Jörmungandr-Ouroborus, the Tesseract

Tesseract

According to legend, Odin threw the serpent Jörmungandr into the great ocean (primal void or akasha) surrounding the plane called Midgard. The serpent grew so large as to surround the Earth.

This tends to confirm the truism that Earth is encompassed by 4-D space-time -- exactly that which is represented by the tesseract.

Ouroborus

Jörmungandr is known in the Greek mythic  tradition as Ouroborus.  Both are commonly represented by a serpent biting (swallowing) its own tail.  Indeed, the name itself means "tail-eating," or self-devouring.  This is an apt description of what we see in the animated graphic at top left.

hypercube pendant

The deeper significance, hinted in the opening paragraph, is that our three spatial dimensions, plus the temporal fourth, constitute an eternal self-devouring subject which acts as our existential domain.
The photograph at left presents a three-dimensional object commonly presented in two-dimensions as a cross within a circle, known as the Sun cross, or Odin's cross.  It is used in astrology to represent the planet Earth.

"Since each vertex of a tesseract is adjacent to four edges, the vertex figure of the tesseract is a regular tetrahedron."  (source)

18 Magic Spells

Having derived candidates for the 18 spells, it remains for us to choose or develop a framework for interpreting them.  The following are what came immediately to mind:

1. divination
2. DNA (genome)

Xian Tian magic square

The 18 collections or "spells" were derived from the xian tian magic square arrangement (discussed elsewhere) and are listed below.  The hyper-linked numbers correspond to the traditional hexagram ordering attributed to King Wen, while the numbers in the graphic at right are numerical values of the same.  The first few "spells" have been loosely interpreted using the divinatory language of the I Ching:



Spells from ROWS
01)     2,44,13,19,15,6,25,11
receptive/field, coupling/meeting, humbling, approaching, arguing, fidelity, peace 
02)     9,51,40,53,61,55,32,20
small harvest, thunder, deliverance, advancement, re-centering, fullness, endurance, viewing
03)     14,3,29,56,38,63,48,35
04)     45,18,22,58,31,4,27,43
05)     23,28,49,41,52,47,17,26
06)     5,21,64,39,60,30,50,8
07)     34,42,59,62,54,37,57,16
08)     12,46,36,10,33,7,24,1

Spells from COLUMNS
09)     2,9,14,45,23,5,34,12
10)     44,51,3,18,28,21,42,46
11)     13,40,29,22,49,64,59,36
12)     19,53,56,58,41,39,62,10
13)     15,61,38,31,52,60,54,33
14)     6,55,63,4,47,30,37,7
15)     25,32,48,27,17,50,57,24
16)     11,20,35,43,26,8,16,1

Spells from DIAGONALS
17)     2,51,29,58,52,30,57,1
18)     11,32,63,31,41,64,42,12

Magic Spells for Odin/Mercury

Ashtāpada: the uncheckered 8x8 board

I Ching ELEMENTS
64 six-line figures or "hexagrams" (e.g. ䷄), each with associated judgments, imagery, and texts.  May be written using line drawings: ⚊ ⚋ or with strings of binary digits; for example: 010111b is a binary expression of the hexagram figure above.


I Ching ARRANGEMENT



Ashtāpada, the uncheckered 8×8 board, of Hindu derivation.  The literal meaning of "ashtapadi" is "eight steps." This word is the origin of the word ashtāpada, an Indian board game, the forerunner of chess. The word now primarily refers to the board itself.

Vastu-Purusha mandala

• Related to the game Chaturanga: (Sanskrit: "four-limbed"), the ancestor of modern chess
• Related to Manduka/Chandita mandala 
• Populated by the 64 hexagrams, it represents 2080 
• Characterized as a "field of action [activity]," indicating the body, or even the Universe suggesting perhaps the identity relation between these 

• related to "kshetra," meaning 'field' or 'body' 
• related to Vastu-Purusha mandala
• q.v. Bhavagad Gita: ch. 13
• q.v. Ta Chuan p1:c11:v5-6 
• q.v. "Symbolism of Chess," Titus Burkhardt

I Ching SYNTAX
Xian Tian, a symmetrical arrangement of 32 pairs of binary complements over the ashtāpada. 

