Thursday, September 20, 2012

Sphere: Squared

In the diagram at right, each figure's cell is identified by two numbers: traditional sequence at upper-right, and scalar quantity at lower-left. 
This 64-cell grid is arranged so that each cell is placed at 180 degrees of rotation from its logical complement.

Other ways of expressing this spatial relationship are that the complementary figures are maximally-separated, that they are antipodally-
positioned, or that they are diametrically-arranged . 

Complementary hexagram pairs comprise a symbolic whole when their scalar values are summed, represented by the number 65.  Seen together, one figure will have YANG lines where the other figure has YIN lines.  

In the diagram following, complementary relationships are also depicted by color pairs: RED with VIOLET, ORANGE with INDIGO, YELLOW with BLUE, or GREEN with GREEN.  Complementary pairs on the 8 x 8 grid are precisely analogous to antipodally-positioned hexagram pairs on any of the thirty-two spherical axes.


RED + VIOLET = 2 polar cells (absolutely polarized)ORANGE + INDIGO = 12 cells (strongly polarized)
YELLOW + BLUE = 30 cells (weakly polarized)
GREEN = 20 equatorial cells (non-polarized)


RED and VIOLET figures establish a pole that vertically bisects the grid
YELLOW and BLUE figures are located close to GREEN and far from RED and VIOLET, thus they are weakly polarized: young, static, stable
ORANGE and INDIGO figures are located close to the RED and VIOLET polar figures, thus they are strongly polarized: old, dynamic, changing
GREEN figures horizontally bisect the grid, are located between BLUE and YELLOW, and are generally farthest from the poles; thus they are neutrally- or non-polarized

Classified per line-type count,

YELLOW and ORANGE have predominantly open lines and have YIN character
GREEN have YANG and YIN in equal measure
BLUE and INDIGO have predominantly closed lines and are YANG in character


Wen's traditional pairing schema is preserved within the xian tian arrangement.  Wen's pairs are defined as 
  1. a figure pairing with its figurate inversion where such inversion produces a figure different from the original; otherwise by
  2. a figure mated to its logical negation, as in the case of (#01, #02).
Icosahedral characteristics                                 
30 EDGES, 20 triangular FACES, 12 VERTICES

Dodecahedral characteristics
30 EDGES, 12 pentagonal FACES, 20 VERTICES

xiang::nucleotide bases: adenine + uracil  & cytosine + guanine
hexagram equivalent in RNA language: three nucleic acids :: codon :: hexagram

xiang::nucleotide pairing schema:

adenine: (INDIGO, old yang, 9) + uracil (ORANGE, old yin, 6)
cytosine (YELLOW, young yin, 8) + guanine (BLUE, young yang, 7)

ROYGBIV Color "octave"

Figures #01 and #02, located in the diagram at the high and low corners, are poles of the reference axis.  The remaining 62 paths, or 31 axes are "free"

Human peripheral nervous system comprises 31 nerve pairs

Tun calendar :: 18 * 20 = 360

Tzolkin calendar :: 13 * 20 = 260

20 * (13 + 18) = 20 * (31) = 620

Wednesday, September 19, 2012

Tun, Tzolkin, and XMS

The Maya are said to have used many calendars, several of which remain unknown to outsiders.  Three of their calendars are fairly-well known; these calendars are based on the standard 24-hour day, called kin.  
  • Tun (literally, stone) is regarded as a prophetic calendar by the Maya people.  It comprises 360 kin partitioned into 18 uinal (or winal, month), each with 20 kin.
  • Tzolkin is the sacred calendar of the Maya; it comprises 20 periods of thirteen kin, or 260 days.  Together these two form the basis of the Maya timekeeping system.
  • Haab (365 kin) reflected the Mayan solar year, and was used as a civil calendar, strictly used for accounting, taxation, and perhaps agriculture.
All Mayan calendar dates, including personal birthdays, dedications and most sacred ceremonial events were measured in 360-day periods called, tuns.  All dates "carved in stone" throughout the Mayan and Toltec world are tun dates...
This Tun, 360 day calendar is the only Maya calendar directly connected to the Tzolkin and they run together like two gears, each day being a tooth on the respective gears.
J. Eric Thompson's paper:
J. T. Goodman wrote of the 360-day period, now called the Tun: “This period is the real basis of the Maya chronological system.”
Furthermore, every known unit in the Maya calendar has in its composition the symbol for the 360-day year....
C. J. Calleman's statements:
The real point to get for the study of the Mayan calendar is however that in its prophetic uses it is based on the 260-day tzolkin or the 360-day tun, cycles that do not have an origin in the physical universe.

