group A, C, F, L, or T: 
boxed pairs are binary complements (65)
boxed pairs constitute a circle (or two triangles) 360 degrees.  An hexagram is 180 degrees or pi, on average.  

Heaven is Round, Earth is Square

The numbers that yield THE CREATIVE total 216; those which yield THE RECEPTIVE total 144, making in all 360. They correspond with the days of the year.
--Ta Chuan pt. 1, ch. 9, v. 4

"[N]umbers" in this context refers to the yao-numbers, while ䷀ (CREATIVE) and ䷁ (RECEPTIVE) are the first two hexagrams in the traditional sequence.

Two useful representations of the Book of Changes include the square and the cube since both can be used to represent the 64 figures:

64 = 8 x 8
64 = 4 x 4 x 4

Observe the yao-numbers of Receptive and Creative compared with the representations of 64 (above):
144 = 12 x 12
216 = 6 x 6 x 6

In each case we presented two expressions of the form

a. X^2
b. (X/2)^3

144 is a perfect square, 216 is a perfect cube; 64 is both perfect square and cube.

Tesseract, Revisited

The four colored line segments span opposite corners of the cube and those of its bounding box.  The axes presume to represent the traditional three dimensions of space, plus a fourth -- time, or Change.  We envision our fourth dimension via notions of expansion or contraction.  In this case, the blue cube expands (in the time domain) to fully occupy the bounding frame.  In our simple example, the time axis is not discrete from three-dimension space, but supplements it.  As the blue cube expands along the colored lines, we measure the elapsed time by noting the positional change of the vertices along their respective axes as a function of elapsed time. Is it but coincidence that the two opposed square pyramids (their tips meeting at [0,0,0]) within the bounding box resemble an hourglass?

Deriving a means to integrate the ashtapada and the hypercube representations of the Book of Changes has remained a goal of the author.  A physical model was fashioned with this end in mind.  While the yellow and green axes represent Earth and Air respectively, the blue and red axes symbolize Fire and Water.  Earth and Fire stand for OLD YIN and YOUNG YIN; Air and Water stand for OLD YANG and YOUNG YANG.  These four symbols (xiang) are the stuff of existence.Our model begins with a depiction of the four symbols (xiang) presented in pairs and forming a 2x2x2 cube (the "bagua cube").  A second, exploded bagua cube extends in space beyond the corners of the first along the major diagonals.  
In total, this produces the framework of 4x4x4 tesseract.  If the ashtapada is a representation of planar/areal dimension ("Heaven is Round, Earth is Square") , here, we have "folded" space into a cubical form.  Imagine the outer eight cubes as an expansion of the inner 2x2x2  along the diagonals, and you will have seen the author's vision of space-time.

Tuesday, April 10, 2012

Exploring the Dodecahedral Model

Per the work introduced at (authored by Mark White, MD et al.), the current author has expanded and refined ideas relating to the Book of Changes.  Briefly, White invites his readers to reappraise the genetic code (and related constructs) as language based on shape, rather than through the conventional reductive viewpoint of nucleotides and amino acids.
Here, we adapt the dodecahedral model to the system of Change in an attempt to discern what insights or refinements to extant notions we might derive.  Hand-held models are dear to the author, particularly for the practical value they provide in exploring the systems thus represented.  This exploration proves no exception.  The model displayed below employs color to characterize the faces.  Each face is blazoned with McNeil notation (q.v.) to assist in diagramming the model.  The blazon allows for quick identification the attached vertices, face, and edges.

Faces stand for symbols (xiang).  Our symbol set is described by the letters [E W F A], standing for the four elements recognized by the ancients (earth, water, fire, and air). In the "alphabet" of Change, these would be represented by xiang di-grams [⚏ ⚎ ⚍ ⚌].  Dodecahedral modeling provides three faces per symbol, possibly to provide for each position in a sequence; that possibly being a defining characteristic of a symbol -- where it falls within a sequence.
We should like to investigate the possible significance of the pentagon as it relates to the symbols.  "Why five sides?"  Five does not often appear in the lexicon of Change.  The Chinese did have a theory of Five Elements (the four mentioned plus 'Metal').  We further observe that the maximal path length connecting any two vertices on the model is five.

the dodecahedral model of Change places hexagrams at the vertices, where faces (symbol triplets) converge.  Since the dodecahedron has twenty vertices, each vertex identifiable by a McNeil symbol triplet, with each of the twenty triplets having six possible symbol sequences, the model appears to allow for 120 possible hexagrams, providing for some redundancy.  In the discussion that follows, 

Isochromous (same color) triplets (e.g. AAA) have only one distinct sequence; four such triplet sequences exist:
EEE    WWW    FFF    AAA

