The Foursome Pairs

You have probably noticed that the hexagrams of the I Ching are arranged in 32 pairs.  Most pairs are created by inverting (turning upside-down) each member.  But four pairs (shaded red in the diagram below), which consist of hexagrams which are symmetrical and do not change when inverted, are created by reversing (turning yin to yang and vice-versa) each member.  And for four pairs (shaded blue), the inverse and reverse are the same.

Each of the 24 pairs whose members are not the reverse of each other (shaded black) has a corresponding pair which consists of the reverse of each of its members.  These hexagrams are thus members of “foursomes” as described by Carrin Dunne in Yijing Wondering and Wandering.  In the diagram below, the King Wen number of each hexagram appears above it, and the number of the reversed hexagram appears below it.  The members of the red and blue pairs are all the reverse of each other; but for the others, the reversed hexagrams are the other members of the foursome.

Note that every hexagram has a reversed opposite; but eight, being symmetrical, do not have an inverted opposite.  For some reason, inversion was given priority in the King Wen arrangement, and reversal was only used for the four symmetrical pairs.

• top number:  King Wen numbers
• bottom number:  the same for each hexagram’s reversed opposite

The hexagrams can also be arranged in binary sequence, as they were by Shao Yong (1011-1077); this came to be known as the Fu Xi order.  Below is such an arrangement; but they are also grouped into alternating reversed pairs.  Above each hexagram is the binary value of the hexagram (yin = 0, yang = 1) followed by the King Wen number for comparison; below are the numbers for the inverted opposite.

Note that although most of these pairs are different from the King Wen pairs, the foursomes are all the same.  Furthermore, a symmetry is preserved since every pair is made of reversed opposites.

• top numbers:  binary value - King Wen numbers
• bottom numbers:  the same for each hexagram’s inverted opposite

63003417 58231602
59555607 47610804
3825483 37410132
39573330 18451428
15600540 43531020
49356231 06242226
46293609 11524413
27541950 51124221

Immanuel Olsvanger (1888-1961) discovered an amazing system of symmetries in the King Wen order, based on the binary values of the 8x8 “magic square” of hexagrams.  This is described in deatail in Zhouyi:  A New Translation with Commentary of the Book of Changes by Richard Rutt.  Briefly, the sum of all the binary values is 2016.  But the square can be divided in two in a number of odd ways (one of which is illustrated to the right) where the sum of the numbers on each side is half the total, or 1008.  This suggests that someone by the time of the Han dynasty understood binary numbers and put a great deal of effort into the King Wen arrangement.

Note that if the hexagrams are arranged in the alternating binary sequence as above, the sum of each pair is 63, and the “magic square”of hexagrams can be divided into equal halves, fourths, eighths, and so on, in any way that preserves the grouping of the pairs.  The symmetries are thus increased manyfold, although in a more mundane manner.

Another symmetry emerges from the alternating binary sequence.  Below, the red and blue pairs are the ones which are their own reversed opposites, as above.  The green pairs in the second column all form inverted foursomes with green pairs in the fourth column; the same goes for the gold pairs in the first and third columns.  The black pairs all form foursomes with the black pair above or below them.  Each foursome is thus a member of a “foursome of foursomes.”

• top numbers:  binary value - King Wen numbers
• bottom numbers:  the same for each hexagram’s inverted opposite

The members of the green pairs are all in reversed order with respect to the other pair of the foursome; the gold ones are not.  The black pairs which are not adjacent are similarly reversed; the adjacent ones are not.

Here it is in a 3-dimensional 4×4×4 arrangement.  (Mentally stack the 4×4 groups on top of each other.)  The axes are based on the four combinations of two-line bigrams.  Each row, and each z-axis column (coming out of the page), has one hexagram of each color.  In this arrangement, the reversed opposites are diametrically opposed.

• top numbers:  binary value - King Wen numbers
• bottom numbers:  the same for each hexagram’s inverted opposite