18 "spells" of xiantian magic square
Xiantian Magic Square
The XMS arrangement comprises eight columns, eight rows, and two diagonals for a total of 18 groups linearly-arranged.  We can also refer to the groups as octets or "spells."  Each of the eighteen "spells" comprises a unique collection of eight numbers all arranged in a line on the grid.  Additionally, each octet sums to 260, the same number of days as in the tzolkin.
  • Each octet may stand for a winal (20-day tun month) since both octets and uinals relate to  eighteen
  • Since an octet represents 20 days (winal) and sums to 260 (tzolkin year), it  associates each day of an octet with the value 13
Considered this way, XMS can be considered as relating to both tun and tzolkin calendars, though our use of metaphor may somewhat exaggerate these relations.  It should be noted that the number 20 is not yet directly discernible within the XMS arrangement, though it is a demonstrable feature of the 64 hexagrams (q.v. yao-groups).  The equatorial plane of the spherical representation of Change comprises twenty complementary pairs. 

Complementary pairs on the XMS grid sum to 65; any four such pairs sum to 260, the same number of days in the tzolkin calendar.  Four pair is an octet, an octet is 20 days; therefore one pair is five days (value 65) and one day has value 13.  QED

Coincidentally, 13 / 8 * 360 (tun) = 585, a fair approximation to the Venus cycle of 584.  Admittedly, 8 / 5 * 365 (haab) = 584; a precise fit.  Both fractions (13/8 and 8/5) are common approximations to the golden ratio, phi (~1.618).

Scientists have calculated that approximately 1 billions of years ago, the Moon was ~25% closer than it is today, the Earth having an ~18-hour day and a 18 - 20-day month as marked by the Moon's circuit.

Linear octants are also called "spells" in acknowledgement of the 18 spells gained by Odin through his self-sacrifice on the World Tree Yggdrasil: two for each day he was hanged.

Tuesday, September 18, 2012

Axes and Reflections on the 8 x 8

16 axial figures
Ashtapada, the 8 x 8 field, is here populated with xian tian (complementary opposition) hexagram arrangement.  We observe that two pairs of orthogonal axes (opposed at 90 degrees) cross the field.  These axes include HORIZONTAL with VERTICAL, and DIAGONAL with SLANT.  
The 48 remaining cells of the field are effectively partitioned into four groups of twelve contiguous cells by the SLANT and DIAGONAL axes.  The VERTICAL and HORIZONTAL axes further bisect these four groups of 12 cells.
  • HORIZONTAL axis (unmarked) divides the field into an upper half and a lower half
  • VERTICAL axis (unmarked) divides the field into left and right halves
  • DIAGONAL axis is defined by the eight cells that span the corners lower left to upper right, or conversely.  
  • SLANT axis is defined by the eight cells that span the corners lower right to upper left, or conversely.
It should be noted that the figures located on the DIAGONAL and SLANT axes exhibit the associated relationships when they are folded or reflected across each other.  Example: figures (#52,#53) located on the SLANT axis (which crosses the DIAGONAL), are related through transposing and complementing both trigrams.  

48 non-axial figures

Imagine our grid printed on a square piece of paper which is then folded across length and width, and across both diagonals.  An axis is equivalent to any singular fold-line; reflection across an axis results in two cells that "mirror" each other on either side of the fold-line.  If the paper were actually folded as described, the mirrored cells would overlap perfectly.  Now with a working definition of axial reflection over our grid, let's look at relationships among the reflected figures.

  1. Reflection across the HORIZONTAL axis complements a figure's lower trigram, leaving the upper trigram unchanged; e.g.: (#46,#24)**
  2. Reflection across the VERTICAL axis complements a figure's  upper trigram, leaving the lower trigram unchanged; e.g.: (#59,#40)
  3. Reflection across the SLANT axis transposes OR complements both trigrams of a figure; e.g.: (#28,#61)
  4. Reflection across the DIAGONAL axis transposes AND complements both trigrams of a figure; e.g.: (#4,#38)  ** parenthesized numbers indicate the traditional hexagram ordering, located at upper-right corner of cells in white text.
Likewise, paired figures (#11,#12) located on the DIAGONAL axis (which crosses the SLANT), are related through transposing trigrams; alternatively, through complementing trigrams.
Reflection across a single axis is equivalent to rotating the grid in place by 90 degrees; therefore, reflection across two orthogonal axes is equivalent to rotating the grid in place by 180 degrees.  