• Any given hexagram on the ashtāpada has a complement at 180 degrees of rotation 
• The binary values of each complementary hexagram pair sums to 65 
• The expression 64!! (double factorial), equal to [64*62*60...8*4*2] computes the xian tian arrangements that can exist

Xian Tian magic square

Our xian tian magic square (XMS) is an arrangement wherein all linear collections of eight numerical elements (rows, columns, and main diagonals) sum to 260. 

• XMS produces two proper partitions of the populated ashtāpada.
• The eight rows constitute a partition, the eight columns constitute another partition, and the diagonals comprise two additional octonary collections for a total of 18 linear octets. 
• The author is presently unclear how to calculate the number of xian tian arrangements that obey the magic square restriction, though it is obviously a subset of total number of xian tian arrangements . 

Explication:

Circular arrangement of the Armanen runes

The 18 linear collections of eight hexagrams within Mercury's magic square are suggestive the 18 spells (also charms, runes) gained by Odin as described in the Runatal and Ljooatal sections of the Hávamál.

The author proposes that the 18 linear octets of hexagrams, arranged on the XMS, are candidates for the eighteen spells of Odin. In the binary language of computing hardware, eight bits (binary digits) commonly constitute a 'word.' 
In this discussion, the sixty-four 6-bit hexagrams serve as the words or runes. Therefore each of the eighteen collections comprises a unique combination of eight 6-bit "words."  Each of our "spells" consists of eight words (alternatively, eight hexagrams or runes), having a numerical or numerological value of 260.


Words certainly imply 'spells' and 'spelling,' as to spell a word is to precisely enumerate its letters in correct sequence. The word 'gospel,' for example, literally means "good spell," or "good word."  


Words and runes are symbolic representations of ideas.  Runes are fairly distinct from letters, it seems, since individual letters seem devoid of semantic value viz. the word in which they appear; while runes preserve a measure of their individual meanings in isolation. Additionally, runes seem more akin to words in a phrase than to letters in a word.

Yao Numbers and the Tree of Life

The following animation is a graphical depiction the nine species of "Yao Numbers," a concept introduced in Ta Chuan, the Great Treatise.  It displays the nine families in forward and reverse sequence.  It is the so-called Fu Xi (also, xiantian) diagram rotated 45 degrees CW.  We suggest that these nine families are analogous to the "Nine Worlds" surrounding Yggdrasil, the Teutonic Tree of Life.
  

The animation appears to depict a signal initiated at one pole (congruent with the realms called Hel and Asgard in the Teutonic tradition), propagated through the medium, then confirmed by the receiving pole.  Conceptually, it bears  resemblance to a common depiction of Yggdrasil, the Teutonic Tree of Life with its "Nine Homeworlds."  


From the poles  the signal passes through either group of three hexagrams with pairs formed by complementarity or "separated pairs".  These are congruent to the planes called Bright Alfheim & Dark Alfheim in the Teutonic scheme. The  purported power of these realms is to mediate between the worlds of Asgard and Midgard:

There is also a transitional world between Asgardhr and Midgardhr, where the energies of the former are transmitted to the latter world. This realm is called Alfheimr (world of the Elves), and is characterised by the higher aspects of light and air (equivalent perhaps to the occult conception of the Etheric planes). In this world, the highest energies of the human world mix with the lowest energies of the Gods. Beneath Midgardhr (the physical world) is a corresponding region intermediate to Midgardr and Hel, called Svart-Alfheim (world of the black elves). In this region are found the mysteries of earthly manifestation, represented by the dwarves.  

Teutonic Cosmology by M. Alan Kazlev

In most detailed depictions of the Tree of Life, a formation comprising a central node bounded by a group of four nodes (q.v. quincunx) is present.  In this presentation, the largest and central of the five families comprises twenty hexagrams, each with three YIN and three YANG lines, and each having yao-number <180>.  This node is congruent to Teutonic Midgard, the home of Men.


