By: Tao Steven Zheng (郑涛)
This article introduces sets of abstract figures found in East Asia that are pictorial representations of combinatorics. These diagrams were probably conceived without the intent of being mathematical, but ended up being mathematical curiosities in modern times.
The 64 hexagrams of the I-Ching or Yijing (易經) is the foundation of an ancient Chinese form of divination that dates from the Western Zhou dynasty (c. 1050 - 771 BCE). Each of the 64 hexagrams is a combination of six bars. The solid bar represents yang (陽), the positive force. The broken bars represent yin (陰), the negative force. Interestingly, the sixty-four hexagrams can represent the set of binary numbers from 0 to 63.
The math behind the I-Ching hexagrams is simple. Just apply the fundamental counting principle to get the number of unique hexagrams.
Genji Monogatari Chapter Symbols
These 54 combinations are the chapter symbols in The Tale of Genji (源氏物語), an 11th century novel from Medieval Japan. It is uncertain when these symbols were invented (most likely after the 11th century CE). Interestingly, 52 out of the 54 of the symbols are a visual representation of the Bell number for partitioning five elements. The Bell numbers were discovered by the Scottish-born American mathematician Eric Temple Bell (1883 - 1960).
The number of symbols can be calculated using the Bell Triangle. It works similarly to the Pascal Triangle.