For any figure selected, crossing two orthogonal axes results in complementing the entire figure.  This fact is intrinsic to the xian tian arrangement whereby pairs of complementary hexagrams are separated on the field by 180 degrees of rotation, or two orthogonal axes.   Since DIAGONAL and SLANT axes cross both HORIZONTAL and DIAGONAL axes, the reflections of the figures on those axes are complementary.

No matter the combination of axial reflections, any selected figure and its several reflections remain confined within the same concentric band of figures on the field.  These are termed xiang probability bands because of their relation to the divinatory probabilities for generating each of the four kinds of lines (xiang) that may appear when consulting the oracle.

XMS (XIANTIAN MAGIC SQUAREelsewhere described in detail) is a magic square that is also a xian tian or complementary opposition arrangement; therefore, the relationships described above likely also hold for XMS.

This diagram embodies many of the same qualities as the 8 x 8 xiantian diagram

xiang probability bands

NB: The two outer bands of the xiang probability square representing static YIN and static YANG contain 28 + 20 = 48 cells, the same number as the non-axial cells Therefore, these cells may stand for static YIN with static YANG, or a stable condition overall.  The same logic represents the axial cells as 12 + 4 = 16, dynamic YANG with dynamic YIN, a changing condition.

Thursday, July 19, 2012

I Ching and Bloch Sphere

Bloch sphere
The Bloch or Poincare sphere seems particularly instrumental to understanding how I Ching may be related to quantum mechanics; perhaps even an ancient depiction of it.  I Ching is founded on two principles, Ch'ien and K'un, or the Creative and the Receptive, commonly known as YANG and YIN, and discussed at length in various appendices to I Ching (namely, appendices III and IV).  Appendix III, the Great Treatise, is quite clear on the distinction between hexagrams #1 and #2 and the remainder of I Ching.  

Stated clearly in several verses, Ch'ien and K'un are the "pole," "axis, or "gates" of Change, without which Change could not be seen and would cease to exist. (*provide  references*).  These passages provide support to the intuition that hexagrams #1 and #2 are intended to act as basis vectors and define the spherical system that represents the I Ching.  

Spherical I Ching
We now set aside Ch'ien and K'un for consideration of the remaining thirty-one pairs of hexagrams. 

Some effort was made earlier to find meaningful correspondence between the human body's 31 pairs of spinal nerves and the 31 (+1) hexagram pairs, but these efforts have not borne fruit.

A presentation from the Vienna Center for Parallel Computing shows us that our complementary or xian tian hexagram pairs having scalar values which sum to 65 are regarded as orthogonal and identical states (except for phase/sign) on the Bloch sphere.  The model of I Ching presented in these blogs accounts not only orthogonal hexagram pairs, but also the pairs given by Wen, Duke of Chou.  
I Ching tradition describes Duke's imprisonment at the hands of the evil tyrant of Shang, and of the pairing scheme he devised during his period of captivity.  

In general, Wen's pairs are distinct figurate inversions (read: 180°rotation) except in the eight cases where inversion does not produce an hexagram distinct from the original:  
[(1,2) (51,57) (29,30) (52,58)]. In those cases, the complementary pair is taken as the Wen pair.

The graphic of the Bloch sphere (at left) highlights its important features:
basis vectors |0 and |1, arbitrary quantum state psi (Ψ), two coordinate systemspolar: denoted by angles theta (Θ) and phi (Φ); and rectangular: denoted by orthogonal axes x,y,z.  

Selecting one of the remaining 56 'non-distinct' hexagrams to approach this question spatially, we see that complementary pairs are opposite and orthogonal points on the Bloch sphere, while Wen pairs lay on the same (or opposite latitude) to the selected hexagram and  may also be complementary to the selected hexagram.  We see in the graphic above the north and south poles (denoted by |0 and |1⟩) along with psi (Ψforming an imaginary triangle.  
Arbitrarily selecting hexagram #41, we get its Wen pair #42, and #31 as its orthogonal point.  These three hexagrams form a corresponding imaginary triangle: #41 and #42 analogous to the poles, and #31 as the surface point.  The hexagram, its Wen inverse, and its opposite form a triplet: two hexagrams share a latitude while two share a diameter.  Selecting any hexagram from the eight pairs listed above produces just a diametric pair instead of a triplet since the Wen pair is also the opposite hexagram.

