In the textual order of 64 hexagrams in the Yijing, we see a process whereby yin and yang move toward interfusion and harmony in several respects. For example, there are more “balanced hexagrams” having three yang and three yin lines in the second half of the sequence than in the first half (14 in second half, 6 in first half). Moreover, the phenomenon of yin-yang “resonance” between inner lines and outer lines is more prevalent in the second half than in the first half. (For example, in Hexagram #31—made up of Marsh over Mountain—the inner trigram’s lower, middle and upper lines are wholly resonant with the lower, middle and upper lines of the outer trigram.) In fact the phenomenon of resonance is a delicate expression of interchange between yin and yang, or one could say it is a relatively graceful dance move in the interplay of Qian and Kun. Looking at yin-yang interaction from another angle, we discover that the average yin or yang clump size of hexagrams is smaller in the second half than in the first half. At the beginning of the first half are Qian and Kun, which are the two hexagrams having the largest clump size (six lines in a row of yin or yang). By the time we get to Already-Across and Not-Yet-Across at the end of the second half, the separateness of yin and yang is reduced to a minimum. This is a phenomenon of gradual harmonization: In the segment #1-#10 the average yin-yang clump size is larger than in #11-20 and #21-30; the average yin-yang clump size of #21-30 is larger than that of #31-40; the average yin-yang clump size of #31-40 is larger than that of #41-50; the average yin-yang clump size of #41-50 is larger than that of #51-60; and the average clump size of #51-60 is larger than that of #61-64. Upon combining these three tendencies of “balance,” “resonance” and “interspersion” (which can collectively be called “harmonization”), what we see is not an arid process of formal permutations: instead, it is a framework upon which we can confer an abundance of imputed meanings. Aside from “unification of contraries,” this framework presents an increasingly finely tuned and poised interplay between Qian and Kun. Upon detailed investigation we will discover that the whole framework possesses numerous subtle symmetrical relations which go beyond description in simple terms of yin and yang. For example, there is a subtle balance in distribution between two trigram groups—heaven, earth, water, fire vs. thunder, wind, mountain marsh. Of 16 hexagrams formed by combining the heaven, earth, water and fire trigrams among themselves, 12 are found in the first half of the classic and 4 are found in the second half. Conversely, of 16 hexagrams formed by combining thunder, wind, mountain and marsh trigrams among themselves, 12 are found in the second half of the classic, and 4 are found in the first half. This balance is remarkable because the number of hexagrams in the first “half” (30) is different from the number of hexagrams in the second “half” (34). There are other symmetries which are purely numerical. For example, when the Israeli mathematician Immanuel Olsvanger treated the hexagrams as binary numbers and laid them out in order on an 8 x 8 grid: he discovered many numerically symmetrical areas across the grid. I feel that to conceive such a formally rigorous framework, while at the same time allowing so much space for imputed meaning is a stroke of masterful creativity. (See my essay: “The Upper and Lower Halves of the Zhouyi: A Study in Contrasts,” Minima Sinica, 2011 Vol. 2. edited by Professor Wolfgang Kubin of BonnUniversity)
Yijing Hexagram Order: Symmetries of the Two Halves
《周易》上经、下经的对称性
在《周易》64卦的排列顺序中我们看到阴、阳以多种方式走向调和交融的过程。比方说下经里三阳三阴爻的“平衡卦”比上经多（下经14、上经6）。而且下经里各卦内爻与外爻之间的阴阳“相应”现象比上经多。（拿泽山“咸”来说，其内卦的底、中、上爻与外卦的底、中、上爻全应。）而“应”的现象其实是阴阳互动的细致表现，也可以说是乾坤之间一种更为优雅的舞姿。从另一个角度看阴阳的互动，还可以发现下经各卦的平均阴阳板块比上经小。上经最初的乾与坤是阳阴板块最大的两个卦（连续六个阳爻或阴爻）；到下经结尾的既济、未济，阴阳的间隔性已经减少到最小的程度。这是一个渐进的调和现象：#1-#10这段的平均阴阳板块比#11-#20与#21-#30的板块大；#21-#30的平均板块比#31-40大；#31-#40比#41-50大；#41-#50比#51-#60大；而#51-#60的平均板块比#61-#64大。如果把“平衡”、“相应”、“无间隔性”三种趋势综合起来（可合称为“调和性”）， 我们看到的不是一个干吧的形式置换过程，而是一部可以赋予丰富寓意的架构。除了“对立面的统一”以外，这个架构还呈现乾、坤之间越来越细致匀称的的互动。如果细究起来，还可以发现整体架构里存在着许多微妙的对称关系，已经超过简单的阴阳论所能描述。比方说，天、地、水、火与雷、风、山、泽两批经卦在分布上呈现微妙平衡。由天、地、水、火四个经卦所配出来的16个别卦，有12个出现在上经、4个出现在下经；与此相反，由雷、风、山、泽配出来的16个别卦，有12个出现在下经、4个出现在上经。【这种平衡很难得，因为上经的卦数（30）与下经的卦数（34）不一样。】还有一些纯数字的对称关系，比方说，以色列数学家Immanuel Olsvanger 把64卦按经文的次序以二进位数字的方式排成方阵，发现整体方阵里有很多互相对称的数字区。我觉得能够建立这么严谨的形式架构同时能留很大的寓意空间，是相当美妙的手笔。（见拙文”The Upper and Lower Halves of the Zhouyi: A Study in Contrasts,” Minima Sinica, 2011 Vol. 2，德国伯恩大学顾彬教授主编）
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