Heterochromous (all different colors) triplets (e.g. AFW) have six valid sequences; twenty-four such triplet sequences exist:
EWF    EFW    EWA    EAW    EFA    EAF
WEF    WFE    WEA    WAE    WFA    WAF
FEW    FWE    FWA    FAW    FEA    FAE
AEW    AWE    AEF    AFE    AWF    AFW

Dichromous (two colors) triplets (e.g. AAE) have only three valid sequences, one for each position of the odd symbol; thirty-six such triplet sequences exist:
EEW    EWE    WEE    EEF    EFE    FEE    EEA    EAE    AEE
WWE    WEW    EWW    WWF    WFW    FWW    WWA    WAW    AWW
FFE    FEF    EFF    FFW    FWF    WFF    FFA    FAF    AFF
AAE    AEA    EAA    AAW    AWA    WAA    AAF    AFA    FAA

Author Mark White, MD proposes that each nucleotide in a triplet is inherently distinct, and that all of the the six possible isochromous sequences, AeAfAw (subscripted for clarity) for example, are meaningfully different from one another.  Likewise treating heterochromous and dichromous triplets, we discover that 56 (120 - 64 = 56) "isotopic" sequences exist that are generally indistinguishable (unless subscripted) from one of the basic 64 triplet sequences.  We should note, however, that this inability to distinguish isotopic sequences derives from our abstraction of the sequence from the physical model and its orientation in space.

Edges define faces, each edge held in common by two adjoining faces.  Edges also connect vertices.  Thirty distinct edges exist on the dodecahedron. Since vertices stand for hexagrams, edges act as paths connecting vertices.  This role suggests that an edge defines a change from one hexagram to another, further implying that the paths are directed, hence numbering sixty in total.  We have elsewhere noted that the hexagrams may be understood as forming thirty-two pairs, under at least two schema.  Efforts to integrate edge-as-pairing would have to account for the missing paths/edges.

It is arguable that the pentagon, with its vertex angle of 108° and central angle of 72°, defines the dodecahedron, as its adjoining faces meet at the same 108°angle.  A pentagram results when the vertices of a pentagon are connected.  The diagram at left shows that 36°angle is fundamental to the pentagon (and pentagram), thus to the dodecahedron.  In fact, the pentagonal net is fractal of a single 72-36-72 isosceles triangle.  
If a side of the pentagon (purple) is taken as unity (1), a diagonal (blue) measures phi.  The red segment equals (phi -1), and the yellow segment equals (2 - phi).

The pentagon's relation to phi is worthy of mention in this section since the author had long-sought a relation between Change and phi, now obtained, albeit indirectly. 

Dodecagonal Depiction of Change

The author revisited yao-numbers, recalculating them based strictly on the divination ritual values of Ch'ien (9) and K'un (6).

The outcome is a distribution that naturally accords with the combinatorics of a 6-bit system.  
There are seven groups with the following distribution: 
[1 6 15 20 15 6 1].
This is a departure from the prior distribution of yao-numbers, which incorporated 9 groups: 
[1 3 6 12 20 12 6 3 1]

the author had the recent fortune to discover a website that treats systems like Change (and the genetic code) in some depth. That author makes the unqualified claim that the genetic code is a dodecahedral system.  The Book of Changes, again, is based on the same underlying grammatical components (four basic elements, arranged in triplets).  One tremendously useful product of the website is the depiction of the genetic code on a dodecahedron (12-faceted Platonic solid).  Several tools and perspectives are additionally presented to facilitate pattern-recognition.

The current author observes that the array [1 6 15 20 15 6 1] can also be depicted as [2 12 30 20] when that array is "folded" on the central value (20).  What we find is that at least three of the values of the resultant array find direct correspondence to attributes of a dodecahedron, perhaps confirming the author's claim.

Dodecahedra have 20 vertices, 30 edges, and 12 pentagonal faces.  The array value of '2' is not nearly as unambiguous, but could correspond to symmetry axes or polarity.  An arrangement suggested by the author of the mentioned website involves four (tetrahedral) poles corresponding to the four genetic nucleotides.  

In the Book of Changes, four corresponding elements exist called symbols (or xiang). When we computed the Wen distribution based on the divination ritual values of xiang (6,7,8,9), no correspondence (to date) is found with any real-world objects.  
Conversely, when we computed (as described in the open paragraph) the Wen distribution based on 6 and 9 (the ritual values of Ch'ien and K'un), the characteristics of a dodecahedron emerge as just described.  This is the rationale for asserting that the value '2' in the array [2 12 30 20] may indicate the number of poles to incorporate in the dodecahedral depiction of the Book of Changes.