Friday, July 8, 2011

Origins of Ashtāpada: the 8 x 8 Grid

Ashtāpada, the uncheckered 8x8 board
Ashtāpada is an Indian board game known for his board have been used in earlier versions of chess, Chaturanga famously appeared sometime around the 6th century in India. It could be played by two to four participants and data used to determine the amount of houses to be moved. Ashtāpada The word comes from Sanskrit and its meaning was established by Patanjali in Mahābhāshya book written in the 2nd century, as a board in which each line has eight squares and the term a familiar object.

Chess was designed for an ashtāpada (Sanskrit for "having eight feet", i.e. an 8x8 squared board), which may have been used earlier for a backgammon-type race game (perhaps related to a dice-driven race game still played in south India where the track starts at the middle of a side and spirals in to the center).[17] 

Ashtāpada, the uncheckered 8×8 board served as the main board for playing Chaturanga.[18] Other Indian boards included the 10×10 Dasapada and the 9×9 Saturankam.[18] Traditional Indian chessboards often have X markings on some or all of squares; these may have been "safe squares" where capturing was not allowed in a dice-driven backgammon-type race game played on the ashtāpada before chess was invented.[17]

Two citations are presented here comparing the ashtāpada to the a "field of action" involving combat, or fortune and misfortune:
It has been pointed out[3] that [the chess-board] symbolizes existence conceived as a "field of action" of the divine powers. The combat which takes place in the game of chess thus represents, in its most universal meaning, the combat of the devas with the asuras, of the "gods" with the "titans", or of the “angels”[4] with the "demons", all other meanings of the game deriving from this one.
"Therefore there is in the Changes the Great Primal Beginning. This generates the two primary forces. The two primary forces generate the four images. The four images generate the eight trigrams. The eight trigrams determine good fortune and misfortune. Good fortune and misfortune create the great field of action."
(Commentary on I Ching, tr. Wilhelm and Baynes 1967: Pt.1, Ch.11)
It is worth mentioning that the battlefield referenced in the opening chapter of Bhagavad Gita is an historical location called Kurukshetra.  Kuru, from the Sanskrit root kri="work, material action" and ksetra="field". This "field of action," Kurukshetra, is a metaphor for the human body with its physical, mental and soul faculties, on which all activities of one's life take place.
Ashtapadi are Indian hymns where the music has eight lines (steps) within each composition. Each ashtapadi song is set in a special raga (an Indian musical mode) and tala. It is a rhyme of eternal love and supreme devotion. The literal meaning of "ashtapadi" is "eight steps." This word is also the source for the word ashtāpada, an Indian board game, the forerunner of chess.

Dr. Anand Bhardwaj explains types, properties and uses of Mandala:

In Hindu cosmology the piece or the surface of land is represented in the form of square. The earth is represented as four cornered with reference to the horizon's relationship with sunrise and sunset, the North and South direction. It is known as Chaturbhuji (four cornered) and represented in the symbolic form of the Prithvi Mandala. Each side of square can be divided from 1 to 32 divisions therefore, the number of squares in mandala may vary from 1, 4, 16 till 1024. Each of these mandalas are used in specific contexts and has a distinct name.Whatever the case may be, each square is sub-divided into smaller squares by drawing parallel lines to the sides.
The central area in all the mandalas is brahmasthana and space occupied by it varies in different mandalas. The most important mandalas are the Manduka/Chandita Mandala with 64 squares and the Paramasaayika Mandala with 81 squares.
Normally, ideal position of Vaastu Purusha is depicted by head in northeast and feet in southwest. While in Manduka Mandala, the ideal position is depicted by keeping head towards east and feet towards west.