If we accept that I Ching is analogous with Bloch's sphere as a working model or depiction of quanta, the question begs, what is to be made of Wen's pairs?  We understand that Wen pairs are figurate inverses, but what does this mean in terms of the Bloch sphere and how might this inform us to quanta?

Thursday, May 10, 2012

The Spherical Model of Change

As promised at the end of the previous entry,  here we present a non-arbitrary arrangement of the hexagrams on the sphere.  Our spherical arrangement is based on a construct called "yao-numbers", the existence of which is attested in an appendix of the Book of Changes known as Ta Chuan (The Great Treatise).

Spherical Model
(equatorial view)
Once the hexagrams were sorted by the size of the yao-group in which they fall [1, 6, 15, or 20], it seemed natural to sort them again by their scalar (xiantian) index. Recall that the xiantian or scalar index of an hexagram is the decimal value of an hexagram figure rendered into binary.  Scalar value is a reasonable indicator of an hexagram's magnitude; thus it appears a reasonable choice for placing the hexagrams of a given yao-group (which all lay on a common latitude) at specific meridians.

Spherical model
(depicted with vectors)
Expressed another way, the problem was to determine how to arrange the hexagrams on each latitude of the sphere in such a way that the ordering remained consistent with the complementarity of the 32 antipodal pairs.  Scalar value was used as a proxy for angular displacement (PHIon a latitude, with larger values corresponding to larger angular measures.

Spherical Model
(polar view)
In the table following, each hexagram is assigned a coordinate pair (theta,phi) on the sphere: THETA is the measure of latitude (declination from the positive vertical) while PHI measures longitude (angular displacement from a given meridian) on a latitude.  The hexagrams are grouped, generally speaking, by yao-number and ordered within a yao-group by scalar value (XT).

As detailed in other entries, the yao-numbers of the paired hexagrams sum to 360.  Additionally, these pairs have XT numbers summing to 65, thus they are complementary pairs.  The spherical model presented here preserves those relationships.
One unexpected outcome of this effort was the discovery of a natural partition of yao-group[20] which, for lack of a self-consistent way to divide it, had been treated only as a unit.  With the aid of the XT index, however, this group of 20 hexagrams falls naturally into halves: one half with XT less than 32; the XT of the other ten hexagrams exceeding 32.  Only yao-group of 20 features this relationship.  Indeed, only this group requires additional means to distinguish pair-mates.  

Whereas they were previously presented as seven groups on discrete latitudes, the 64 hexagrams can now be portrayed as four groups of varying sizes, each group with equal-sized halves as shown in the table.  These eight sections comprise an octo-partition of spherical space, bringing us full-circle (pardon the pun).  

The model is expected to serve as an anti-stereographic projection of the xiantian magic square onto the unit sphere and may prove useful for visualizing projections of XMS subsets onto 3-space.  For example, how do the XMS main diagonals -- or any of the "18 spells" for that matter -- appear when displayed on the sphere?  The projection may also help with transforming the XMS field into a magic 4x4x4 cube (with magic constant 130).

[NB: Consequently, the yao-groups should perhaps be redefined for the sake of clarity in designation:
[2 12 30 20] with modifiers '+' or '-' to indicate latitudes above or below the equator, respectively.  Again, this coincides with the specification of a dodecahedron]

Tuesday, May 8, 2012

Progressing Toward a Spherical Model of Change

Yao-numbers depicted as radial vectors
Having determined a set of coordinates for each hexagram allows us to  generate 32 pairs of complementary vectors with which vector-analysis may be performed.  
The spherical model also enables us to derive insight from quantum mechanical systems by using the Bloch/Poincare model of the qubit as an entry-point.
The basis vectors in our model are hexagrams #1 and #2, found at the "poles" of our model; the remaining 31 pairs of  hexagram figures are said to derive from these.  Prior to performing an instance of divination, the answer to the posed question is like a superposition of the basis vectors, entailing 64 x 64 = 4,096 possible results.  Once the oracle is consulted, the superposition collapses to a single result.  

Future entries on this topic intend to determine the meridian lines in order to produce a model that fits the 64 hexagrams onto the unit sphere in a non-arbitrary fashion.  That is, the particular assignment of hexagrams to points on the surface of the sphere will be based a scheme that preserves the integrity of known relationships.

