Bagua is a Group, Most Likely Z8

Binary Interpretation

The Bagua consists of eight trigrams mapping one-one to the eight integers between 0 and 7 representable with three binary bits

Bagua trigrams are also commonly manipulated -- meaning that they can morph from one trigram into other trigrams based on specific rules.  These manipulations, however, do not appear to be numeric in nature -- in other words, the manipulations of trigrams (with operations like flipping a bit -- turning a broken line into a solid line and vice versa) do not map closely to integer arithmethic.

Group Theoretical Interpretation

If one treats the bagua trigrams as a group, however, the manipulations appear to make more sense.  All the requirements of groups appear to match well with the characteristics of the bagua:

• The eight trigrams are mutually exclusive
• The eight trigrams are collectively exhaustive and closed (i.e. there is no possible trigram outside the eight)
• Manipulations of trigrams yield other trigrams
• There is a trigram mapping to identity (most possibly 乾 ☰ or 坤 ☷)
• There is an inverse to each trigram (negative image, flipped, etc.)
• There is an ordering of the 64 hexgrams into an 8x8 matrix (分宮卦序) used in the Four Pillars system, with columns matching the bagua trigrams (shown below) and rows definitely not matching, which almost begs to suggest that the 64 hexgrams is the product of two non-identical groups of order 8 (although it can obviously be also made out of the Cartesian product of two identical groups)

Possible Groups With Order Eight

There are only five groups with order eight up to isomorphism:

• Z8 -- cyclical group
• Z2 x Z4 -- cyclical version of dihedral group
• E8 or Z2 x Z2 x Z2 -- elementary abelian group
• D8 -- dihedral group of eight
• Q8 -- quaternion group

Let us look at all five possibilities in turn.

Trigram Inverses

Classical literature often refer to the inverse of a bagua trigram, almost as if the bagua is a group.  There are two primary systems to describe the negative or inverse of a trigram:

• Flipping all the lines
• Flipping the trigram upside-down (i.e. the first line becomes the last and vice versa, with the middle line staying put)

Flipping all the lines is an obvious way of matching a trigram with an inverse.  Under such a system, each of the three lines in a trigram can be interpreted as independent of each other, like three orthogonal axes (with the eight bagua trigrams on the eight corners of a cube).

Turning the trigram upside-down to find its inverse is also an extremely interesting idea, not merely due to the fact that it is outlined in I-Ching's 雜卦傳 (the one document primarily concerned with inverses), which pairs up the 64 hexgrams as inverse pairs, and most of them are upside-down versions (e.g. 比 and 師, 損 and 益, 震 and 艮).

The "upside-down" concept of finding inverse is also apparent if one treats the three lines in a trigram as three operations to be performed one after another, and if the operation of each line is its own inverse -- which means that the inverse of the combination operation resulting from the three trigram lines is merely the three same operations applied in reverse (and thus flipping the trigram upside-down).

These two possible manners of finding an inverse of a trigram may shed light towards the group structure of the Bagua.

Z8

The simple cyclical group of order eight has one element (say `a`) that generates the entire group:

`a`, `a^2`, `a^3`, `a^4`, `a^5`, `a^6`, `a^7`, `a^8 = e`

It immediately suggests the Binary Order, with 1 being the generator and group product being addition modulo 8 and the identity element being 0:

☰ = `e` (0)
☱ = `a` (1), inverse = ☷ (`a^7` or 7)
☲ = `a^2` (2), inverse = ☶ (`a^6` or 6)
☳ = `a^3` (3), inverse = ☵ (`a^5` or 5)
☴ = `a^4` (4), self-inverse
☵ = `a^5` (5), inverse = ☳ (`a^3` or 3)
☶ = `a^6` (6), inverse = ☲ (`a^2` or 2)
☷ = `a^7` (7), inverse = ☱ (`a` or 1)

It is obvious that this mapping of trigrams to group elements is not attractive, as there is no apparent relation between pair trigrams (e.g. ☰ and ☷, ☲ and ☵ etc.). It is probably because of this that the Binary Order of trigrams was not invented until much later in A.D.

Another mapping of Z8 to trigrams, however, suggests itself, based on the Sibling Order. Notice that `a^4` is its own inverse, and if we map 坤 ☷ to `a^4` we get the following:

☰ = `e` (Father)
☶ = `a` (Youngest son), inverse = ☴ (`a^7`)
☵ = `a^2` (Middle son), inverse = ☲ (`a^6`)
☳ = `a^3` (Eldest son), inverse = ☱ (`a^5`)

☷ = `a^4` (Mother) = -1, self-inverse
☱ = `a^5` (Youngest daughter), inverse = ☳ (`a^3`)
☲ = `a^6` (Middle daughter), inverse = ☵ (`a^2`)
☴ = `a^7` (Eldest daughter), inverse = ☶ (`a`)

Some attractive features of this mapping:

