Quaternion Group Q8 and Yi Jing (I Ching) Hexagrams

In mathematics the quaternion group $\mathbf{Q_8}$ is formed by the basis quaternions $\mathbf{1, i, j, k}$. The multiplication table is given by

compare the table above to the Yi Jing (I Ching) hexagrams below.

The table of Yi Jing (I Ching) hexagrams is a representation of the quaternion group $\mathbf{Q_8}$. There is a one-to-one correspondence between the Yi Jing trigrams and the basis quaternions. When we identify the resultant hexagrams in the 64 cells of the table above with the resultant quaternions in the first table we come up with a correspondence that lead us to new aspects of the hexagrams. I would call these the reflection poles.

Reflection Poles of the 64 Hexagrams in the Q8 Interpretation

Hexagram 01 (The Creative), Hexagram 32 (Duration), Hexagram 63 (After Completion), Hexagram 64 (Before Completion), Hexagram 41 (Decrease), Hexagram 42 (Increase), Hexagram 2 (The Receptive), Hexagram 31 (Influence/Wooing) map to quaternion 1; therefore the reflection pole of these hexagrams will be the KUN (Earth).

Hexagram 34 (Power of the Great), Hexagram 25 (Innocence/The Unexpected), Hexagram 4 (Youthful Folly), Hexagram 56 (The Wanderer), Hexagram 20 (Contemplation), Hexagram 46 (Pushing Upward), Hexagram 49 (Revolution), Hexagram 60 (Limitation) map to quaternion i; therefore the reflection pole of these hexagrams will be the SUN (Wind).

Hexagram 5 (Nourishment), Hexagram 17 (Following), Hexagram 6 (Conflict), Hexagram 62 (Preponderance of the Small), Hexagram 35 (Progress), Hexagram 18 (Work On What Has Been Spoiled), Hexagram 36 (Darkening of the Light), Hexagram 61 (Inner Truth) map to quaternion j; therefore the reflection pole of these hexagrams will be the LI (Fire).

Hexagram 26 (The Taming Power of the Great), Hexagram 3 (Difficulty at the Beginning), Hexagram 45 (Gathering Together), Hexagram 33 (Retreat), Hexagram 55 (Abundance), Hexagram 50 (The Caldron), Hexagram 19 (Approach), Hexagram 59 (Dispersion) map to quaternion k; therefore the reflection pole of these hexagrams will be the TUI (Lake).

Hexagram 11 (Peace), Hexagram 51 (The Arousing), Hexagram 29 (The Abysmal), Hexagram 52 (Keeping Still), Hexagram 12 (Standstill), Hexagram 57 (The Gentle), Hexagram 30 (The Clinging), Hexagram 58 (The Joyous) map to quaternion -1; therefore the reflection pole of these hexagrams will be the CHIEN (Heaven).

Hexagram 9 (The Taming Power of the Small), Hexagram 24 (Return), Hexagram 47 (Oppression), Hexagram 39 (Obstruction), Hexagram 16 (Enthusiasm), Hexagram 44 (Coming to Meet), Hexagram 22 (Grace), Hexagram 38 (Opposition) map to quaternion -i; therefore the reflection pole of these hexagrams will be the CHEN (Thunder).

Hexagram 14 (Possession in Great Measure), Hexagram 27 (Providing Nourishment), Hexagram 7 (The Army), Hexagram 53 (Development), Hexagram 8 (Union), Hexagram 28 (Preponderance of the Great), Hexagram 13 (Fellowship with Others), Hexagram 54 (The Marrying Maiden) map to quaternion –j;
therefore the reflection pole of these hexagrams will be the KAN (Water).

Hexagram 43 (Breakthrough/Resoluteness), Hexagram 21 (Biting Through), Hexagram 40 (Deliverance), Hexagram 15 (Modesty), Hexagram 23 (Splitting Apart), Hexagram 48 (The Well), Hexagram 37 (The Family), Hexagram 10 (Conduct) map to quaternion –k; therefore the reflection pole of these hexagrams will be the KEN (Mountain).

Reflection Poles are Elemental

Note that all reflection poles are elemental, in other words, they refer to the fundamental trigrams: the KUN (Earth), the SUN (Wind), the LI (Fire), the TUI (Lake), the CHIEN (Heaven), the CHEN (Thunder), the KAN (Water), the KEN (Mountain).