Monday, May 7, 2012

Subspatial Scaffolding, Aetheric Architecture

This graphic attempt to convey the notions of subspace, or the architecture of volumetric space.  
It is easy to take space for granted, like we assume a fish takes water for granted.  Indeed, were there no water, there'd be no fish -- likewise for space and ourselves.  
It's not so easy to wrap one's mind around non-volumetric reality, though some are familiar with 'Flatland,' a fictional 2-D world.  Space normally provides an habitat for objects to occupy, but under certain conditions, intense gravitation does not allow space to exist as we understand it.  Under those conditions, space is presumed to collapse.  The Big Bang cosmological theory proposes that the universe as we know it evolved from a singularity; otherwise there was no space nor time.  From this we may infer that some 'thing' or force exists that gives space the rigidity to resist gravitationally-induced collapse.  Some physicists have proposed that hypothesized (but undetected) "dark energy" fill this function.  To this point, physicist David Bohm once remarked in The Holographic Universe that "every cubic centimeter of empty space contains more energy than the total energy of all the matter in the known universe."   We propose that this role is fulfilled by an architecture, hypothesized in the above graphic.

In the diagram, mutually complementary elements intersect on orthogonal axes.  This arrangement is presumed to produce a framework that 'erects' space and maintains its volume, similar to the way that a balloon maintains its shape from the force exerted by the gas within.
Complementarity is depicted via the axes of 'opposing' color pairs: red and blue, yellow and green.  These opposing pairs span opposite corners of the bounding box. 
The nexus of the four axes at the center of the figure suggests a combination of the four axes that intersect there.  This is perhaps where the proposed structive force emerges.
We note that the central nexus is adjoined by the apexes of six square pyramids, the bases of which comprise the  faces of the bounding box and its enclosed cube.

Saturday, May 5, 2012

11,520: "The Number of All Things"

This entry treats the connection between the 8x8 grid and 11,520, the "number of all things," attested in Ta Chuan (the "Great Treatise").  The 64 hexagram are commonly depicted on an 8 x 8 grid, known in Hindu culture as ashtapada. Its metric (generic term for measure) is 2080, and is thus related to 11,520:

1) Begin by enumerating its cells beginning with 1, through 64, noting that the cells sum to 2080.
2) Arrange the numbered cells such that the numbers within each row, column, and major diagonal sum to 260. 
Under this arrangement, the 18 columns, rows, and diagonals sum to 4680 = 18 * 260 = 13 * 360 = 18 * 13 * 20.  This number seems to link the Mayan 360-day tun calendar to the Mayan 260-day tzolkin calendar.  
3) Convert the numbers into hexagrams (6-line binary figures) , the least-valued hexagram valued having the value one (1).
4) Transform each figure into a yao-number by substituting its lines with the corresponding divination ritual numbers: '6' for YIN lines and '9' for YANG lines, summing the substitutions, then scaling each sum by 4.  This step is derived from instructions given in Ta Chuan, and has the effect of:

  • "flattening" the hexagram (like a logarithm), which may be regarded as a "stack" of binary exponents 
  • shifting the range of values from a continuum of [1..64] to [144..216] in discrete intervals of 12 
The 18 rows, columns, and diagonals noted above now sum to 1440, and produce a grand sum of 18 * 1440 = 25,920, equal to the duration of a precession cycle.
5) Finally, the hexagrams and their associated yao-numbers from steps #3 and #4 are sorted to produce seven groups with the following membership and distribution:
The 64 resulting yao-numbers are summed to produce 11,520, the "number of all things."

Thursday, May 3, 2012

XMS and Genetic Code

German medical scientist Dr. Martin Schönberger is attributed with the initial observation that the 64 hexagrams of the genetic code are analogous to the 64 codons of the DNA genetic code.  His book, I Ching and the Genetic Code: The Hidden Key to Life, is regarded as the inaugural work on this topic.
 Other authors have convincingly discussed the similarity between the I Ching and the genetic code, including Steve Krakowskiwhose work borrowed from Schönberger's, as well as Mark White M.D. Both these authors' work is cited by the current author.
Krakowski's work seeks to integrate Hebrew language, 22 Tarot trumps, genetic code, and I Ching.  Dr. White's introduction of "dodecahedral language" brings the discussion squarely into the realm of tangible objects by mapping the 64 hexagrams onto the twelve-faceted, twenty-verticed Platonic solid lovingly-called a "12-tope." 

xiantian magic square
The figure at right poses a comparison between the XMS (xiantian magic square, at right), and the tzolkin calendar (lower right) of the Maya. The author holds that there is a deep connection between these two objects, and uses math and art in his work to promote said thesis.