The four groups surrounding the central group have {sizes} and <yao numbers> as shown: ({6} <172>, {12} <176>, {12} <184>, {6} <188>).
The congruence of these groups to the Teutonic tradition is less clear than the others.  The candidates are: Jontunheim, Vanaheim, Niflheim, & Muspelheim.

The following crude table displays the 64 hexagrams and the nine yao-number "families" into which they fall, the quantity of hexagrams per family, as well as yao-numbers and line-analysis (#YIN lines and #YANG lines) for each hexagram.  Wen pairs are shown within parentheses where possible.


0 YANG,6 YIN <144> { 1} [2]*
2 YANG,4 YIN <168> { 3} [27 29 62]*
5 YANG,1 YIN <172> { 6} [(9 10) (13 14) (43 44)]
4 YANG,2 YIN <176> {12} [(5 6) (25 26) (33 34) (37 38) (49 50) (57 58)]
3 YANG,3 YIN <180> {20} [(11 12) (17 18) (21 22) (31 32) (41 42) (47 48) (53 54) (55 56) (59 60) (63 64)]
2 YANG,4 YIN <184> {12} [(3 4) (19 20) (35 36) (39 40) (45 46) (51 52)]
1 YANG,5 YIN <188> { 6} [(7 8) (15 16) (23 24)]
4 YANG,2 YIN <192> { 3} [28 30 61]*
6 YANG,0 YIN <216> { 1} [1]*


*indicates Wen pairs formed via complementary opposition; Wen pairs of this nature fall into separate families.

Introduction to Yao-numbers

ABSTRACT
Yao-numbers or "stick-numbers" are the product of a mathematical transformation of the I Ching's 32 hexagram pairs as presented by King Wen.  In the case of twenty-eight pairs of hexagrams, the mates of a pair differ only by 180 degrees of rotation.  The last four pairs are complementary opposites per Fu Xi's hexagram and trigram arrangements.  


Yao-numbers seem to call attention to the four pairs of complementary opposites: pair-mates generally have identical yao-numbers except in four cases: (01 02) (27 28) (29 30) (61 62), where the mates have "unbalanced" yao-numbers.  Again, these four pairs bear the distinction of having been formed from complementary opposition, as opposed to pair formation from figurative inversion (180 degrees of rotation) in the remaining 28 pairs of hexagrams.


The transformation procedure (described below) produces nine discrete yao-numbers from the 32 Wen pairs of hexagrams. The following crude table displays the 64 hexagrams and the yao-number "families" into which they fall, the quantity of hexagrams per family, as well as yao-numbers and algebraic line-analysis (#YIN lines and #YANG lines) for each hexagram.


0 YANG,6 YIN (144) { 1} [2]
2 YANG,4 YIN (168) { 3} [27 29 62]
5 YANG,1 YIN (172) { 6} [(9 10) (13 14) (43 44)]
4 YANG,2 YIN (176) {12} [(5 6) (25 26) (33 34) (37 38) (49 50) (57 58)]
3 YANG,3 YIN (180) {20} [(11 12) (17 18) (21 22) (31 32) (41 42) (47 48) (53 54) (55 56) (59 60) (63 64)]
2 YANG,4 YIN (184) {12} [(3 4) (19 20) (35 36) (39 40) (45 46) (51 52)]
1 YANG,5 YIN (188) { 6} [(7 8) (15 16) (23 24)]
4 YANG,2 YIN (192) { 3} [28 30 61]
6 YANG,0 YIN (216) { 1} [1]

INTRODUCTION
In Taoist cosmology, the SINGULARITY splits into YIN and YANG; YIN and YANG then combine to produce Four Images, xiang: Heaven (9), Earth (6), Fire (8), and Water (7).  For use in divination ritual, the Taoist tradition assigned "ritual numbers" to the xiang as detailed in parentheses.  Also notable is the traditional association between xiang and the four cardinal directions.


