Gray Codes
The reflected binary code, also known as the Gray code after Frank Gray, is a binary numeral system in which consecutive values differ by only one bit. The reflected binary code was originally designed to prevent spurious output from electromechanical switches. Today, Gray codes are widely used to facilitate error correction in digital communications such as digital terrestrial television and some cable TV systems. —Wikipedia
A six-bit Gray code represents an ordering of the 64 hexagrams such that each is transformed into the next by changing exactly one line. It is thus an extreme example of hexagram transitions, a single-line transition pathway that traverses all 64 hexagrams. The Gray code hexagram sequence is illustrated below, including the actual code in the left column, and the hexagram reference on the right, including the line that changes to make the next transition. Note that the sequence is cyclic; the last hexagram changes into the first.
The original reflected binary code can be generated with a simple algorithm. But it is not unique; many other Gray codes exist. Here are 10 six-bit Gray codes, generated by an unscientific random walk:
|
000000 000100 000101 000001 000011 100011 110011 110010 010010 010000 110000 111000 111100 111101 101101 100101 110101 110100 110110 100110 000110 000111 100111 110111 111111 011111 011110 111110 111010 101010 100010 100000 100100 101100 101110 101111 001111 001110 001100 001101 001001 001011 101011 111011 111001 011001 011000 001000 101000 101001 100001 110001 010001 010101 011101 011100 010100 010110 010111 010011 011011 011010 001010 000010 |
000000 000001 100001 100000 100100 100101 000101 000100 001100 101100 101101 001101 001111 001110 001010 000010 010010 010011 110011 111011 111001 011001 011101 010101 010001 110001 110101 111101 111100 110100 010100 010000 110000 110010 111010 111000 101000 101001 101011 101010 101110 101111 100111 110111 010111 010110 000110 000111 000011 100011 100010 100110 110110 111110 111111 011111 011110 011100 011000 011010 011011 001011 001001 001000 |
000000 100000 100010 110010 110011 110001 110101 111101 101101 101001 111001 011001 011011 111011 111111 101111 100111 110111 010111 011111 011101 011100 111100 111110 101110 101010 001010 001000 011000 011010 111010 111000 101000 101100 001100 001101 000101 010101 010001 000001 001001 001011 101011 100011 100001 100101 100100 100110 110110 110100 110000 010000 010100 000100 000110 010110 011110 001110 001111 000111 000011 010011 010010 000010 |
000000 000100 000110 000111 000011 010011 010111 010101 000101 001101 011101 111101 111001 110001 110011 100011 100111 110111 110101 100101 101101 101111 101110 001110 001111 011111 011110 010110 110110 110100 110000 100000 101000 101010 001010 000010 010010 010000 010100 011100 001100 101100 100100 100110 100010 110010 111010 011010 011011 011001 010001 000001 100001 101001 001001 001011 101011 111011 111111 111110 111100 111000 011000 001000 |
000000 001000 101000 111000 110000 110100 110110 110010 100010 101010 001010 001011 001111 001110 011110 011100 111100 101100 101101 111101 111001 111011 111111 110111 110101 110001 010001 000001 001001 101001 100001 100000 100100 100101 000101 001101 001100 000100 010100 010101 010111 010110 000110 100110 100111 000111 000011 000010 010010 011010 111010 111110 101110 101111 101011 100011 110011 010011 011011 011111 011101 011001 011000 010000 |
000000 010000 011000 111000 110000 110001 111001 011001 001001 001101 101101 101111 101110 101100 101000 101001 101011 001011 001111 000111 010111 010101 000101 000100 000110 001110 001100 001000 001010 011010 010010 010110 010100 110100 111100 111110 111111 111011 011011 011111 011110 011100 011101 111101 110101 110111 110110 110010 111010 101010 100010 000010 000011 000001 010001 010011 110011 100011 100111 100110 100100 100101 100001 100000 |
000000 100000 100001 000001 000011 000111 001111 101111 101101 100101 000101 000100 010100 011100 011000 011010 001010 001000 101000 101001 111001 111011 111111 110111 100111 100110 000110 000010 100010 110010 110000 110100 100100 101100 001100 001101 011101 111101 110101 110001 010001 010101 010111 010110 110110 111110 111100 111000 111010 101010 101110 001110 011110 011111 011011 011001 001001 001011 101011 100011 110011 010011 010010 010000 |
000000 000001 010001 010000 010100 110100 100100 100101 000101 000100 000110 100110 100010 100000 101000 101100 111100 011100 011101 010101 110101 110001 100001 101001 111001 011001 001001 001011 001010 000010 010010 110010 110000 111000 011000 011010 011110 011111 011011 010011 110011 111011 101011 101111 111111 111101 101101 001101 001111 000111 000011 100011 100111 110111 010111 010110 110110 111110 111010 101010 101110 001110 001100 001000 |
000000 001000 001010 001011 000011 000001 100001 100000 110000 110010 111010 111110 111100 111000 011000 011010 011011 010011 010111 000111 001111 101111 101101 100101 000101 001101 001001 101001 101000 101100 100100 000100 001100 001110 101110 101010 101011 111011 111001 011001 010001 010000 010010 010110 000110 100110 100111 110111 110110 110100 010100 010101 011101 011100 011110 011111 111111 111101 110101 110001 110011 100011 100010 000010 |
000000 010000 011000 011010 111010 111110 011110 011100 011101 010101 010111 011111 011011 010011 110011 110111 100111 100101 101101 001101 000101 000100 010100 110100 100100 100000 110000 111000 101000 101010 101011 111011 111111 101111 101110 001110 001010 001000 001100 101100 111100 111101 110101 110001 111001 011001 010001 000001 001001 101001 100001 100011 000011 001011 001111 000111 000110 010110 110110 100110 100010 110010 010010 000010 |
Listing codes like this quickly becomes very cumbersome. Another way to represent a code sequence is to simply list the lines that change (a transition sequence). The original Gray code sequence then looks like this (note the multiple reflected symmetries, and the preponderance of small numbers):
1213121412131215121312141213121612131214121312151213121412131216
Below are 1000 unique transition sequences. As they were generated by random walks, the numbers tend to be more balanced. Also note that, when listing the changed lines like this, the changes can begin on any hexagram (not just 2 like the ones above), and the resulting sequences are all still unique. Reversing the string also creates a unique sequence as there are no reversed duplicates on the list. So these 1000 transition sequences can generate 128,000 (1000 × 64 × 2) unique hexagram sequences, by beginning each sequence on every hexagram as well as reversing them.
If that weren’t enough, a Gray code can be rearranged by reordering its columns. Imagine cutting one of the codes above into six vertical strips, then rearranging the strips. Six strips can be arranged 720 different ways, and every one of them is still a Gray code. However, they are no longer guaranteed to be unique. That means that up to 92,160,000 (128,000 × 720) hexagram sequences can be generated by changing the starting hexagram, reversing the sequence, and permuting the columns, of the codes below.
And because I knew you wanted to see it, here is the 12-bit Gray code, which traverses all 4096 hexagram pairs. (It occupies all 8 columns below.)