The XMS is a particular arrangement of the 64 hexagrams on the 8x8 grid whereby any hexagram is mated to its complementary opposite located at 180 degrees of rotation.  Each hexagram on the grid is both indexed, and assigned a scalar value; the scalar values of complementary pairs sum to 65.  This relationship is also proven by the fact that when the hexagrams are rendered into their numerical equivalents, all rows, columns, and main diagonals sum to 4 * 65 = 260.  The latter number is emblematic of the tzolkin calendar of the Maya people.

Arguelles and "His" Tzolkin

The 260-day tzolkin (at right) is displayed as a 13x20 grid with a highlighted subset of 52 days identified by Tony Shearer as galactic "portal days" or by Jose Arguelles as the "Loom of the Maya."  [The author is seeking/awaiting an detailed account of their derivation].  Arguelles' greatest achievement, arguably, was the popularization of Mayan calendrics.  This was also his greatest folly as well, since his efforts -- even post-mortem -- have taken on the trappings of new-age religion.  Nonetheless, Arguelles was possessed of a peculiar spiritual intuition which makes the current author reluctant to throw out the bathwater for fear that the baby may still be in it.  References to the Loom of the Maya viz. tzolk'in often involves references to DNA or the genetic code.  The pattern formed by the portal days is held by some to resemble the helical shape of DNA.  The following passage is from a site that promotes Arguelles' work:
This form of the Tzolkin as the "Harmonic Module," (shown above), inclusive of the 52 shaded squares which form what is called the "Loom of the Maya," is based on Dr. Jose Arguelles' presentation in The Mayan Factor, and is distinct from the form of the Tzolkin as taught and followed by the Quiche Maya of Guatemala. For instance, the 52-unit loom is a bi-lateral symmetry pattern which reflects the basic pattern of our DNA double helix, and was passed down from a secret lineage of Yucatec Mayan shamans, received and revealed by the works of Dr. Jose Arguelles. Integrating the galactic code of light into the genetic code of life, this "portal" formation is a resonant structure linked to the activation of our full DNA potential. Find out when these galactic activation portal days occur, and receive indepth descriptions of the 13 Tones of Creation and the 20 Solar Seals by utilizing the 13-Moon Natural Time Calendar.
So, while the Arguelles camp appears to recognize a connection between time and inner space (body its subtler structures), no evidences are offered to ground its bold assertions.


The colored partitions of the XMS grid originate from a simple mathematical transform of the hexagrams that assigns a yao-number to each.  [The origin of the concept of yao-numbers is discussed in greater detail elsewhere; suffices it to say, they were not invented by the author.]  
Yao-groups and Yao-numbers
Yao-numbers range from 144 (hexagram #2, all YIN) to 216 (hexagram #1, all YANG), with intervals of 12 between the seven groupings.  The midpoint of the yao-number scale is found at 180.    
Yao-numbers measure the YIN- or YANG-ness of an hexagram on a continuum.  Thus, the 22 YIN-dominant hexagrams (having yao-numbers less than 180) are red-colored, while the 22 YANG-dominant hexagrams (having yao-numbers greater than 180) are blue-colored.  The remaining 20 hexagrams that are balanced in terms of YIN and YANG (yao-number 180) are white-colored; they also divide the grid in half.

While the similarity of the figures formed by the 20 white-colored cells and that of the 52 portal days forming the Loom of the Maya is superficial and debatable, what is less-debatable is that the 20 white-colored cells partition the grid into 22 pairs, curiously identical to the number of human autosomes, which are indistinguishable by sex.  The uncounted (23rd) pair of human chromosomes is responsible for sex-differentiation, and is necessarily different between males and females.  
The white-colored group of 20 hexagrams is fairly representative of the 20 essential amino acids of the human genome.

Thus, in a single diagram we have represented salient characteristics of the genetic code:

22 yang hexagrams <==> 22 chromosomes contributed by the male
22 yin hexagrams <==> 22 chromosomes contributed by the female
20 neutral hexagrams <==> 20 essential amino acids
An admittedly-vague helical or chromosomal shape

The analogy seems fairly solid so far; Argulles, we're not done with you just yet.

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.