Note: xiang are subtly distinct from the trigrams with the same name; the xiang emerge into being before the bagua (trigrams).  


The transformation that produces yao-numbers is described as follows:
Given the King Wen sequence of thirty-two hexagram pairs as input, employ the xiang bi-grams to measure any changes that occur within a given Wen pair at any of the six line positions.  Four possible measurement outcomes exist:


⚌ EARTH: YIN alternating to YANG (6), 
⚏ SKY: YANG alternating to YIN (9), 
⚍ FIRE: stable YIN (8), or 
⚎ WATER: stable YANG (7)


Note: the xiang bi-grams are represented using Unicode and may not render properly on browsers or systems using UTF-8 or other character encoding


We may contextualize and clarify the transformation procedure by interpreting Wen pairs as though they were products of ritual divination.  For example, We can pretend that, using yarrow stalks, we cast hexagram #1 with six moving lines; this normally indicates a situation initially characterized by hexagram #1 subsequently changing to one represented by hexagram #2.   We now encode the differences between #1 and #2 using xiang.  


Line-by-line comparison of hexagram #2 to #1, beginning at position 1 through position 6 shows YANG changing to YIN at all six places.

䷀ ==> ䷁

This implies moving YANG (ritual number 9).  We therefore encode Wen pair (1 2) as [999999].  


If instead we had cast #2 changing at all six places to #1, we would use moving YIN (ritual number 6), to encode Wen anti-pair (2 1) as [666666].  

䷁ ==> ䷀  

The significance of anti-pairs is discussed elsewhere, but as change is represented by movement, it seems to suggest reversal of direction or perhaps reversal of time.


The encoding for Wen pair (3 4) is [968896]; its anti-pair encodes as [698869].  

䷂ ==> ䷃

At positions 3 and 4 both hexagrams indicate YIN -- a non-changing condition -- encoded by stable YIN (8).


The encoding for Wen pair (5 6) is [979676]; [676979] is the encoding for its anti-pair (6 5).  

䷄ ==> ䷅

At positions 2 and 5, both hexagrams indicate YANG.  This, again, is a non-changing condition encoded by stable YANG (7).


Iterating this transformation procedure over the 32 Wen pairs produces 32 vectors of six xiang.  We call them 'vectors' because they depict ordered arrays of data.  The relevant section of Ta Chuan, the Great Treatise (Pt.1, Ch. 9, Sec. 4) provides three clues for interpreting these vectors: 

• Hexagram #1 has stick-number 216, 
• Hexagram #2 has stick number 144, 
• All 64 stick-numbers summed yield 11520, "the number of the myriad things."  

The rule for obtaining the correct stick-numbers from the vectors involves summing the ritual numbers of the xiang for a given vector and multiplying by four.  The derivation of this rule is discussed in the translators' notes to the Great Treatise.  


The divination ritual involves manipulating 49 yarrow stalks to produce a remainder of stalks, then dividing that remainder by four to produce a ritual number (6|7|8|9). Each ritual number corresponds to one of the four xiang (as described in the INTRODUCTION).  Each xiang produced by ritual divination represents a single hexagram line, which may be dynamic or static.  This procedure is conducted six times to produce a complete hexagram.  


We may infer from this that the rule for obtaining the correct yao-numbers involves the inverse operation to dividing by four, i.e., multiplying by four.


As shown in the following illustration, when all 32 pairs of vectors are thus treated, the combined stick-number sum calculates to 11,520 as stipulated in the Great Treatise.

So, what to do with this information?  What is its use?  What does it tell us?  A simple categorization of the 64 stick-numbers shows them falling into 9 discrete "families" (numbered 1 through 9, left to right) with all members of a family having a particular stick-number.  Additionally, Wen pairs tend to fall together within a given family -- excepting the two triplet groups of hexagrams (#2 & #8), and the two singleton hexagram groups (#1 & #9), where the four Wen pairs are separated from each other.

