I Ching Memory Wheels

Sixty-four digits can be arranged to represent all the hexagrams of the I Ching in an overlapping sequence.  The sequence wraps around, like beads on a circular string; one could use such a string of beads to select hexagrams.  I have illustrated the concept for one sequence below following the example in 64 bit I Ching Binary Loop / Wheel in the Abrahadabra.com forum; please see The Changes and the Bruijn sequence on the Clarity site for more information.

 
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
0000001010001001011000011001101001110001111010101110110111111001
wheel

Here are 892 memory wheel sequences.  They are all topologically unique, both shifted and reversed.  If each were connected end-to-end, they would all be different, whether you rotated or flipped them.


If you have a real need, here are 222,539 unique sequences, in zipped format (uncompressed file size is 14.0 MB):

222539memorywheels.zip (986 KB).  [Woo!  Got the time down to 13 minutes.]

If there are 22n-1-n different memory wheels for binary n-tuples, then as I understand it there are 226 = 67,108,864 different I Ching memory wheels.  Given a total of 264 = 18,446,744,073,709,551,616 different sequences of 64 digits, that means that only 1/274,877,906,944 of the sequences are memory wheels.



Using a memory wheel to consult the I Ching

Selecting one bead on a memory wheel determines one hexagram.  How then to determine the moving lines?  One way would be to make two or three more selections to mimic the coin methods described on the coins page.  For the yarrow stalk probabilities, three more selections make a total of four beads per hexagram line; exactly three yang beads among the four means that the line is moving.  For the traditional coin probabilities, make two more selections; for each line, if both of the second two beads are yang, the line is moving.

The beads below represent the first of the two hexagrams illustrated at the top of the page.  Yang is white, yin is black.

        yin
    moving yin
    yang
    yin
    moving yang
    yin

Another way to do it would be to use the memory wheel to make three- or four-bead selections to determine each line separately, in the same way that coins are used.  A 64-bead memory wheel would work for this, but a 16-bead wheel could be used as well.  Here are a few: