I Ching/Octal Bagua

The eight trigrams of ancient China provide a portal to elemental thinking. A trigram is symmetric when the top and bottom lines agree, regardless of the middle line. The symmetric trigrams with yang at core are Air &#9776; and Water &#9781;. The symmetric trigrams with yin at core Earth &#9783; and Fire &#9778;. These are the elements of Empedocles, adopted by Aristotle and Plato. The asymmetric trigrams with yang at core are Lake &#9777; and Wood/Wind &#9780;. The asymmetric trigrams with yin at core are Thunder &#9779; and Mountain &#9782;.

As the trigrams or bagua are derived from the yin and yang duality, there is always the correspondence of a trigram with its complement. Fire and Water are complements, as are Mountain and Lake. It is absorbtion of yin and yang to the sixth degree that is manifested in the I Ching, and the entrance to structure comes by way of the bagua. The sixty-four hexagrams correspond to a complete, directed graph on the eight bagua. The graph includes eight loops, one at each of the bagua, which correspond to doubled trigrams, so eight articles of the I Ching illuminate intensity of a doubled bagua, and the names of these articles can be used  to designate them.

G.W. Leibniz showed in 1703 that a yin line can be taken as zero (0) and a yang line as one (1) so that an arbitrary line is a binary digit (bit). Three lines are then three bits or an octal number. The octal digits 0 to 7 provide a brief designation of a bagua. The complementation is n &rarr; 7&minus;n. For example:

Fire = &#9778; = 101 = 5 &rarr; 2 = 010 = &#9781; = Water.

Hexagram sequence
The Book of Changes is a tool of divination and is consulted on a chance-selection basis, traditionally with yarrow stalks or coins. Nevertheless, the sixty-four hexagrams have an order, starting from absolute separation of yin and yang and ending with complete intermingling of yin and yang lines. The rationale for the order remains a mystery. The hexagrams of doubled trigrams have the following order:


 * &#9776; air (1), &#9783; earth (2), &#9778; fire (29), &#9781; water (30).
 * &#9779; thunder (51), &#9782; mountain (52), &#9780; wood (57), &#9777; lake (58).

Consecutive hexagrams can be compared by counting their differences. This count is called the Hamming distance in coding theory of bit strings. For consecutive hexagrams the maximum distance is 6, corresponding to the case where each trigram is replaced by its complement. Of the sixty-three consecutive pairs, 23 are at distance 2, 17 are at distance 4, and 12 at distance 3. The nine cases of distance 6 (complementary hexagrams) are these: 1 & 2, 11 & 12, 17 & 18, 27 & 28, 29 & 30, 38 & 39, 53 & 54, 61 & 62, 63 & 64. There are only two pairs at a distance of 1, and none at distance 5.

For the twelve months of the year expressed with hexagrams there is a definite algorithm: The darkest month corresponds to the all yin hexagram (#2). Then light begins to appear one month later with a yang line at the base (#24). The next month two yang lines (#19) invade the hexagram from below. Each month a new yang line appears until all lines are yang, so after six months the brightest days appear (summer solstice, #1). The algorithm continues by introducing yin lines: #44 has one yin line at the base represents one month into summer. With advancing yin lines from below the darkest month #2 returns.

Can an algorithm for the sequence from 1 to 64 be found? It represents a depolarization from the first pair to the last pair. The derived sequence of Hamming distances has some features: absence of 5 and few distances 1 and 3, so the distance is mostly even in the steps through the I Ching. In terms of the bagua, six instances of transposition occur: 5 to 6, 7 to 8, 11 to 12, 13 to 14, 35 to 36, and 63 to 64. In 19 cases just one trigram changes place. Only six steps have a constant trigram in place. As the commentaries lean heavily on trigram images, these characteristics are noted in the text.

The transpositions
Only the Greek elements partake in transposition in the sequence.
 * Water &#9781; above and below Air &#9776; are #5 and #6.
 * Earth &#9783; above and below Water &#9781; are #7 and #8.
 * Earth &#9783; above and below Air &#9776; are #11 and #12.
 * Fire &#9778; above and below Air &#9776; are #13 and #14.
 * Fire &#9778; above and below Earth &#9783; are #35 and #36.
 * Fire &#9778; above and below Water &#9781; are #63 and #64.

Group theory
The meaning of this section may be enhanced by reference to the wikibook Abstract Algebra, where the basic concepts of binary operation, abstract group and cyclic group are presented.

The interpretation of bagua as binary numbers according to Leibnitz associates them with the cyclic group of order eight $$C_8 .$$ In fact, there are five different groups of order eight that might have their elements represented by bagua.

For instance, the three lines of a trigram may be considered a binary register with three parallel operations, performed with binary arithmetic of the integers modulo 2 where 1 + 1 = 0. As a group with this operation, the bagua represent
 * $$C_2 \times C_2 \times C_2 ,$$ the cube of the cyclic group of order 2.

Another group structure on the set of 8 bagua may use the subset {&#9783;, &#9779;, &#9781;, &#9782;} or {earth, thunder, water, mountain} to represent the cyclic group of order 4: $$C_4 .$$ The other four bagua arise as complements of these four, with complementation generating the cyclic group of order 2. With these operations the bagua represent the product group $$C_4 \times C_2 .$$

The groups $$C_8, \ C_4 \times C_2, \ (C_2)^3$$ represent the commutative groups of order 8 (see link to Groupprops below). But in group theory the binary operation may not be symmetric: order matters: some groups have elements p and q where pq ≠ qp. There are two of these of order 8 called the quaternion group and the dihedral group.

The complement of a bagua will represent the negative element. Take &#9776; to represent 1, then its complement &#9783; represents &minus;1. Since bagua represent three bits of information, they can be combined with logical AND and OR operations. Multiplying by &#9776; uses ^ (AND), and multiplying by &#9783; means invoking the complementary bagua, written T.

It is natural to associate &#9779; = i, &#9781; = j, and &#9782; = k by reason of order, thus connecting to traditional representation of quaternion and dihedral groups. The v (OR) operation is used to obtain the product of two distinct bagua, thus obtaining a product with two yang lines. The quaternion group takes the complement of this product in order to obtain one of i, j, or k. The dihedral group does not use the complement in the case jk = &#9781; v &#9782; = &#9780; = T&#9779; = &minus; i. The order of the two distinct factors determines whether the product is simple or complemented, so kj = i in the dihedral group.

Since the negative of an element is its complement, the OR of an element with its negative gives &#9776; = 1. For the quaternion group the minus can be moved to the other side of the equation giving &minus;1 for the square of i, j, or k. However, in the dihedral group j2 = +1 = k2. The associative property of group operations can be confirmed with either notation for elements. These groups are known for the four-dimensional algebras they underpin, quaternions and split-quaternions.

In summary, the eight bagua, with their structures of order and complementation, provide the means to represent any of the groups of order eight. The cyclic group has no particular claim on bagua for representation. In fact, these bagua structures can be taken as precursors to abstract algebra, as well as to binary numeration.