 # Fantasy FactorialHexagrams The I Ching, , "Book of Changes," is a Classic of Chinese literature. In it the two forces of Nature, and , are represented by a broken line, , and an unbroken line, , respectively, in order to model the different possible states of reality. Diagrams of three lines, "trigrams," are put together in pairs to make diagrams of six lines, "hexagrams." There are 23 or 8 possible trigrams, and 26 or 64 possible hexagrams.

When I was living in Hawai'i in 1974, I began to think about diagrams that might represent Yin and Yang in a different way, using, not broken and unbroken lines, but crossed and uncrossed lines. This would leave a single line free to represent the Tao itself, which is beyond Yin and Yang (although, according to Taoism, with an affinity for Yin, while Confucianism is regarded as having an affinity for Yang). At left we see how this looks. A decision needed to be make, of course, about which was Yin and which was Yang. The broken line represented Yin because, being broken, it would be yielding, , which is an attribute of Yin. Similarly, I took the crossed lines, which might also look like the straight lines bent into each other, as equally suggestive of the yielding.

The crossing of the lines in , were it to involve a crossing of string or rope in the real world, could conceals various kinds of interactions. The ways this can happen are called "braids." The serious study of this is part of Group Theory in mathematics. Thus, as we see in the diagram at right, the red line can go under the black line, or over it. Or the lines can be twisted, so that each ends on the same side where began, after going around the other one. But this can happen two different ways. We can take the red line and put it over the black one, and then back under. Or we can take the red line and put it under the black one, and then back over. One becomes familiar with these alternatives in learning to tie knots, or, indeed, to braid hair or rope. These complications, however, will not concern me here. The lines cross at an intersection, and that is that. Otherwise, the permutations would expand vastly, to unclear significance, and, as we shall see, they will be numerous enough as it is. On the other hand, if one wanted to make this all a lot more complicated, braids are an easy way to do it. At the same time, and the will have no variations. Perhaps the "infinite variety" of Cleopatra is what goes with .

With the broken and unbroken lines, there are eight possible combinations of three lines, and so eight trigrams. With crossed and uncrossed lines, there were only six possible combinations of three lines. This leaves two of the trigrams with no possible match in the new diagrams of six lines. I've let the Second Son and Second Daughter trigrams get bumped out at left. Pure Yang, the "Father," and pure Yin, the "Mother" seem pretty obvious for the diagrams with no crossed lines and all crossed lines, respectively. Exactly how the others match up is bound to be rather arbitrary. The colored boxes now indicate the symmetries of the diagrams. Purple boxes are for those symmetrical in three axes, i.e. they can be flipped left to right (in the y axis), flipped top to bottom (in the x axis), or spun around 180 degrees (in the z axis), and the result is identical in appearance to the way it started. The green boxes indicate the diagrams only symmetrical around the y axis, and the red those only symmetrical around the z axis.

The next question was going to be how many diagrams we get with larger numbers of lines. With the broken and unbroken line diagrams, this will always be the power of 2 equal to the number of lines. With the hexagrams, that will then be 26 or 64. For the crossed and uncrossed lines, things ended up looking rather different. I soon realized that the number of possibilities increased much faster than in the traditional diagrams. For each number of lines, there would be possible combinations equal to the factorial of that number -- i.e. all the integers up to and including that number multiplied together. So with six lines, we would get 6! diagrams, or 720. That sentence needs an exclamation point, but then that would be the mathematical notation indicating a factorial! If we make up the diagrams with four lines, we get 4! diagrams, or 24, still a very manageable number. But now we get some emerging patterns. There is a symmetry between the diagrams that are symmetrical in three axes, and also between those symmetrical in the y axis and those symmetrical in the z axis. We also now get a couple of diagrams, in blue boxes, that are symmetrical around the x axis alone. This higher level symmetry involves what would happen if we take one side of a particular diagram and flip it over, top to bottom, relative to the other side. Thus, if we take the diagram at upper left, where no lines cross, and flip over one side, we generate the diagram at the lower right, where all the lines cross. The diagrams symmetrical in three axes always transform into diagrams with similar symmetry. Transforming a diagram symmetrical in the y axis alone generates one symmetrical in the z axis. That works the other way around also. Transforming a diagram symmetrical in the x axis alone generates another diagram symmetrical in the x axis alone. Other relationships become evident on inspection. The 720 diagrams of six lines, which I might call "factorial hexagrams," now represent a vast number of possibilities. I have organized them in 30 panels of 24 diagrams each. The first is at left. Unlike traditional hexagrams, I have started their construction from the top rather than the bottom. The last two lines at the bottom represent the Yang and Yin varation of a hexagram defined by the first four lines. This is the most obvious with with the diagram Aa1a, where we have clearly uncrossed and then crossed lines at the bottom. Rather than crowd the other 29 panels onto this webpage, links for pop-ups are given for them below. To construct a hexagram, as is done for traditional hexagrams with yarrow stalks or coins, one needs only six counters numbered one to six. Blindly picking the first counter, this will definine a line that begins at the first point at upper left and connects to the point corresponding to the number on the counter on the right. Aa1a thus would represent the draw "123456." The lower right hexagram on the panel would represent the draw "126543" The very last hexagram, Fe4c, would represent the draw "654321" (see below). I have not considered how a provision for changing lines would work. One line could not change on its own. You would need to pick two numbers, the first to identify the first line, the second for the one it will change with. The whole process could be done with a single die, thrown until all six numbers come up. It is unlikely that all six numbers would be obtained before other numbers are thrown again, so those duplications could be used, in some manner, to define the changing lines. What these hexagrams are going to mean is a tall order of interpretation. Part of the interpretation of traditional hexagrams involves the meanings of the trigram above and the trigram below. Here, where lines from above and below may be crossed over each other, the hexagram may not easily separate into two halves. Indeed, this could not be done with most of the hexagrams just in the first panel. A notion of how interpretation would work must emerge after long consideration. Meanwhile, I think that there is a beauty and fascination in the patterns, which of course are much more elaborate than traditional hexagrams. Gazing at them, they start to seem like Chinese characters, and I start to get the sense that each should indeed mean something -- even as many as 720 pales next to the thousands of Chinese characters. But after doing a bit of that for more than thirty years, not much has come of it.

Below are links for pop-ups of the other panels of hexagrams. Yin & Yáng and the I Ching