We find there is very strong correlation/correspondence between these nine families and the seven groups (below) produced by algebraically sorting the hexagrams; that is, according to the quantity of YIN and YANG lines in a given hexagram. Alternating between views of the two schema, we observe that the triplet groups move from their positions next to the poles to a position adjacent to the central group of twenty figures.  Both groups of twelve temporarily gain three, bringing their totals to fifteen.

We also observe that the four "unbalanced" hexagram pairs: (01,02) (27,28) (29,30) (61,62) [families #1, #2, #8, & #9]  all share a peculiar trait -- none of them has a proper Wen inverse as do the other twenty-eight pairs of hexagrams.   The pairs formed from these eight hexagrams are not figurative inverses, but complementary opposites.  Complementary opposition is the basis of the xiantian (Earlier Heaven) trigram arrangement, and the square (8x8) hexagram arrangement, both attributed to Fu Xi.


CONCLUSION
To recap, twenty-eight pairs of hexagrams are formed by 180 degrees of rotation per King Wen's ordering.  The remaining four pairs are complementary opposites as per Fu Xi's arrangement.  Pair-mates generally have identical stick-numbers excepting four cases: (01,02) (27,28) (29,30) (61,62). 


The stick-numbers (calculated above) appear to highlight the complementary opposition of these four pairs against a background of twenty-eight pairs formed from 180 degrees of rotation ("figure inversion").

Although we observe that pairings based on complementary opposition, it is currently unclear how output of the transformation procedure significantly correlates to output of the algebraic sort.

Lexicon:

Position refers to lines of an hexagram, numbered 1 (bottom-most), to 6 (top-most).  

Movement refers to change between YIN and YANG at any of the positions.  

Movement can be dynamic (or alternating) when members of a Wen pair disagree at any of the six positions.  Dynamic also refers to moving lines produced from ritual divination (YANG <==> YIN).

Movement can also be static, as when both pair-members agree at a given position, or when no moving lines appear in a divination.

Wheel Cross: Navigation Tool of Antiquity

Its overall appearance is reminiscent of the comic book character "X-Men" insignia (a cross within a circle); less recently, of a religious symbol.  Wikipedia's article on religious symbols provided the first pointer (below right) but symbols.com provided clear confirmation of this symbol's place within our overall field of investigation:

The wheel cross, sun cross, Odin's cross or Woden's cross. Nordic Odin and Teutonic Wotan or Woden was the supreme god of the Nordic religion before Christianity. Odin was the god of art, culture, warfare, and the dead; depicted as an old, one-eyed man with two ravens as his intelligence agents and messengers.  

Sun Cross 

 The structure  is one of the first non-pictorial graphs to appear when humankind was on the threshold of the Bronze Age. It is common on rock carvings. It appears in ancient Egypt, China, pre-Columbian America, and the Near East. From the facts available it seems as if  is associated with the wheel, not so much with its invention as with its revolutionary effect on the existing society. In ancient China this sign was associated with thunder, power, energy, head, and respect. 

This figure is also known as the gamma cross, as the Greek letter 'gamma' appears four times within the circle, radially rotated.  It is shown to play a prominent role in the Norse, Egyptian, Chinese, and pre-Columbian culture.  Perhaps excepting the Egyptian, this observation is consistent with our thesis that the religion of these cultures is based on representations the Tree of Life. 

The sun cross embodies several notable features: the circle, the right angle, the line segment, and the point.  These figures are all associated with measurement, surveying, mathematics, Platonic geometry, and Freemasonry.  Another feature is the fact that the sun cross can be reassembled into the Ankh cross, by remaking the Greek cross into a Tau cross, and placing the circle atop.

Crichton Miller's research on the Celtic cross convincingly demonstrates its use in antiquity as an instrument of spherical/nautical navigation based on the impeccably-supported theory that working models of the true Celtic cross embody a weighted rotating wheel.  as the circular portion. Moreover, the Celtic cross was used as a tool of determining longitude and latitude.  That Wotan's wheel cross and the Celtic cross bear both a figurative and literal resemblance is worthy of remark.