• There are clear choices for the father and mother trigrams (which map to the two most important pure trigrams among the eight)
• The inverse of each trigram is the negative image (i.e. flipping each line) of itself flipped upside-down
• Each trigram is related to one other trigram which has all the lines flipped by the factor -1 -- i.e., any trigram multiplied by ☷ yields its negative image

Z2 x Z2 x Z2 or E8

The elementary abelian group of order eight is one that maps straight to the trigram representation, with each line mapping to one Z2 subgroup.  However, the interactions form an exclusive-or system, since a broken line may map to `a` and a solid line to `a^2`, with `a^2 = e`.  The broken line is thus a generator, meaning one broken line applied against another broken line yields a solid line, with the following multiplication table:

broken x broken --> solid
broken x solid --> broken
solid x broken --> broken
solid x solid --> solid

In other words, all trigrams are self-inverses, and that applying one trigram to another merely flips all the lines in the first trigram where the second trigram has broken lines, and 乾 ☰ is e (i.e. 1,1,1) and 坤 ☷ is (-1,-1,-1).

Z2 x Z4

On the surface, the group Z2 x Z4 looks unlikely to be a strong candidate -- it is abelian and cyclical, thus limiting the possibility of complex interactions.

However, this group is almost a one-to-one map against the strange (but ancient) Sibling Order, as shown below:

☰ = `(e, e)` (Father) = `e`
☶ = `(e, a)` (Youngest son), inverse = ☳
☵ = `(e, a^2)` (Middle son), self-inverse
☳ = `(e, a^3)` (Eldest son), inverse = ☶

☷ = `(-1, e)` (Mother) = -1
☱ = `(-1, a)` (Youngest daughter), inverse = ☴
☲ = `(-1, a^2)` (Middle daughter), self-inverse
☴ = `(-1, a^3)` (Eldest daughter), inverse = ☱


Notice the unexpected features of this mapping:

• There are clear choices for the father and mother trigrams (which map to the two most important pure trigrams among the eight)
• The inverse of each trigram is the trigram flipped upside-down (no other group mapping can boast this feature), a concept outlined in I-Ching's 雜卦傳 (another ancient text)
• There are two special trigrams which look the same when flipped upside-down (i.e. fire ☲ and water ☵) -- they are self-inverses, but also mirror each other by a factor of -1
• Each trigram is related to one other trigram which has all the lines flipped (another common inverse algorithm) by the factor -1

Dihedral Group of Eight (D8)

The dihedral group models 90° planar rotations and planar reflections -- usually denoted by `a` and `r` respectively.  A common representation is:

`e`, `a`, `a^2`, `a^3`, `r`, `ar`, `a^2r`, `a^3r`

where `e` = identity, `a` = rotation by 90° and `r` = reflection.

Only two elements are inverses of each other: `a` and `a^3`.  All other elements are self-inverses.
This makes D8 a less attractive candidate group for the bagua trigrams because:

• so many elements are self-inverses
• it is not apparent to suggest which element should be 坤 ☷ if 乾 ☰ is taken as `e` and vice versa
• it is not apparent how to map each of the three lines in a trigram

One possible mapping of the trigrams is:

☰ = `e`
☱ = `r`, self-inverse
☲ = `a`, inverse = ☵
☳ = `a^2r`, self-inverse
☴ = `ar`, self-inverse
☵ = `a^3`, inverse = ☲
☶ = `a^3r`, self-inverse
☷ = `a2`, self-inverse

Under this mapping, fire 火 ☲ and water 水 ☵ are inverses of each other, representing rotations of +90° and -90°.

Multiplication rules based on manipulating solid and broken lines, however, are difficult to come up with.

Quaternion Group (Q8)

The quaternion group is promising. Not only is it non-abelian, it is anti-commutative and thus opens up much flexibility in terms of manipulation.

One possible mapping of the quaternion group is:

☰ = e
☴ = i
☲ = j
☱ = k
☳ = -i
☵ = -j
☶ = -k
☷ = -1

Notice that this mapping has much going for it:

• There are only two elements which are self-inverses, namely 乾 ☰ and 坤 ☷, incidentally the "father" and "mother" trigrams.
• Each quaternion axis is represented by one line -- three of them (i, j, k), three lines for each trigram.
• The inverse of each quaternion axis is simply the trigram with all lines flipped.
• Since the three quaternion axes are related by anti-commutative rules (e.g. ij = k and ij = -ji), the inverse of each trigram makes some sense (i.e. the inverse of each axis is a combination of the other two axes, or e.g. -i = kj, -j = ik, -k = ji)

Multiplication rules based on manipulating solid and broken lines, however, are difficult to come up with.

Groups Mapping Summary

In order to decide which of the five different groupings map to the eight Bagua trigrams (or the 64 hexgrams), it is useful to identify, for each group, how many elements are self-inverses and how many elements are inverse pairs:

Group	Self-Inverses	Inverse Pairs	Abelian?	Anti-Commutative
E8 (Z2 x Z2 x Z2)	8	0	Y	N
Z2 x Z4	4	2 x 2	Y	N
Z8	2	3 x 2	Y	N
D8	6	2	N	N
Q	2	3 x 2	N	Y