Thursday, July 7, 2011

I Ching Divination Ritual


To employ the yarrow stalk method, one begins with a bundle of 50 straight, dried stalks of the yarrow herb.  
One stalk is set aside and not used again. 
  • The bundle of the remaining 49 stalks is then divided into two bundles.
  • One stalk is taken from the bundle on the right and placed between the ring finger and little finger of the left hand.
  • The bundle on the left is then taken up in the left hand, and stalks are removed therefrom and set aside in groups of 4 using the right hand, until 4 or fewer remain. The remaining stalks are placed between the middle finger and ring finger of the left hand.
  • The bundle on the right is then taken up in the left hand, and stalks are removed therefrom and set aside in groups of 4 using the right hand, until 4 or fewer remain. The remaining stalks are placed between the index finger and middle finger of the left hand.
  • The total number of stalks between the fingers of the left hand are then counted and noted. The possibilities are 9 and 5. Nine stalks results in the value of 2, and five stalks results in the value of 3. Here are the possible outcomes of the first operation:
Between index and middle fingers:4312
Between middle and ring fingers:4132
Between ring and little fingers:1111
Assigned Value:2333

  • These 9 or 5 stalks are then set aside and the remaining bundle of 40 or 44 stalks is divided and counted out in the same manner. This time, eight stalks results in the value of 2, and four stalks results in the value of 3. Here are the possible outcomes of the second operation:

Between index and middle fingers:4312
Between middle and ring fingers:3421
Between ring and little fingers:1111
Assigned Value:2233

  • These 8 or 4 stalks are then set aside and the remaining bundle of 32, 36 or 40 stalks is divided and counted out in the same manner for a third operation, the possible outcomes of which are identical to those of the second operation.

Each operation has now produced a value of either 2 or 3, and these three values are now added together to produce a total of 6, 7, 8, or 9. The totals of 6 and 8 yield a (static) Yin line (-- --), the totals 7 and 9 yield a (static) Yang line (-----). A total of 6 is considered a moving Yin line (--x--), and a total of 9 is considered a moving Yang line (--o--).

This first line is placed at the bottom of the hexagram and the entire operation is repeated five times to produce the remaining five lines. The sixth line is placed at the top of the hexagram.

ed. note: after the third manipulation, the bundle contains one of (24|28|32|36) stalks, which when divided by four gives the corresponding ritual number (6,7,8,9)

Tuesday, July 5, 2011

Octopartition pt. II

This uncommon term is used to describe the act of dividing a thing into eight sections, called octants; colloquially, "pieces of eight."  I Ching lends itself well to illustrating the notion since it can be octo-partitioned twice in succession as the following graphic depicts.

The largest cube represents I Ching as a holism.  The mid-sized cube represents Ba Gua (one of the Eight Diagrams), a three-line figure or trigram, two of which, when arranged in superposition, produce an hexagram.  The smallest cube, its six faces standing for the six lines, represents an individual hexagram out of the 64 that comprise the I Ching.

Taoist Cosmogony pt. II

To recap, the two First Powers interact to produce the Four Symbols.  The Four Symbols intermingle with the two First Powers again to produce the eight diagrams, or Ba Gua.

arbitrary Ba Gua
Observe that the Ba Gua effectively combine to produce an octo-partitioned space in three dimensions.  The binary metric space allows for two values per dimension.  One of the Ba Gua is depicted here, with each of its corner given as a triplet of binary digits.

Three binary axes combine to yield 2 x 2 x 2 = 8 partitions in the metric space we've defined.  Each of the eight partitions may then be visualized as a cube.  The eight partitions also can represent the Ba Gua as eight cubes.

Eight Ba Gua comprise an hypercube, 

These 8 cubes come together at a common vertex (ORIGIN) to form a composite cube of a higher order, or hypercube.    Each of the eight Ba Gua cubes contributes one distinct corner to the higher-order composite cube.  A vector (path) drawn from the ORIGIN to each of the 8 corners of the hypercube defines direction in space and the bounds of the space.  If all eight paths measure equally, a spherical space results.

Hexagram, Trigram, I Ching

Movement between corners indicates motion, or CHANGE.  Note that all observable phenomena are intrinsically based on motion.  Consider that we can't really observe something that does not change, and if we were to try, what would be the point?  Most interesting matters involve CHANGE, and understanding CHANGE is the whole point of I Ching.

Complementary Pairs

When displayed on the xiantian field, complementary (antipodal) pairs appear separated by 180° of rotation.  Such pairs, insofar as their binary values are considered, are complementary pairs, meaning that one member always has YANG where the other has YIN.  Complementary pairs sum to 65.  In the following animation, xiantian is rotated 45° clockwise.

32 Complementary pairs

Saturday, July 2, 2011

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.